{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 10 with capacity 1,\nan edge from node 3 to node 0 with capacity 17,\nan edge from node 3 to node 2 with capacity 15,\nan edge from node 4 to node 0 with capacity 13,\nan edge from node 4 to node 2 with capacity 14,\nan edge from node 5 to node 1 with capacity 7,\nan edge from node 5 to node 4 with capacity 16,\nan edge from node 6 to node 7 with capacity 19,\nan edge from node 6 to node 10 with capacity 3,\nan edge from node 6 to node 1 with capacity 18,\nan edge from node 6 to node 8 with capacity 10,\nan edge from node 7 to node 8 with capacity 4,\nan edge from node 7 to node 3 with capacity 15,\nan edge from node 8 to node 7 with capacity 18,\nan edge from node 8 to node 0 with capacity 7,\nan edge from node 8 to node 4 with capacity 19,\nan edge from node 8 to node 2 with capacity 14,\nan edge from node 9 to node 1 with capacity 8,\nan edge from node 9 to node 8 with capacity 12,\nan edge from node 9 to node 4 with capacity 7,\nan edge from node 10 to node 8 with capacity 9,\nan edge from node 10 to node 6 with capacity 1,\nan edge from node 10 to node 5 with capacity 18.\nQ: What is the maximum flow from node 3 to node 7?\nA:", "answer": "The maximum flow from node 3 to node 7 is 1.", "difficulty": "hard", "doc_id": "0"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 8 with capacity 9,\nan edge from node 0 to node 6 with capacity 8,\nan edge from node 0 to node 5 with capacity 1,\nan edge from node 1 to node 11 with capacity 12,\nan edge from node 3 to node 5 with capacity 6,\nan edge from node 3 to node 0 with capacity 17,\nan edge from node 4 to node 11 with capacity 7,\nan edge from node 4 to node 8 with capacity 17,\nan edge from node 4 to node 9 with capacity 4,\nan edge from node 5 to node 11 with capacity 19,\nan edge from node 6 to node 7 with capacity 11,\nan edge from node 6 to node 10 with capacity 15,\nan edge from node 6 to node 1 with capacity 12,\nan edge from node 6 to node 0 with capacity 3,\nan edge from node 7 to node 3 with capacity 3,\nan edge from node 7 to node 0 with capacity 12,\nan edge from node 9 to node 2 with capacity 7,\nan edge from node 9 to node 1 with capacity 19,\nan edge from node 10 to node 5 with capacity 4,\nan edge from node 10 to node 9 with capacity 13,\nan edge from node 10 to node 1 with capacity 12,\nan edge from node 10 to node 3 with capacity 2,\nan edge from node 11 to node 10 with capacity 15.\nQ: What is the maximum flow from node 10 to node 11?\nA:", "answer": "The maximum flow from node 10 to node 11 is 18.", "difficulty": "hard", "doc_id": "1"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 6 with capacity 3,\nan edge from node 0 to node 1 with capacity 4,\nan edge from node 0 to node 4 with capacity 7,\nan edge from node 0 to node 5 with capacity 2,\nan edge from node 1 to node 6 with capacity 8,\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 1 to node 4 with capacity 9,\nan edge from node 2 to node 6 with capacity 7,\nan edge from node 2 to node 0 with capacity 5,\nan edge from node 2 to node 3 with capacity 5,\nan edge from node 3 to node 4 with capacity 2,\nan edge from node 3 to node 5 with capacity 1,\nan edge from node 4 to node 6 with capacity 7,\nan edge from node 4 to node 3 with capacity 6,\nan edge from node 5 to node 0 with capacity 9,\nan edge from node 5 to node 1 with capacity 6,\nan edge from node 5 to node 4 with capacity 2,\nan edge from node 6 to node 0 with capacity 10,\nan edge from node 6 to node 3 with capacity 7,\nan edge from node 6 to node 2 with capacity 2,\nan edge from node 6 to node 1 with capacity 9.\nQ: What is the maximum flow from node 0 to node 3?\nA:", "answer": "The maximum flow from node 0 to node 3 is 15.", "difficulty": "easy", "doc_id": "2"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 7 with capacity 20,\nan edge from node 0 to node 10 with capacity 7,\nan edge from node 0 to node 17 with capacity 4,\nan edge from node 0 to node 5 with capacity 12,\nan edge from node 1 to node 2 with capacity 14,\nan edge from node 1 to node 12 with capacity 12,\nan edge from node 1 to node 10 with capacity 2,\nan edge from node 2 to node 7 with capacity 3,\nan edge from node 2 to node 4 with capacity 1,\nan edge from node 2 to node 14 with capacity 16,\nan edge from node 2 to node 16 with capacity 11,\nan edge from node 2 to node 9 with capacity 11,\nan edge from node 3 to node 13 with capacity 19,\nan edge from node 3 to node 1 with capacity 9,\nan edge from node 3 to node 12 with capacity 19,\nan edge from node 3 to node 0 with capacity 4,\nan edge from node 4 to node 13 with capacity 18,\nan edge from node 4 to node 8 with capacity 5,\nan edge from node 4 to node 17 with capacity 11,\nan edge from node 4 to node 15 with capacity 20,\nan edge from node 4 to node 5 with capacity 17,\nan edge from node 5 to node 11 with capacity 9,\nan edge from node 5 to node 16 with capacity 5,\nan edge from node 5 to node 9 with capacity 18,\nan edge from node 6 to node 2 with capacity 11,\nan edge from node 6 to node 4 with capacity 2,\nan edge from node 6 to node 0 with capacity 5,\nan edge from node 6 to node 10 with capacity 20,\nan edge from node 6 to node 14 with capacity 10,\nan edge from node 7 to node 13 with capacity 5,\nan edge from node 7 to node 1 with capacity 9,\nan edge from node 7 to node 2 with capacity 1,\nan edge from node 7 to node 3 with capacity 12,\nan edge from node 7 to node 9 with capacity 3,\nan edge from node 8 to node 7 with capacity 13,\nan edge from node 8 to node 2 with capacity 2,\nan edge from node 8 to node 11 with capacity 15,\nan edge from node 8 to node 6 with capacity 15,\nan edge from node 8 to node 9 with capacity 13,\nan edge from node 8 to node 5 with capacity 20,\nan edge from node 9 to node 7 with capacity 10,\nan edge from node 9 to node 1 with capacity 1,\nan edge from node 9 to node 6 with capacity 1,\nan edge from node 9 to node 14 with capacity 9,\nan edge from node 9 to node 16 with capacity 6,\nan edge from node 10 to node 13 with capacity 8,\nan edge from node 10 to node 1 with capacity 14,\nan edge from node 10 to node 11 with capacity 9,\nan edge from node 10 to node 15 with capacity 11,\nan edge from node 11 to node 7 with capacity 20,\nan edge from node 11 to node 12 with capacity 20,\nan edge from node 12 to node 13 with capacity 16,\nan edge from node 12 to node 3 with capacity 14,\nan edge from node 12 to node 6 with capacity 15,\nan edge from node 13 to node 0 with capacity 9,\nan edge from node 13 to node 17 with capacity 5,\nan edge from node 13 to node 9 with capacity 2,\nan edge from node 14 to node 1 with capacity 4,\nan edge from node 14 to node 8 with capacity 5,\nan edge from node 14 to node 5 with capacity 17,\nan edge from node 15 to node 1 with capacity 4,\nan edge from node 15 to node 0 with capacity 9,\nan edge from node 15 to node 10 with capacity 2,\nan edge from node 16 to node 7 with capacity 6,\nan edge from node 16 to node 1 with capacity 18,\nan edge from node 16 to node 3 with capacity 5,\nan edge from node 16 to node 17 with capacity 7,\nan edge from node 16 to node 5 with capacity 4,\nan edge from node 17 to node 13 with capacity 5,\nan edge from node 17 to node 7 with capacity 7,\nan edge from node 17 to node 2 with capacity 7,\nan edge from node 17 to node 12 with capacity 6.\nQ: What is the maximum flow from node 1 to node 0?\nA:", "answer": "The maximum flow from node 1 to node 0 is 27.", "difficulty": "hard", "doc_id": "3"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 1 with capacity 19,\nan edge from node 0 to node 8 with capacity 13,\nan edge from node 0 to node 7 with capacity 1,\nan edge from node 0 to node 2 with capacity 19,\nan edge from node 1 to node 9 with capacity 5,\nan edge from node 1 to node 14 with capacity 5,\nan edge from node 1 to node 5 with capacity 17,\nan edge from node 2 to node 7 with capacity 19,\nan edge from node 2 to node 0 with capacity 12,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 10 with capacity 11,\nan edge from node 3 to node 6 with capacity 17,\nan edge from node 3 to node 16 with capacity 18,\nan edge from node 4 to node 1 with capacity 3,\nan edge from node 4 to node 3 with capacity 17,\nan edge from node 4 to node 9 with capacity 14,\nan edge from node 4 to node 0 with capacity 1,\nan edge from node 4 to node 11 with capacity 4,\nan edge from node 4 to node 5 with capacity 12,\nan edge from node 5 to node 8 with capacity 13,\nan edge from node 5 to node 10 with capacity 5,\nan edge from node 5 to node 13 with capacity 1,\nan edge from node 6 to node 0 with capacity 13,\nan edge from node 7 to node 1 with capacity 17,\nan edge from node 8 to node 1 with capacity 17,\nan edge from node 8 to node 15 with capacity 6,\nan edge from node 8 to node 14 with capacity 17,\nan edge from node 8 to node 13 with capacity 10,\nan edge from node 8 to node 16 with capacity 5,\nan edge from node 8 to node 4 with capacity 8,\nan edge from node 9 to node 7 with capacity 11,\nan edge from node 9 to node 14 with capacity 11,\nan edge from node 9 to node 12 with capacity 4,\nan edge from node 9 to node 0 with capacity 12,\nan edge from node 9 to node 4 with capacity 6,\nan edge from node 10 to node 1 with capacity 18,\nan edge from node 11 to node 1 with capacity 6,\nan edge from node 11 to node 3 with capacity 18,\nan edge from node 11 to node 8 with capacity 13,\nan edge from node 11 to node 12 with capacity 14,\nan edge from node 11 to node 16 with capacity 6,\nan edge from node 11 to node 4 with capacity 10,\nan edge from node 11 to node 5 with capacity 13,\nan edge from node 12 to node 7 with capacity 14,\nan edge from node 12 to node 10 with capacity 9,\nan edge from node 12 to node 16 with capacity 1,\nan edge from node 12 to node 4 with capacity 1,\nan edge from node 13 to node 8 with capacity 2,\nan edge from node 13 to node 9 with capacity 5,\nan edge from node 13 to node 15 with capacity 19,\nan edge from node 13 to node 6 with capacity 5,\nan edge from node 13 to node 5 with capacity 5,\nan edge from node 14 to node 10 with capacity 6,\nan edge from node 14 to node 11 with capacity 6,\nan edge from node 14 to node 5 with capacity 9,\nan edge from node 15 to node 7 with capacity 19,\nan edge from node 15 to node 11 with capacity 9,\nan edge from node 16 to node 1 with capacity 7,\nan edge from node 16 to node 7 with capacity 14,\nan edge from node 16 to node 9 with capacity 4,\nan edge from node 16 to node 10 with capacity 19,\nan edge from node 16 to node 2 with capacity 12,\nan edge from node 16 to node 5 with capacity 20.\nQ: What is the maximum flow from node 7 to node 1?\nA:", "answer": "The maximum flow from node 7 to node 1 is 17.", "difficulty": "hard", "doc_id": "4"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 6 with capacity 5,\nan edge from node 0 to node 3 with capacity 7,\nan edge from node 1 to node 7 with capacity 7,\nan edge from node 1 to node 2 with capacity 3,\nan edge from node 2 to node 7 with capacity 7,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 5 with capacity 7,\nan edge from node 3 to node 1 with capacity 8,\nan edge from node 4 to node 0 with capacity 3,\nan edge from node 5 to node 7 with capacity 5,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 6 to node 7 with capacity 6,\nan edge from node 6 to node 0 with capacity 9,\nan edge from node 7 to node 4 with capacity 4,\nan edge from node 7 to node 1 with capacity 8.\nQ: What is the maximum flow from node 1 to node 6?\nA:", "answer": "The maximum flow from node 1 to node 6 is 3.", "difficulty": "easy", "doc_id": "5"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 18 with capacity 16,\nan edge from node 0 to node 11 with capacity 10,\nan edge from node 1 to node 12 with capacity 3,\nan edge from node 1 to node 4 with capacity 1,\nan edge from node 1 to node 8 with capacity 18,\nan edge from node 2 to node 9 with capacity 9,\nan edge from node 2 to node 16 with capacity 15,\nan edge from node 2 to node 17 with capacity 2,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 18 with capacity 1,\nan edge from node 3 to node 10 with capacity 15,\nan edge from node 3 to node 6 with capacity 12,\nan edge from node 3 to node 0 with capacity 10,\nan edge from node 4 to node 11 with capacity 15,\nan edge from node 4 to node 0 with capacity 11,\nan edge from node 4 to node 7 with capacity 20,\nan edge from node 5 to node 15 with capacity 2,\nan edge from node 5 to node 12 with capacity 13,\nan edge from node 6 to node 16 with capacity 13,\nan edge from node 6 to node 17 with capacity 17,\nan edge from node 6 to node 18 with capacity 9,\nan edge from node 6 to node 14 with capacity 11,\nan edge from node 6 to node 13 with capacity 13,\nan edge from node 6 to node 0 with capacity 19,\nan edge from node 6 to node 3 with capacity 6,\nan edge from node 7 to node 17 with capacity 9,\nan edge from node 7 to node 18 with capacity 19,\nan edge from node 7 to node 12 with capacity 20,\nan edge from node 8 to node 15 with capacity 10,\nan edge from node 8 to node 12 with capacity 1,\nan edge from node 8 to node 4 with capacity 12,\nan edge from node 8 to node 3 with capacity 1,\nan edge from node 9 to node 6 with capacity 6,\nan edge from node 9 to node 0 with capacity 1,\nan edge from node 9 to node 4 with capacity 4,\nan edge from node 9 to node 8 with capacity 11,\nan edge from node 9 to node 3 with capacity 16,\nan edge from node 10 to node 6 with capacity 6,\nan edge from node 10 to node 11 with capacity 5,\nan edge from node 10 to node 13 with capacity 17,\nan edge from node 10 to node 3 with capacity 13,\nan edge from node 11 to node 10 with capacity 11,\nan edge from node 11 to node 6 with capacity 20,\nan edge from node 11 to node 12 with capacity 3,\nan edge from node 12 to node 6 with capacity 16,\nan edge from node 13 to node 2 with capacity 16,\nan edge from node 13 to node 17 with capacity 7,\nan edge from node 13 to node 18 with capacity 4,\nan edge from node 13 to node 10 with capacity 13,\nan edge from node 13 to node 5 with capacity 4,\nan edge from node 13 to node 12 with capacity 1,\nan edge from node 14 to node 2 with capacity 2,\nan edge from node 14 to node 9 with capacity 3,\nan edge from node 14 to node 17 with capacity 6,\nan edge from node 14 to node 18 with capacity 10,\nan edge from node 14 to node 5 with capacity 16,\nan edge from node 14 to node 4 with capacity 10,\nan edge from node 15 to node 18 with capacity 1,\nan edge from node 15 to node 10 with capacity 1,\nan edge from node 15 to node 5 with capacity 4,\nan edge from node 15 to node 14 with capacity 18,\nan edge from node 15 to node 13 with capacity 3,\nan edge from node 16 to node 2 with capacity 17,\nan edge from node 16 to node 9 with capacity 7,\nan edge from node 16 to node 14 with capacity 3,\nan edge from node 16 to node 13 with capacity 3,\nan edge from node 16 to node 12 with capacity 20,\nan edge from node 17 to node 9 with capacity 4,\nan edge from node 17 to node 4 with capacity 1,\nan edge from node 17 to node 3 with capacity 2,\nan edge from node 18 to node 9 with capacity 19,\nan edge from node 18 to node 16 with capacity 17,\nan edge from node 18 to node 17 with capacity 6,\nan edge from node 18 to node 14 with capacity 9.\nQ: What is the maximum flow from node 18 to node 7?\nA:", "answer": "The maximum flow from node 18 to node 7 is 20.", "difficulty": "hard", "doc_id": "6"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 4 with capacity 9,\nan edge from node 0 to node 7 with capacity 17,\nan edge from node 0 to node 14 with capacity 15,\nan edge from node 0 to node 5 with capacity 17,\nan edge from node 1 to node 15 with capacity 10,\nan edge from node 1 to node 10 with capacity 16,\nan edge from node 1 to node 6 with capacity 11,\nan edge from node 1 to node 12 with capacity 7,\nan edge from node 2 to node 10 with capacity 17,\nan edge from node 2 to node 11 with capacity 20,\nan edge from node 2 to node 5 with capacity 6,\nan edge from node 3 to node 1 with capacity 18,\nan edge from node 3 to node 2 with capacity 1,\nan edge from node 3 to node 0 with capacity 4,\nan edge from node 3 to node 11 with capacity 2,\nan edge from node 4 to node 16 with capacity 5,\nan edge from node 4 to node 12 with capacity 20,\nan edge from node 4 to node 5 with capacity 11,\nan edge from node 5 to node 4 with capacity 11,\nan edge from node 5 to node 7 with capacity 5,\nan edge from node 5 to node 2 with capacity 9,\nan edge from node 5 to node 0 with capacity 12,\nan edge from node 6 to node 1 with capacity 7,\nan edge from node 6 to node 10 with capacity 8,\nan edge from node 6 to node 2 with capacity 9,\nan edge from node 6 to node 0 with capacity 17,\nan edge from node 6 to node 5 with capacity 16,\nan edge from node 7 to node 13 with capacity 4,\nan edge from node 7 to node 6 with capacity 6,\nan edge from node 8 to node 15 with capacity 7,\nan edge from node 8 to node 10 with capacity 18,\nan edge from node 8 to node 13 with capacity 16,\nan edge from node 9 to node 7 with capacity 19,\nan edge from node 9 to node 3 with capacity 6,\nan edge from node 9 to node 13 with capacity 8,\nan edge from node 9 to node 5 with capacity 20,\nan edge from node 10 to node 16 with capacity 11,\nan edge from node 10 to node 11 with capacity 15,\nan edge from node 10 to node 12 with capacity 11,\nan edge from node 10 to node 5 with capacity 19,\nan edge from node 11 to node 4 with capacity 16,\nan edge from node 11 to node 10 with capacity 18,\nan edge from node 11 to node 2 with capacity 8,\nan edge from node 11 to node 13 with capacity 3,\nan edge from node 12 to node 14 with capacity 2,\nan edge from node 13 to node 1 with capacity 8,\nan edge from node 13 to node 4 with capacity 13,\nan edge from node 13 to node 15 with capacity 12,\nan edge from node 13 to node 16 with capacity 12,\nan edge from node 13 to node 9 with capacity 6,\nan edge from node 14 to node 4 with capacity 8,\nan edge from node 14 to node 2 with capacity 20,\nan edge from node 14 to node 13 with capacity 1,\nan edge from node 14 to node 9 with capacity 11,\nan edge from node 14 to node 6 with capacity 13,\nan edge from node 14 to node 8 with capacity 3,\nan edge from node 14 to node 5 with capacity 16,\nan edge from node 15 to node 10 with capacity 8,\nan edge from node 15 to node 8 with capacity 6,\nan edge from node 16 to node 7 with capacity 5.\nQ: What is the maximum flow from node 15 to node 4?\nA:", "answer": "The maximum flow from node 15 to node 4 is 14.", "difficulty": "hard", "doc_id": "7"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 9 with capacity 15,\nan edge from node 0 to node 1 with capacity 20,\nan edge from node 0 to node 7 with capacity 16,\nan edge from node 1 to node 3 with capacity 10,\nan edge from node 1 to node 8 with capacity 15,\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 1 to node 5 with capacity 2,\nan edge from node 2 to node 4 with capacity 13,\nan edge from node 2 to node 8 with capacity 4,\nan edge from node 2 to node 1 with capacity 7,\nan edge from node 2 to node 7 with capacity 16,\nan edge from node 4 to node 3 with capacity 20,\nan edge from node 4 to node 8 with capacity 16,\nan edge from node 4 to node 0 with capacity 12,\nan edge from node 4 to node 9 with capacity 12,\nan edge from node 5 to node 11 with capacity 18,\nan edge from node 5 to node 3 with capacity 10,\nan edge from node 5 to node 10 with capacity 14,\nan edge from node 6 to node 4 with capacity 3,\nan edge from node 6 to node 8 with capacity 19,\nan edge from node 6 to node 5 with capacity 10,\nan edge from node 6 to node 1 with capacity 10,\nan edge from node 7 to node 3 with capacity 17,\nan edge from node 7 to node 8 with capacity 12,\nan edge from node 7 to node 0 with capacity 4,\nan edge from node 7 to node 1 with capacity 6,\nan edge from node 7 to node 10 with capacity 10,\nan edge from node 8 to node 11 with capacity 17,\nan edge from node 8 to node 3 with capacity 11,\nan edge from node 8 to node 0 with capacity 1,\nan edge from node 8 to node 5 with capacity 15,\nan edge from node 8 to node 2 with capacity 2,\nan edge from node 8 to node 6 with capacity 14,\nan edge from node 8 to node 1 with capacity 14,\nan edge from node 8 to node 10 with capacity 5,\nan edge from node 9 to node 4 with capacity 6,\nan edge from node 9 to node 8 with capacity 7,\nan edge from node 9 to node 5 with capacity 5,\nan edge from node 10 to node 11 with capacity 18,\nan edge from node 10 to node 3 with capacity 1,\nan edge from node 10 to node 8 with capacity 11,\nan edge from node 10 to node 2 with capacity 2,\nan edge from node 10 to node 7 with capacity 13,\nan edge from node 11 to node 0 with capacity 3,\nan edge from node 11 to node 10 with capacity 5.\nQ: What is the maximum flow from node 2 to node 3?\nA:", "answer": "The maximum flow from node 2 to node 3 is 40.", "difficulty": "hard", "doc_id": "8"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 7 with capacity 14,\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 0 to node 3 with capacity 5,\nan edge from node 1 to node 6 with capacity 15,\nan edge from node 1 to node 2 with capacity 3,\nan edge from node 1 to node 10 with capacity 5,\nan edge from node 1 to node 5 with capacity 16,\nan edge from node 1 to node 17 with capacity 18,\nan edge from node 1 to node 4 with capacity 20,\nan edge from node 2 to node 15 with capacity 2,\nan edge from node 2 to node 11 with capacity 19,\nan edge from node 2 to node 12 with capacity 16,\nan edge from node 3 to node 10 with capacity 12,\nan edge from node 3 to node 18 with capacity 9,\nan edge from node 3 to node 5 with capacity 16,\nan edge from node 3 to node 13 with capacity 12,\nan edge from node 3 to node 8 with capacity 11,\nan edge from node 3 to node 4 with capacity 5,\nan edge from node 4 to node 2 with capacity 7,\nan edge from node 4 to node 16 with capacity 12,\nan edge from node 4 to node 12 with capacity 13,\nan edge from node 5 to node 1 with capacity 5,\nan edge from node 5 to node 11 with capacity 16,\nan edge from node 5 to node 12 with capacity 7,\nan edge from node 6 to node 2 with capacity 5,\nan edge from node 6 to node 11 with capacity 13,\nan edge from node 6 to node 3 with capacity 10,\nan edge from node 6 to node 17 with capacity 6,\nan edge from node 7 to node 2 with capacity 6,\nan edge from node 7 to node 3 with capacity 9,\nan edge from node 7 to node 0 with capacity 10,\nan edge from node 7 to node 12 with capacity 4,\nan edge from node 8 to node 7 with capacity 9,\nan edge from node 8 to node 18 with capacity 16,\nan edge from node 8 to node 16 with capacity 5,\nan edge from node 8 to node 5 with capacity 3,\nan edge from node 8 to node 3 with capacity 8,\nan edge from node 9 to node 6 with capacity 16,\nan edge from node 9 to node 14 with capacity 7,\nan edge from node 9 to node 10 with capacity 15,\nan edge from node 9 to node 3 with capacity 12,\nan edge from node 9 to node 8 with capacity 9,\nan edge from node 10 to node 14 with capacity 12,\nan edge from node 10 to node 18 with capacity 6,\nan edge from node 10 to node 16 with capacity 16,\nan edge from node 10 to node 5 with capacity 17,\nan edge from node 10 to node 17 with capacity 9,\nan edge from node 11 to node 6 with capacity 10,\nan edge from node 11 to node 2 with capacity 12,\nan edge from node 11 to node 16 with capacity 8,\nan edge from node 11 to node 3 with capacity 15,\nan edge from node 11 to node 12 with capacity 19,\nan edge from node 12 to node 6 with capacity 5,\nan edge from node 12 to node 11 with capacity 10,\nan edge from node 12 to node 8 with capacity 16,\nan edge from node 13 to node 6 with capacity 15,\nan edge from node 13 to node 4 with capacity 2,\nan edge from node 14 to node 10 with capacity 8,\nan edge from node 14 to node 18 with capacity 13,\nan edge from node 14 to node 5 with capacity 11,\nan edge from node 14 to node 0 with capacity 4,\nan edge from node 14 to node 4 with capacity 18,\nan edge from node 15 to node 14 with capacity 7,\nan edge from node 15 to node 7 with capacity 5,\nan edge from node 15 to node 1 with capacity 7,\nan edge from node 15 to node 0 with capacity 2,\nan edge from node 15 to node 12 with capacity 18,\nan edge from node 16 to node 2 with capacity 9,\nan edge from node 16 to node 10 with capacity 19,\nan edge from node 16 to node 17 with capacity 1,\nan edge from node 16 to node 8 with capacity 7,\nan edge from node 16 to node 4 with capacity 6,\nan edge from node 17 to node 2 with capacity 7,\nan edge from node 17 to node 14 with capacity 19,\nan edge from node 17 to node 1 with capacity 14,\nan edge from node 17 to node 18 with capacity 14,\nan edge from node 17 to node 5 with capacity 4,\nan edge from node 17 to node 12 with capacity 2,\nan edge from node 18 to node 10 with capacity 19,\nan edge from node 18 to node 11 with capacity 10,\nan edge from node 18 to node 3 with capacity 6,\nan edge from node 18 to node 12 with capacity 12.\nQ: What is the maximum flow from node 8 to node 3?\nA:", "answer": "The maximum flow from node 8 to node 3 is 41.", "difficulty": "hard", "doc_id": "9"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 5 with capacity 20,\nan edge from node 0 to node 15 with capacity 11,\nan edge from node 0 to node 12 with capacity 12,\nan edge from node 0 to node 14 with capacity 12,\nan edge from node 1 to node 9 with capacity 2,\nan edge from node 1 to node 12 with capacity 12,\nan edge from node 1 to node 17 with capacity 6,\nan edge from node 1 to node 14 with capacity 14,\nan edge from node 2 to node 9 with capacity 9,\nan edge from node 2 to node 4 with capacity 15,\nan edge from node 2 to node 8 with capacity 3,\nan edge from node 3 to node 18 with capacity 3,\nan edge from node 3 to node 15 with capacity 2,\nan edge from node 3 to node 4 with capacity 13,\nan edge from node 3 to node 7 with capacity 13,\nan edge from node 3 to node 14 with capacity 3,\nan edge from node 3 to node 10 with capacity 1,\nan edge from node 4 to node 12 with capacity 6,\nan edge from node 4 to node 14 with capacity 1,\nan edge from node 5 to node 8 with capacity 12,\nan edge from node 5 to node 14 with capacity 2,\nan edge from node 5 to node 10 with capacity 12,\nan edge from node 6 to node 11 with capacity 16,\nan edge from node 6 to node 9 with capacity 14,\nan edge from node 6 to node 7 with capacity 16,\nan edge from node 6 to node 17 with capacity 17,\nan edge from node 6 to node 10 with capacity 7,\nan edge from node 7 to node 11 with capacity 1,\nan edge from node 7 to node 3 with capacity 14,\nan edge from node 8 to node 6 with capacity 8,\nan edge from node 8 to node 11 with capacity 6,\nan edge from node 8 to node 3 with capacity 2,\nan edge from node 8 to node 9 with capacity 10,\nan edge from node 8 to node 10 with capacity 5,\nan edge from node 9 to node 16 with capacity 17,\nan edge from node 9 to node 5 with capacity 3,\nan edge from node 9 to node 3 with capacity 12,\nan edge from node 9 to node 15 with capacity 8,\nan edge from node 9 to node 8 with capacity 5,\nan edge from node 9 to node 13 with capacity 16,\nan edge from node 10 to node 3 with capacity 11,\nan edge from node 10 to node 4 with capacity 16,\nan edge from node 10 to node 17 with capacity 6,\nan edge from node 10 to node 14 with capacity 4,\nan edge from node 11 to node 6 with capacity 13,\nan edge from node 11 to node 16 with capacity 19,\nan edge from node 11 to node 3 with capacity 3,\nan edge from node 11 to node 8 with capacity 19,\nan edge from node 11 to node 0 with capacity 6,\nan edge from node 12 to node 2 with capacity 13,\nan edge from node 12 to node 5 with capacity 5,\nan edge from node 12 to node 7 with capacity 12,\nan edge from node 12 to node 14 with capacity 4,\nan edge from node 13 to node 1 with capacity 18,\nan edge from node 13 to node 5 with capacity 13,\nan edge from node 13 to node 12 with capacity 17,\nan edge from node 14 to node 18 with capacity 4,\nan edge from node 14 to node 2 with capacity 1,\nan edge from node 14 to node 16 with capacity 7,\nan edge from node 14 to node 4 with capacity 8,\nan edge from node 14 to node 12 with capacity 11,\nan edge from node 14 to node 10 with capacity 4,\nan edge from node 15 to node 9 with capacity 10,\nan edge from node 15 to node 8 with capacity 15,\nan edge from node 15 to node 0 with capacity 11,\nan edge from node 16 to node 11 with capacity 2,\nan edge from node 16 to node 12 with capacity 20,\nan edge from node 16 to node 8 with capacity 5,\nan edge from node 17 to node 12 with capacity 8,\nan edge from node 17 to node 7 with capacity 20,\nan edge from node 17 to node 8 with capacity 3,\nan edge from node 18 to node 6 with capacity 9,\nan edge from node 18 to node 5 with capacity 20,\nan edge from node 18 to node 3 with capacity 6,\nan edge from node 18 to node 4 with capacity 17.\nQ: What is the maximum flow from node 6 to node 16?\nA:", "answer": "The maximum flow from node 6 to node 16 is 43.", "difficulty": "hard", "doc_id": "10"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 1 to node 5 with capacity 8,\nan edge from node 2 to node 0 with capacity 8,\nan edge from node 3 to node 6 with capacity 4,\nan edge from node 4 to node 6 with capacity 3,\nan edge from node 5 to node 1 with capacity 3,\nan edge from node 5 to node 3 with capacity 8,\nan edge from node 6 to node 5 with capacity 4,\nan edge from node 6 to node 1 with capacity 10,\nan edge from node 6 to node 7 with capacity 10,\nan edge from node 7 to node 5 with capacity 8,\nan edge from node 7 to node 2 with capacity 10,\nan edge from node 7 to node 3 with capacity 2.\nQ: What is the maximum flow from node 6 to node 5?\nA:", "answer": "The maximum flow from node 6 to node 5 is 20.", "difficulty": "easy", "doc_id": "11"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 1 to node 0 with capacity 9,\nan edge from node 2 to node 1 with capacity 4,\nan edge from node 3 to node 1 with capacity 4,\nan edge from node 4 to node 3 with capacity 1,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 1 with capacity 4,\nan edge from node 5 to node 0 with capacity 7.\nQ: What is the maximum flow from node 4 to node 0?\nA:", "answer": "The maximum flow from node 4 to node 0 is 3.", "difficulty": "easy", "doc_id": "12"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 3 with capacity 16,\nan edge from node 0 to node 15 with capacity 5,\nan edge from node 0 to node 4 with capacity 11,\nan edge from node 1 to node 10 with capacity 5,\nan edge from node 1 to node 7 with capacity 4,\nan edge from node 1 to node 15 with capacity 17,\nan edge from node 1 to node 12 with capacity 16,\nan edge from node 1 to node 0 with capacity 15,\nan edge from node 2 to node 13 with capacity 16,\nan edge from node 2 to node 3 with capacity 9,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 3 to node 4 with capacity 20,\nan edge from node 3 to node 11 with capacity 2,\nan edge from node 4 to node 13 with capacity 14,\nan edge from node 4 to node 3 with capacity 16,\nan edge from node 4 to node 12 with capacity 20,\nan edge from node 5 to node 16 with capacity 18,\nan edge from node 5 to node 15 with capacity 1,\nan edge from node 5 to node 4 with capacity 2,\nan edge from node 5 to node 17 with capacity 12,\nan edge from node 6 to node 13 with capacity 5,\nan edge from node 6 to node 3 with capacity 17,\nan edge from node 6 to node 15 with capacity 12,\nan edge from node 6 to node 4 with capacity 20,\nan edge from node 6 to node 0 with capacity 10,\nan edge from node 7 to node 8 with capacity 15,\nan edge from node 7 to node 16 with capacity 1,\nan edge from node 7 to node 1 with capacity 13,\nan edge from node 7 to node 5 with capacity 13,\nan edge from node 7 to node 15 with capacity 17,\nan edge from node 7 to node 4 with capacity 10,\nan edge from node 7 to node 14 with capacity 9,\nan edge from node 7 to node 6 with capacity 1,\nan edge from node 8 to node 7 with capacity 10,\nan edge from node 8 to node 14 with capacity 17,\nan edge from node 8 to node 6 with capacity 12,\nan edge from node 9 to node 8 with capacity 16,\nan edge from node 9 to node 10 with capacity 16,\nan edge from node 9 to node 2 with capacity 15,\nan edge from node 9 to node 7 with capacity 18,\nan edge from node 10 to node 8 with capacity 18,\nan edge from node 10 to node 15 with capacity 8,\nan edge from node 10 to node 12 with capacity 8,\nan edge from node 10 to node 0 with capacity 13,\nan edge from node 11 to node 16 with capacity 17,\nan edge from node 11 to node 1 with capacity 14,\nan edge from node 11 to node 17 with capacity 7,\nan edge from node 12 to node 13 with capacity 1,\nan edge from node 12 to node 16 with capacity 3,\nan edge from node 12 to node 15 with capacity 7,\nan edge from node 12 to node 17 with capacity 2,\nan edge from node 12 to node 6 with capacity 1,\nan edge from node 13 to node 16 with capacity 16,\nan edge from node 13 to node 9 with capacity 5,\nan edge from node 13 to node 12 with capacity 8,\nan edge from node 14 to node 8 with capacity 11,\nan edge from node 14 to node 17 with capacity 16,\nan edge from node 14 to node 0 with capacity 18,\nan edge from node 14 to node 11 with capacity 19,\nan edge from node 15 to node 13 with capacity 14,\nan edge from node 15 to node 10 with capacity 17,\nan edge from node 15 to node 4 with capacity 2,\nan edge from node 15 to node 9 with capacity 19,\nan edge from node 16 to node 13 with capacity 18,\nan edge from node 16 to node 8 with capacity 10,\nan edge from node 16 to node 15 with capacity 2,\nan edge from node 16 to node 17 with capacity 10,\nan edge from node 16 to node 12 with capacity 16,\nan edge from node 16 to node 6 with capacity 9,\nan edge from node 16 to node 11 with capacity 3,\nan edge from node 17 to node 10 with capacity 19,\nan edge from node 17 to node 2 with capacity 13.\nQ: What is the maximum flow from node 11 to node 17?\nA:", "answer": "The maximum flow from node 11 to node 17 is 38.", "difficulty": "hard", "doc_id": "13"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 0 to node 8 with capacity 14,\nan edge from node 0 to node 2 with capacity 3,\nan edge from node 0 to node 3 with capacity 14,\nan edge from node 0 to node 16 with capacity 6,\nan edge from node 1 to node 12 with capacity 18,\nan edge from node 1 to node 6 with capacity 7,\nan edge from node 1 to node 5 with capacity 19,\nan edge from node 2 to node 11 with capacity 14,\nan edge from node 2 to node 5 with capacity 3,\nan edge from node 2 to node 16 with capacity 9,\nan edge from node 2 to node 10 with capacity 6,\nan edge from node 3 to node 8 with capacity 10,\nan edge from node 4 to node 9 with capacity 14,\nan edge from node 4 to node 15 with capacity 15,\nan edge from node 4 to node 11 with capacity 16,\nan edge from node 4 to node 8 with capacity 3,\nan edge from node 5 to node 11 with capacity 16,\nan edge from node 5 to node 13 with capacity 7,\nan edge from node 6 to node 9 with capacity 2,\nan edge from node 6 to node 7 with capacity 5,\nan edge from node 6 to node 8 with capacity 20,\nan edge from node 6 to node 4 with capacity 6,\nan edge from node 6 to node 5 with capacity 13,\nan edge from node 7 to node 17 with capacity 20,\nan edge from node 7 to node 15 with capacity 3,\nan edge from node 7 to node 8 with capacity 3,\nan edge from node 7 to node 2 with capacity 15,\nan edge from node 7 to node 5 with capacity 15,\nan edge from node 8 to node 2 with capacity 14,\nan edge from node 8 to node 13 with capacity 10,\nan edge from node 9 to node 7 with capacity 3,\nan edge from node 9 to node 15 with capacity 4,\nan edge from node 9 to node 14 with capacity 10,\nan edge from node 9 to node 2 with capacity 17,\nan edge from node 10 to node 17 with capacity 17,\nan edge from node 10 to node 9 with capacity 8,\nan edge from node 10 to node 14 with capacity 20,\nan edge from node 10 to node 13 with capacity 12,\nan edge from node 10 to node 5 with capacity 6,\nan edge from node 10 to node 16 with capacity 13,\nan edge from node 11 to node 1 with capacity 20,\nan edge from node 11 to node 12 with capacity 16,\nan edge from node 11 to node 4 with capacity 18,\nan edge from node 11 to node 10 with capacity 20,\nan edge from node 12 to node 1 with capacity 10,\nan edge from node 12 to node 17 with capacity 14,\nan edge from node 12 to node 2 with capacity 13,\nan edge from node 12 to node 4 with capacity 1,\nan edge from node 12 to node 6 with capacity 9,\nan edge from node 12 to node 0 with capacity 20,\nan edge from node 12 to node 3 with capacity 15,\nan edge from node 13 to node 1 with capacity 20,\nan edge from node 13 to node 17 with capacity 18,\nan edge from node 13 to node 14 with capacity 8,\nan edge from node 13 to node 8 with capacity 17,\nan edge from node 15 to node 11 with capacity 13,\nan edge from node 15 to node 4 with capacity 8,\nan edge from node 16 to node 7 with capacity 12,\nan edge from node 16 to node 15 with capacity 16,\nan edge from node 16 to node 8 with capacity 16,\nan edge from node 16 to node 4 with capacity 14,\nan edge from node 16 to node 10 with capacity 14,\nan edge from node 17 to node 15 with capacity 10,\nan edge from node 17 to node 11 with capacity 1,\nan edge from node 17 to node 14 with capacity 11,\nan edge from node 17 to node 2 with capacity 18,\nan edge from node 17 to node 13 with capacity 11.\nQ: What is the maximum flow from node 3 to node 8?\nA:", "answer": "The maximum flow from node 3 to node 8 is 10.", "difficulty": "hard", "doc_id": "14"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 3 with capacity 2,\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 0 to node 9 with capacity 3,\nan edge from node 0 to node 6 with capacity 3,\nan edge from node 0 to node 5 with capacity 12,\nan edge from node 0 to node 8 with capacity 11,\nan edge from node 1 to node 0 with capacity 11,\nan edge from node 1 to node 3 with capacity 11,\nan edge from node 1 to node 10 with capacity 10,\nan edge from node 1 to node 5 with capacity 17,\nan edge from node 2 to node 1 with capacity 15,\nan edge from node 2 to node 11 with capacity 18,\nan edge from node 2 to node 6 with capacity 17,\nan edge from node 3 to node 11 with capacity 17,\nan edge from node 3 to node 9 with capacity 2,\nan edge from node 3 to node 12 with capacity 10,\nan edge from node 4 to node 0 with capacity 19,\nan edge from node 4 to node 7 with capacity 8,\nan edge from node 4 to node 6 with capacity 15,\nan edge from node 4 to node 12 with capacity 2,\nan edge from node 5 to node 0 with capacity 13,\nan edge from node 5 to node 10 with capacity 16,\nan edge from node 5 to node 9 with capacity 4,\nan edge from node 5 to node 6 with capacity 9,\nan edge from node 5 to node 12 with capacity 18,\nan edge from node 6 to node 2 with capacity 17,\nan edge from node 6 to node 1 with capacity 8,\nan edge from node 6 to node 8 with capacity 5,\nan edge from node 7 to node 3 with capacity 17,\nan edge from node 7 to node 9 with capacity 9,\nan edge from node 7 to node 5 with capacity 20,\nan edge from node 7 to node 4 with capacity 10,\nan edge from node 8 to node 2 with capacity 2,\nan edge from node 8 to node 0 with capacity 4,\nan edge from node 8 to node 1 with capacity 13,\nan edge from node 8 to node 4 with capacity 14,\nan edge from node 9 to node 5 with capacity 19,\nan edge from node 10 to node 9 with capacity 18,\nan edge from node 10 to node 4 with capacity 11,\nan edge from node 12 to node 10 with capacity 2,\nan edge from node 12 to node 11 with capacity 8,\nan edge from node 12 to node 7 with capacity 17,\nan edge from node 12 to node 4 with capacity 6.\nQ: What is the maximum flow from node 12 to node 4?\nA:", "answer": "The maximum flow from node 12 to node 4 is 25.", "difficulty": "hard", "doc_id": "15"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 1 with capacity 16,\nan edge from node 0 to node 7 with capacity 10,\nan edge from node 0 to node 3 with capacity 18,\nan edge from node 0 to node 9 with capacity 8,\nan edge from node 0 to node 5 with capacity 1,\nan edge from node 1 to node 7 with capacity 15,\nan edge from node 1 to node 6 with capacity 7,\nan edge from node 1 to node 9 with capacity 9,\nan edge from node 1 to node 4 with capacity 13,\nan edge from node 1 to node 13 with capacity 7,\nan edge from node 2 to node 6 with capacity 9,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 6 with capacity 19,\nan edge from node 3 to node 4 with capacity 2,\nan edge from node 3 to node 11 with capacity 17,\nan edge from node 4 to node 6 with capacity 2,\nan edge from node 4 to node 11 with capacity 9,\nan edge from node 4 to node 8 with capacity 6,\nan edge from node 5 to node 7 with capacity 2,\nan edge from node 5 to node 0 with capacity 8,\nan edge from node 5 to node 3 with capacity 5,\nan edge from node 5 to node 4 with capacity 7,\nan edge from node 6 to node 0 with capacity 8,\nan edge from node 6 to node 3 with capacity 8,\nan edge from node 6 to node 4 with capacity 7,\nan edge from node 6 to node 13 with capacity 8,\nan edge from node 6 to node 8 with capacity 18,\nan edge from node 7 to node 3 with capacity 16,\nan edge from node 7 to node 9 with capacity 2,\nan edge from node 7 to node 10 with capacity 15,\nan edge from node 7 to node 12 with capacity 5,\nan edge from node 8 to node 1 with capacity 15,\nan edge from node 8 to node 3 with capacity 13,\nan edge from node 8 to node 13 with capacity 13,\nan edge from node 8 to node 2 with capacity 20,\nan edge from node 8 to node 5 with capacity 9,\nan edge from node 8 to node 10 with capacity 9,\nan edge from node 8 to node 12 with capacity 6,\nan edge from node 9 to node 1 with capacity 18,\nan edge from node 9 to node 4 with capacity 10,\nan edge from node 9 to node 13 with capacity 14,\nan edge from node 10 to node 1 with capacity 15,\nan edge from node 10 to node 11 with capacity 3,\nan edge from node 10 to node 5 with capacity 5,\nan edge from node 11 to node 3 with capacity 14,\nan edge from node 11 to node 9 with capacity 12,\nan edge from node 11 to node 2 with capacity 7,\nan edge from node 12 to node 13 with capacity 1,\nan edge from node 13 to node 7 with capacity 16,\nan edge from node 13 to node 11 with capacity 1,\nan edge from node 13 to node 8 with capacity 13,\nan edge from node 13 to node 12 with capacity 13.\nQ: What is the maximum flow from node 1 to node 12?\nA:", "answer": "The maximum flow from node 1 to node 12 is 24.", "difficulty": "hard", "doc_id": "16"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 1 with capacity 19,\nan edge from node 0 to node 3 with capacity 13,\nan edge from node 0 to node 2 with capacity 9,\nan edge from node 0 to node 8 with capacity 12,\nan edge from node 2 to node 6 with capacity 17,\nan edge from node 2 to node 3 with capacity 7,\nan edge from node 3 to node 0 with capacity 13,\nan edge from node 3 to node 1 with capacity 12,\nan edge from node 4 to node 6 with capacity 1,\nan edge from node 4 to node 1 with capacity 17,\nan edge from node 4 to node 3 with capacity 4,\nan edge from node 5 to node 4 with capacity 20,\nan edge from node 5 to node 2 with capacity 3,\nan edge from node 6 to node 9 with capacity 13,\nan edge from node 6 to node 4 with capacity 18,\nan edge from node 6 to node 3 with capacity 12,\nan edge from node 6 to node 7 with capacity 4,\nan edge from node 6 to node 10 with capacity 11,\nan edge from node 7 to node 9 with capacity 16,\nan edge from node 7 to node 0 with capacity 9,\nan edge from node 7 to node 1 with capacity 7,\nan edge from node 8 to node 9 with capacity 14,\nan edge from node 9 to node 1 with capacity 1,\nan edge from node 10 to node 6 with capacity 20,\nan edge from node 10 to node 8 with capacity 12.\nQ: What is the maximum flow from node 6 to node 0?\nA:", "answer": "The maximum flow from node 6 to node 0 is 17.", "difficulty": "hard", "doc_id": "17"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 4 with capacity 1,\nan edge from node 0 to node 10 with capacity 17,\nan edge from node 0 to node 8 with capacity 19,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 5 with capacity 14,\nan edge from node 3 to node 7 with capacity 5,\nan edge from node 3 to node 6 with capacity 3,\nan edge from node 3 to node 8 with capacity 19,\nan edge from node 3 to node 11 with capacity 12,\nan edge from node 3 to node 2 with capacity 13,\nan edge from node 3 to node 5 with capacity 13,\nan edge from node 4 to node 1 with capacity 1,\nan edge from node 4 to node 0 with capacity 11,\nan edge from node 5 to node 10 with capacity 6,\nan edge from node 5 to node 3 with capacity 3,\nan edge from node 6 to node 4 with capacity 5,\nan edge from node 6 to node 1 with capacity 12,\nan edge from node 6 to node 5 with capacity 2,\nan edge from node 7 to node 6 with capacity 1,\nan edge from node 8 to node 1 with capacity 8,\nan edge from node 9 to node 7 with capacity 6,\nan edge from node 9 to node 3 with capacity 15,\nan edge from node 9 to node 0 with capacity 2,\nan edge from node 10 to node 4 with capacity 6,\nan edge from node 10 to node 1 with capacity 2,\nan edge from node 11 to node 7 with capacity 7,\nan edge from node 11 to node 6 with capacity 18.\nQ: What is the maximum flow from node 2 to node 6?\nA:", "answer": "The maximum flow from node 2 to node 6 is 4.", "difficulty": "hard", "doc_id": "18"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 6 with capacity 1,\nan edge from node 0 to node 7 with capacity 10,\nan edge from node 2 to node 1 with capacity 9,\nan edge from node 2 to node 7 with capacity 8,\nan edge from node 3 to node 1 with capacity 9,\nan edge from node 3 to node 6 with capacity 10,\nan edge from node 3 to node 4 with capacity 7,\nan edge from node 4 to node 6 with capacity 4,\nan edge from node 5 to node 3 with capacity 7,\nan edge from node 5 to node 2 with capacity 9,\nan edge from node 6 to node 4 with capacity 1,\nan edge from node 7 to node 5 with capacity 8,\nan edge from node 7 to node 6 with capacity 5.\nQ: What is the maximum flow from node 2 to node 4?\nA:", "answer": "The maximum flow from node 2 to node 4 is 8.", "difficulty": "easy", "doc_id": "19"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 1 to node 0 with capacity 4,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 1 to node 4 with capacity 1,\nan edge from node 3 to node 1 with capacity 3,\nan edge from node 3 to node 4 with capacity 6,\nan edge from node 3 to node 2 with capacity 6,\nan edge from node 4 to node 1 with capacity 7,\nan edge from node 4 to node 2 with capacity 4.\nQ: What is the maximum flow from node 1 to node 2?\nA:", "answer": "The maximum flow from node 1 to node 2 is 7.", "difficulty": "easy", "doc_id": "20"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 6,\nan edge from node 0 to node 6 with capacity 1,\nan edge from node 1 to node 5 with capacity 1,\nan edge from node 1 to node 0 with capacity 7,\nan edge from node 3 to node 1 with capacity 6,\nan edge from node 3 to node 6 with capacity 5,\nan edge from node 4 to node 2 with capacity 8,\nan edge from node 4 to node 0 with capacity 8,\nan edge from node 5 to node 3 with capacity 6.\nQ: What is the maximum flow from node 1 to node 6?\nA:", "answer": "The maximum flow from node 1 to node 6 is 6.", "difficulty": "easy", "doc_id": "21"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 3 with capacity 17,\nan edge from node 0 to node 1 with capacity 16,\nan edge from node 1 to node 3 with capacity 8,\nan edge from node 1 to node 0 with capacity 11,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 8 with capacity 2,\nan edge from node 1 to node 9 with capacity 4,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 8 with capacity 7,\nan edge from node 3 to node 10 with capacity 10,\nan edge from node 4 to node 1 with capacity 8,\nan edge from node 5 to node 6 with capacity 13,\nan edge from node 5 to node 10 with capacity 12,\nan edge from node 5 to node 9 with capacity 18,\nan edge from node 6 to node 3 with capacity 14,\nan edge from node 6 to node 4 with capacity 3,\nan edge from node 6 to node 13 with capacity 9,\nan edge from node 6 to node 9 with capacity 13,\nan edge from node 6 to node 1 with capacity 13,\nan edge from node 7 to node 4 with capacity 11,\nan edge from node 7 to node 0 with capacity 8,\nan edge from node 7 to node 5 with capacity 12,\nan edge from node 7 to node 10 with capacity 13,\nan edge from node 7 to node 1 with capacity 3,\nan edge from node 8 to node 11 with capacity 9,\nan edge from node 8 to node 4 with capacity 11,\nan edge from node 8 to node 7 with capacity 1,\nan edge from node 8 to node 10 with capacity 11,\nan edge from node 9 to node 2 with capacity 2,\nan edge from node 9 to node 5 with capacity 18,\nan edge from node 9 to node 8 with capacity 7,\nan edge from node 10 to node 0 with capacity 20,\nan edge from node 10 to node 1 with capacity 14,\nan edge from node 11 to node 0 with capacity 11,\nan edge from node 11 to node 6 with capacity 13,\nan edge from node 11 to node 7 with capacity 14,\nan edge from node 12 to node 2 with capacity 8,\nan edge from node 12 to node 5 with capacity 13,\nan edge from node 12 to node 8 with capacity 15,\nan edge from node 12 to node 10 with capacity 10,\nan edge from node 13 to node 3 with capacity 2,\nan edge from node 13 to node 0 with capacity 19.\nQ: What is the maximum flow from node 6 to node 0?\nA:", "answer": "The maximum flow from node 6 to node 0 is 50.", "difficulty": "hard", "doc_id": "22"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 1 with capacity 7,\nan edge from node 0 to node 3 with capacity 5,\nan edge from node 0 to node 2 with capacity 1,\nan edge from node 1 to node 5 with capacity 8,\nan edge from node 2 to node 5 with capacity 8,\nan edge from node 3 to node 1 with capacity 1,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 5?\nA:", "answer": "The maximum flow from node 0 to node 5 is 9.", "difficulty": "easy", "doc_id": "23"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 10 with capacity 10,\nan edge from node 1 to node 13 with capacity 16,\nan edge from node 2 to node 11 with capacity 2,\nan edge from node 2 to node 9 with capacity 20,\nan edge from node 2 to node 14 with capacity 9,\nan edge from node 2 to node 3 with capacity 20,\nan edge from node 3 to node 0 with capacity 16,\nan edge from node 3 to node 13 with capacity 10,\nan edge from node 3 to node 11 with capacity 3,\nan edge from node 3 to node 14 with capacity 6,\nan edge from node 3 to node 1 with capacity 1,\nan edge from node 4 to node 7 with capacity 15,\nan edge from node 4 to node 13 with capacity 9,\nan edge from node 5 to node 0 with capacity 13,\nan edge from node 5 to node 13 with capacity 7,\nan edge from node 5 to node 4 with capacity 14,\nan edge from node 5 to node 15 with capacity 15,\nan edge from node 6 to node 7 with capacity 20,\nan edge from node 6 to node 9 with capacity 17,\nan edge from node 6 to node 5 with capacity 2,\nan edge from node 6 to node 14 with capacity 12,\nan edge from node 6 to node 3 with capacity 6,\nan edge from node 7 to node 0 with capacity 9,\nan edge from node 7 to node 12 with capacity 13,\nan edge from node 7 to node 3 with capacity 8,\nan edge from node 8 to node 6 with capacity 10,\nan edge from node 8 to node 14 with capacity 18,\nan edge from node 8 to node 15 with capacity 5,\nan edge from node 8 to node 2 with capacity 15,\nan edge from node 9 to node 13 with capacity 9,\nan edge from node 9 to node 11 with capacity 9,\nan edge from node 9 to node 4 with capacity 14,\nan edge from node 9 to node 5 with capacity 2,\nan edge from node 10 to node 6 with capacity 7,\nan edge from node 10 to node 4 with capacity 18,\nan edge from node 10 to node 14 with capacity 10,\nan edge from node 10 to node 12 with capacity 19,\nan edge from node 11 to node 0 with capacity 14,\nan edge from node 11 to node 4 with capacity 7,\nan edge from node 11 to node 14 with capacity 5,\nan edge from node 11 to node 15 with capacity 2,\nan edge from node 11 to node 1 with capacity 17,\nan edge from node 11 to node 8 with capacity 20,\nan edge from node 11 to node 2 with capacity 20,\nan edge from node 12 to node 7 with capacity 20,\nan edge from node 12 to node 13 with capacity 16,\nan edge from node 12 to node 9 with capacity 3,\nan edge from node 12 to node 4 with capacity 3,\nan edge from node 13 to node 10 with capacity 14,\nan edge from node 13 to node 11 with capacity 17,\nan edge from node 13 to node 9 with capacity 13,\nan edge from node 14 to node 10 with capacity 6,\nan edge from node 14 to node 11 with capacity 3,\nan edge from node 14 to node 12 with capacity 16,\nan edge from node 14 to node 3 with capacity 20,\nan edge from node 14 to node 15 with capacity 4,\nan edge from node 14 to node 2 with capacity 14,\nan edge from node 15 to node 4 with capacity 10,\nan edge from node 15 to node 3 with capacity 9.\nQ: What is the maximum flow from node 3 to node 9?\nA:", "answer": "The maximum flow from node 3 to node 9 is 30.", "difficulty": "hard", "doc_id": "24"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 1 with capacity 4,\nan edge from node 1 to node 7 with capacity 4,\nan edge from node 1 to node 2 with capacity 3,\nan edge from node 1 to node 6 with capacity 5,\nan edge from node 2 to node 8 with capacity 5,\nan edge from node 2 to node 0 with capacity 10,\nan edge from node 2 to node 1 with capacity 4,\nan edge from node 3 to node 5 with capacity 1,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 0 with capacity 9,\nan edge from node 4 to node 1 with capacity 6,\nan edge from node 4 to node 3 with capacity 2,\nan edge from node 5 to node 8 with capacity 1,\nan edge from node 5 to node 1 with capacity 8,\nan edge from node 6 to node 2 with capacity 3,\nan edge from node 6 to node 9 with capacity 2,\nan edge from node 7 to node 4 with capacity 7,\nan edge from node 8 to node 2 with capacity 8,\nan edge from node 8 to node 6 with capacity 9.\nQ: What is the maximum flow from node 6 to node 8?\nA:", "answer": "The maximum flow from node 6 to node 8 is 3.", "difficulty": "easy", "doc_id": "25"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 5 with capacity 9,\nan edge from node 0 to node 3 with capacity 4,\nan edge from node 0 to node 6 with capacity 4,\nan edge from node 0 to node 2 with capacity 9,\nan edge from node 1 to node 5 with capacity 5,\nan edge from node 2 to node 0 with capacity 1,\nan edge from node 4 to node 2 with capacity 3,\nan edge from node 5 to node 3 with capacity 5,\nan edge from node 5 to node 2 with capacity 4,\nan edge from node 6 to node 5 with capacity 2,\nan edge from node 6 to node 1 with capacity 7.\nQ: What is the maximum flow from node 4 to node 3?\nA:", "answer": "The maximum flow from node 4 to node 3 is 1.", "difficulty": "easy", "doc_id": "26"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 10 with capacity 15,\nan edge from node 0 to node 9 with capacity 1,\nan edge from node 0 to node 3 with capacity 18,\nan edge from node 1 to node 5 with capacity 2,\nan edge from node 1 to node 8 with capacity 2,\nan edge from node 1 to node 4 with capacity 3,\nan edge from node 2 to node 5 with capacity 16,\nan edge from node 2 to node 10 with capacity 14,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 9 with capacity 3,\nan edge from node 3 to node 4 with capacity 5,\nan edge from node 3 to node 12 with capacity 6,\nan edge from node 4 to node 10 with capacity 7,\nan edge from node 4 to node 6 with capacity 4,\nan edge from node 4 to node 9 with capacity 8,\nan edge from node 5 to node 10 with capacity 18,\nan edge from node 5 to node 1 with capacity 6,\nan edge from node 5 to node 4 with capacity 8,\nan edge from node 6 to node 0 with capacity 17,\nan edge from node 6 to node 1 with capacity 3,\nan edge from node 6 to node 12 with capacity 13,\nan edge from node 7 to node 5 with capacity 5,\nan edge from node 7 to node 6 with capacity 7,\nan edge from node 7 to node 11 with capacity 1,\nan edge from node 7 to node 12 with capacity 6,\nan edge from node 8 to node 5 with capacity 17,\nan edge from node 8 to node 6 with capacity 14,\nan edge from node 8 to node 9 with capacity 4,\nan edge from node 8 to node 4 with capacity 6,\nan edge from node 9 to node 0 with capacity 9,\nan edge from node 9 to node 6 with capacity 13,\nan edge from node 9 to node 4 with capacity 2,\nan edge from node 10 to node 5 with capacity 10,\nan edge from node 10 to node 8 with capacity 3,\nan edge from node 10 to node 6 with capacity 8,\nan edge from node 10 to node 9 with capacity 14,\nan edge from node 10 to node 3 with capacity 19,\nan edge from node 11 to node 5 with capacity 9,\nan edge from node 11 to node 2 with capacity 15,\nan edge from node 11 to node 7 with capacity 3,\nan edge from node 11 to node 4 with capacity 10,\nan edge from node 12 to node 11 with capacity 15,\nan edge from node 12 to node 9 with capacity 8,\nan edge from node 12 to node 1 with capacity 11,\nan edge from node 12 to node 3 with capacity 7.\nQ: What is the maximum flow from node 12 to node 5?\nA:", "answer": "The maximum flow from node 12 to node 5 is 32.", "difficulty": "hard", "doc_id": "27"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 15 with capacity 4,\nan edge from node 0 to node 11 with capacity 2,\nan edge from node 0 to node 16 with capacity 8,\nan edge from node 0 to node 14 with capacity 18,\nan edge from node 1 to node 4 with capacity 20,\nan edge from node 2 to node 4 with capacity 20,\nan edge from node 2 to node 7 with capacity 14,\nan edge from node 2 to node 11 with capacity 3,\nan edge from node 2 to node 8 with capacity 17,\nan edge from node 3 to node 0 with capacity 3,\nan edge from node 3 to node 6 with capacity 12,\nan edge from node 4 to node 16 with capacity 5,\nan edge from node 5 to node 2 with capacity 9,\nan edge from node 5 to node 15 with capacity 14,\nan edge from node 5 to node 12 with capacity 11,\nan edge from node 5 to node 3 with capacity 6,\nan edge from node 5 to node 11 with capacity 12,\nan edge from node 5 to node 9 with capacity 19,\nan edge from node 6 to node 2 with capacity 9,\nan edge from node 6 to node 15 with capacity 7,\nan edge from node 6 to node 12 with capacity 1,\nan edge from node 6 to node 1 with capacity 12,\nan edge from node 6 to node 16 with capacity 2,\nan edge from node 6 to node 14 with capacity 20,\nan edge from node 7 to node 4 with capacity 18,\nan edge from node 7 to node 5 with capacity 9,\nan edge from node 7 to node 10 with capacity 6,\nan edge from node 7 to node 11 with capacity 5,\nan edge from node 7 to node 16 with capacity 16,\nan edge from node 7 to node 6 with capacity 3,\nan edge from node 8 to node 15 with capacity 8,\nan edge from node 8 to node 4 with capacity 11,\nan edge from node 9 to node 2 with capacity 6,\nan edge from node 9 to node 4 with capacity 6,\nan edge from node 9 to node 12 with capacity 10,\nan edge from node 9 to node 7 with capacity 20,\nan edge from node 9 to node 5 with capacity 18,\nan edge from node 9 to node 11 with capacity 15,\nan edge from node 10 to node 5 with capacity 11,\nan edge from node 10 to node 3 with capacity 6,\nan edge from node 10 to node 11 with capacity 15,\nan edge from node 10 to node 13 with capacity 5,\nan edge from node 11 to node 15 with capacity 9,\nan edge from node 11 to node 5 with capacity 3,\nan edge from node 11 to node 1 with capacity 8,\nan edge from node 11 to node 9 with capacity 2,\nan edge from node 11 to node 13 with capacity 14,\nan edge from node 11 to node 14 with capacity 4,\nan edge from node 12 to node 6 with capacity 13,\nan edge from node 13 to node 7 with capacity 16,\nan edge from node 14 to node 15 with capacity 20,\nan edge from node 14 to node 7 with capacity 15,\nan edge from node 14 to node 0 with capacity 7,\nan edge from node 14 to node 5 with capacity 19,\nan edge from node 14 to node 1 with capacity 10,\nan edge from node 14 to node 9 with capacity 13,\nan edge from node 15 to node 1 with capacity 17,\nan edge from node 15 to node 9 with capacity 6,\nan edge from node 15 to node 14 with capacity 4,\nan edge from node 15 to node 8 with capacity 15,\nan edge from node 15 to node 6 with capacity 1,\nan edge from node 16 to node 12 with capacity 12,\nan edge from node 16 to node 5 with capacity 11,\nan edge from node 16 to node 10 with capacity 1,\nan edge from node 16 to node 3 with capacity 16,\nan edge from node 16 to node 9 with capacity 6,\nan edge from node 16 to node 6 with capacity 16.\nQ: What is the maximum flow from node 10 to node 14?\nA:", "answer": "The maximum flow from node 10 to node 14 is 31.", "difficulty": "hard", "doc_id": "28"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 2 with capacity 9,\nan edge from node 2 to node 1 with capacity 8,\nan edge from node 2 to node 4 with capacity 8,\nan edge from node 3 to node 0 with capacity 5,\nan edge from node 3 to node 1 with capacity 4,\nan edge from node 3 to node 2 with capacity 5,\nan edge from node 4 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 1.\nQ: What is the maximum flow from node 2 to node 1?\nA:", "answer": "The maximum flow from node 2 to node 1 is 16.", "difficulty": "easy", "doc_id": "29"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 0 to node 6 with capacity 2,\nan edge from node 0 to node 15 with capacity 13,\nan edge from node 0 to node 16 with capacity 17,\nan edge from node 0 to node 5 with capacity 12,\nan edge from node 1 to node 10 with capacity 8,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 1 to node 2 with capacity 15,\nan edge from node 1 to node 0 with capacity 7,\nan edge from node 1 to node 6 with capacity 9,\nan edge from node 1 to node 9 with capacity 19,\nan edge from node 1 to node 5 with capacity 4,\nan edge from node 2 to node 10 with capacity 15,\nan edge from node 2 to node 4 with capacity 8,\nan edge from node 2 to node 9 with capacity 5,\nan edge from node 2 to node 8 with capacity 2,\nan edge from node 2 to node 3 with capacity 8,\nan edge from node 2 to node 5 with capacity 6,\nan edge from node 2 to node 11 with capacity 7,\nan edge from node 3 to node 17 with capacity 13,\nan edge from node 3 to node 14 with capacity 4,\nan edge from node 3 to node 2 with capacity 13,\nan edge from node 3 to node 0 with capacity 5,\nan edge from node 3 to node 8 with capacity 13,\nan edge from node 3 to node 7 with capacity 12,\nan edge from node 4 to node 17 with capacity 20,\nan edge from node 4 to node 10 with capacity 3,\nan edge from node 4 to node 15 with capacity 20,\nan edge from node 5 to node 12 with capacity 20,\nan edge from node 5 to node 14 with capacity 13,\nan edge from node 5 to node 9 with capacity 2,\nan edge from node 5 to node 11 with capacity 6,\nan edge from node 6 to node 10 with capacity 20,\nan edge from node 6 to node 2 with capacity 11,\nan edge from node 6 to node 15 with capacity 13,\nan edge from node 6 to node 9 with capacity 14,\nan edge from node 6 to node 8 with capacity 16,\nan edge from node 6 to node 3 with capacity 8,\nan edge from node 6 to node 7 with capacity 3,\nan edge from node 7 to node 1 with capacity 15,\nan edge from node 7 to node 19 with capacity 15,\nan edge from node 8 to node 12 with capacity 18,\nan edge from node 8 to node 15 with capacity 2,\nan edge from node 8 to node 18 with capacity 10,\nan edge from node 8 to node 19 with capacity 11,\nan edge from node 8 to node 5 with capacity 7,\nan edge from node 8 to node 7 with capacity 10,\nan edge from node 9 to node 12 with capacity 17,\nan edge from node 9 to node 4 with capacity 11,\nan edge from node 9 to node 1 with capacity 5,\nan edge from node 9 to node 6 with capacity 15,\nan edge from node 9 to node 15 with capacity 20,\nan edge from node 9 to node 18 with capacity 14,\nan edge from node 9 to node 8 with capacity 8,\nan edge from node 9 to node 7 with capacity 19,\nan edge from node 10 to node 12 with capacity 4,\nan edge from node 10 to node 17 with capacity 19,\nan edge from node 10 to node 8 with capacity 12,\nan edge from node 10 to node 11 with capacity 13,\nan edge from node 11 to node 13 with capacity 18,\nan edge from node 11 to node 6 with capacity 15,\nan edge from node 12 to node 17 with capacity 7,\nan edge from node 12 to node 14 with capacity 19,\nan edge from node 12 to node 10 with capacity 1,\nan edge from node 12 to node 2 with capacity 1,\nan edge from node 12 to node 5 with capacity 20,\nan edge from node 13 to node 17 with capacity 9,\nan edge from node 13 to node 6 with capacity 16,\nan edge from node 13 to node 18 with capacity 18,\nan edge from node 13 to node 8 with capacity 20,\nan edge from node 13 to node 19 with capacity 8,\nan edge from node 13 to node 7 with capacity 20,\nan edge from node 14 to node 10 with capacity 7,\nan edge from node 14 to node 6 with capacity 3,\nan edge from node 14 to node 18 with capacity 13,\nan edge from node 14 to node 8 with capacity 13,\nan edge from node 14 to node 11 with capacity 20,\nan edge from node 15 to node 6 with capacity 18,\nan edge from node 15 to node 9 with capacity 18,\nan edge from node 15 to node 19 with capacity 9,\nan edge from node 15 to node 7 with capacity 3,\nan edge from node 16 to node 17 with capacity 4,\nan edge from node 16 to node 14 with capacity 17,\nan edge from node 16 to node 2 with capacity 5,\nan edge from node 16 to node 0 with capacity 18,\nan edge from node 16 to node 3 with capacity 12,\nan edge from node 16 to node 11 with capacity 5,\nan edge from node 17 to node 3 with capacity 15,\nan edge from node 18 to node 12 with capacity 7,\nan edge from node 18 to node 10 with capacity 6,\nan edge from node 18 to node 4 with capacity 10,\nan edge from node 18 to node 0 with capacity 5,\nan edge from node 18 to node 15 with capacity 18,\nan edge from node 18 to node 11 with capacity 7,\nan edge from node 19 to node 12 with capacity 9,\nan edge from node 19 to node 2 with capacity 4,\nan edge from node 19 to node 0 with capacity 9,\nan edge from node 19 to node 9 with capacity 16,\nan edge from node 19 to node 8 with capacity 11,\nan edge from node 19 to node 16 with capacity 17,\nan edge from node 19 to node 11 with capacity 8,\nan edge from node 19 to node 7 with capacity 14.\nQ: What is the maximum flow from node 5 to node 0?\nA:", "answer": "The maximum flow from node 5 to node 0 is 41.", "difficulty": "hard", "doc_id": "30"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 14 with capacity 9,\nan edge from node 0 to node 13 with capacity 12,\nan edge from node 0 to node 2 with capacity 5,\nan edge from node 0 to node 1 with capacity 12,\nan edge from node 0 to node 5 with capacity 7,\nan edge from node 1 to node 14 with capacity 5,\nan edge from node 1 to node 3 with capacity 8,\nan edge from node 2 to node 4 with capacity 19,\nan edge from node 3 to node 8 with capacity 11,\nan edge from node 3 to node 0 with capacity 3,\nan edge from node 3 to node 11 with capacity 5,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 3 to node 12 with capacity 8,\nan edge from node 4 to node 14 with capacity 12,\nan edge from node 4 to node 3 with capacity 16,\nan edge from node 4 to node 10 with capacity 11,\nan edge from node 4 to node 11 with capacity 18,\nan edge from node 4 to node 7 with capacity 10,\nan edge from node 4 to node 6 with capacity 11,\nan edge from node 5 to node 13 with capacity 16,\nan edge from node 5 to node 0 with capacity 8,\nan edge from node 5 to node 15 with capacity 18,\nan edge from node 5 to node 11 with capacity 5,\nan edge from node 6 to node 5 with capacity 8,\nan edge from node 6 to node 12 with capacity 13,\nan edge from node 7 to node 14 with capacity 12,\nan edge from node 7 to node 0 with capacity 1,\nan edge from node 7 to node 3 with capacity 10,\nan edge from node 7 to node 9 with capacity 17,\nan edge from node 7 to node 17 with capacity 16,\nan edge from node 7 to node 1 with capacity 18,\nan edge from node 7 to node 4 with capacity 3,\nan edge from node 7 to node 12 with capacity 13,\nan edge from node 8 to node 3 with capacity 6,\nan edge from node 8 to node 4 with capacity 9,\nan edge from node 9 to node 13 with capacity 12,\nan edge from node 9 to node 0 with capacity 13,\nan edge from node 9 to node 2 with capacity 2,\nan edge from node 9 to node 12 with capacity 6,\nan edge from node 10 to node 16 with capacity 20,\nan edge from node 10 to node 0 with capacity 18,\nan edge from node 10 to node 3 with capacity 4,\nan edge from node 10 to node 2 with capacity 7,\nan edge from node 10 to node 9 with capacity 13,\nan edge from node 10 to node 6 with capacity 9,\nan edge from node 11 to node 14 with capacity 14,\nan edge from node 11 to node 0 with capacity 7,\nan edge from node 11 to node 9 with capacity 5,\nan edge from node 11 to node 17 with capacity 18,\nan edge from node 12 to node 13 with capacity 7,\nan edge from node 12 to node 0 with capacity 17,\nan edge from node 12 to node 17 with capacity 3,\nan edge from node 12 to node 1 with capacity 10,\nan edge from node 12 to node 6 with capacity 18,\nan edge from node 13 to node 0 with capacity 6,\nan edge from node 13 to node 9 with capacity 2,\nan edge from node 13 to node 1 with capacity 20,\nan edge from node 14 to node 0 with capacity 3,\nan edge from node 14 to node 3 with capacity 20,\nan edge from node 14 to node 10 with capacity 18,\nan edge from node 14 to node 11 with capacity 13,\nan edge from node 15 to node 8 with capacity 14,\nan edge from node 15 to node 0 with capacity 8,\nan edge from node 15 to node 2 with capacity 2,\nan edge from node 15 to node 11 with capacity 15,\nan edge from node 15 to node 1 with capacity 1,\nan edge from node 15 to node 6 with capacity 6,\nan edge from node 15 to node 4 with capacity 14,\nan edge from node 16 to node 14 with capacity 1,\nan edge from node 16 to node 0 with capacity 2,\nan edge from node 16 to node 15 with capacity 16,\nan edge from node 16 to node 2 with capacity 1,\nan edge from node 16 to node 9 with capacity 11,\nan edge from node 16 to node 11 with capacity 1,\nan edge from node 16 to node 5 with capacity 15,\nan edge from node 17 to node 14 with capacity 11,\nan edge from node 17 to node 8 with capacity 14,\nan edge from node 17 to node 12 with capacity 6.\nQ: What is the maximum flow from node 16 to node 1?\nA:", "answer": "The maximum flow from node 16 to node 1 is 47.", "difficulty": "hard", "doc_id": "31"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 6 with capacity 9,\nan edge from node 1 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 7,\nan edge from node 3 to node 1 with capacity 6,\nan edge from node 6 to node 3 with capacity 8,\nan edge from node 7 to node 0 with capacity 10,\nan edge from node 7 to node 8 with capacity 5,\nan edge from node 8 to node 3 with capacity 5,\nan edge from node 8 to node 1 with capacity 10.\nQ: What is the maximum flow from node 7 to node 1?\nA:", "answer": "The maximum flow from node 7 to node 1 is 11.", "difficulty": "easy", "doc_id": "32"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 2 with capacity 18,\nan edge from node 0 to node 12 with capacity 10,\nan edge from node 0 to node 1 with capacity 14,\nan edge from node 0 to node 4 with capacity 1,\nan edge from node 1 to node 9 with capacity 10,\nan edge from node 1 to node 8 with capacity 12,\nan edge from node 1 to node 3 with capacity 16,\nan edge from node 2 to node 0 with capacity 6,\nan edge from node 2 to node 3 with capacity 13,\nan edge from node 2 to node 14 with capacity 10,\nan edge from node 2 to node 10 with capacity 13,\nan edge from node 2 to node 6 with capacity 14,\nan edge from node 3 to node 12 with capacity 8,\nan edge from node 3 to node 8 with capacity 20,\nan edge from node 3 to node 4 with capacity 12,\nan edge from node 4 to node 0 with capacity 19,\nan edge from node 4 to node 6 with capacity 18,\nan edge from node 5 to node 11 with capacity 11,\nan edge from node 5 to node 3 with capacity 13,\nan edge from node 5 to node 1 with capacity 1,\nan edge from node 5 to node 14 with capacity 8,\nan edge from node 5 to node 4 with capacity 1,\nan edge from node 5 to node 7 with capacity 10,\nan edge from node 5 to node 6 with capacity 18,\nan edge from node 6 to node 11 with capacity 7,\nan edge from node 6 to node 3 with capacity 8,\nan edge from node 6 to node 14 with capacity 10,\nan edge from node 6 to node 10 with capacity 20,\nan edge from node 7 to node 0 with capacity 11,\nan edge from node 7 to node 8 with capacity 6,\nan edge from node 7 to node 1 with capacity 9,\nan edge from node 7 to node 10 with capacity 17,\nan edge from node 8 to node 3 with capacity 6,\nan edge from node 9 to node 11 with capacity 9,\nan edge from node 9 to node 2 with capacity 12,\nan edge from node 9 to node 4 with capacity 4,\nan edge from node 10 to node 2 with capacity 14,\nan edge from node 10 to node 0 with capacity 17,\nan edge from node 10 to node 3 with capacity 9,\nan edge from node 11 to node 2 with capacity 8,\nan edge from node 11 to node 3 with capacity 11,\nan edge from node 11 to node 4 with capacity 17,\nan edge from node 11 to node 7 with capacity 4,\nan edge from node 12 to node 5 with capacity 13,\nan edge from node 12 to node 3 with capacity 2,\nan edge from node 13 to node 9 with capacity 13,\nan edge from node 14 to node 0 with capacity 7,\nan edge from node 14 to node 12 with capacity 10,\nan edge from node 14 to node 10 with capacity 6,\nan edge from node 14 to node 7 with capacity 2.\nQ: What is the maximum flow from node 11 to node 7?\nA:", "answer": "The maximum flow from node 11 to node 7 is 16.", "difficulty": "hard", "doc_id": "33"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 6,\nan edge from node 1 to node 5 with capacity 2,\nan edge from node 2 to node 4 with capacity 7,\nan edge from node 2 to node 5 with capacity 2,\nan edge from node 3 to node 4 with capacity 2,\nan edge from node 3 to node 5 with capacity 2,\nan edge from node 4 to node 3 with capacity 3,\nan edge from node 5 to node 2 with capacity 10,\nan edge from node 5 to node 0 with capacity 3.\nQ: What is the maximum flow from node 3 to node 4?\nA:", "answer": "The maximum flow from node 3 to node 4 is 4.", "difficulty": "easy", "doc_id": "34"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 5,\nan edge from node 1 to node 0 with capacity 5,\nan edge from node 2 to node 3 with capacity 4,\nan edge from node 2 to node 1 with capacity 1,\nan edge from node 3 to node 0 with capacity 7,\nan edge from node 3 to node 4 with capacity 8,\nan edge from node 4 to node 0 with capacity 1.\nQ: What is the maximum flow from node 2 to node 1?\nA:", "answer": "The maximum flow from node 2 to node 1 is 5.", "difficulty": "easy", "doc_id": "35"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 1,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 2 to node 3 with capacity 2,\nan edge from node 2 to node 0 with capacity 10,\nan edge from node 3 to node 0 with capacity 9,\nan edge from node 4 to node 0 with capacity 6.\nQ: What is the maximum flow from node 2 to node 4?\nA:", "answer": "The maximum flow from node 2 to node 4 is 5.", "difficulty": "easy", "doc_id": "36"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 11 with capacity 5,\nan edge from node 0 to node 9 with capacity 19,\nan edge from node 0 to node 2 with capacity 9,\nan edge from node 2 to node 10 with capacity 17,\nan edge from node 2 to node 6 with capacity 20,\nan edge from node 3 to node 9 with capacity 9,\nan edge from node 3 to node 0 with capacity 8,\nan edge from node 4 to node 11 with capacity 15,\nan edge from node 4 to node 8 with capacity 20,\nan edge from node 4 to node 7 with capacity 5,\nan edge from node 4 to node 2 with capacity 9,\nan edge from node 4 to node 6 with capacity 15,\nan edge from node 4 to node 0 with capacity 14,\nan edge from node 5 to node 11 with capacity 13,\nan edge from node 5 to node 3 with capacity 12,\nan edge from node 5 to node 1 with capacity 15,\nan edge from node 6 to node 8 with capacity 18,\nan edge from node 6 to node 4 with capacity 7,\nan edge from node 6 to node 1 with capacity 4,\nan edge from node 6 to node 0 with capacity 4,\nan edge from node 7 to node 11 with capacity 19,\nan edge from node 7 to node 4 with capacity 10,\nan edge from node 7 to node 10 with capacity 5,\nan edge from node 8 to node 2 with capacity 10,\nan edge from node 8 to node 3 with capacity 14,\nan edge from node 9 to node 2 with capacity 6,\nan edge from node 9 to node 0 with capacity 9,\nan edge from node 10 to node 6 with capacity 12,\nan edge from node 10 to node 3 with capacity 8,\nan edge from node 10 to node 0 with capacity 12,\nan edge from node 11 to node 8 with capacity 18,\nan edge from node 11 to node 7 with capacity 19,\nan edge from node 11 to node 1 with capacity 9,\nan edge from node 11 to node 0 with capacity 1.\nQ: What is the maximum flow from node 10 to node 6?\nA:", "answer": "The maximum flow from node 10 to node 6 is 32.", "difficulty": "hard", "doc_id": "37"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 6,\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 1 to node 3 with capacity 1,\nan edge from node 2 to node 0 with capacity 5,\nan edge from node 2 to node 3 with capacity 5,\nan edge from node 4 to node 3 with capacity 5,\nan edge from node 4 to node 2 with capacity 4.\nQ: What is the maximum flow from node 4 to node 3?\nA:", "answer": "The maximum flow from node 4 to node 3 is 9.", "difficulty": "easy", "doc_id": "38"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 8 with capacity 7,\nan edge from node 0 to node 3 with capacity 13,\nan edge from node 0 to node 5 with capacity 18,\nan edge from node 0 to node 9 with capacity 3,\nan edge from node 0 to node 10 with capacity 11,\nan edge from node 1 to node 3 with capacity 18,\nan edge from node 1 to node 9 with capacity 15,\nan edge from node 1 to node 10 with capacity 20,\nan edge from node 2 to node 8 with capacity 4,\nan edge from node 2 to node 3 with capacity 5,\nan edge from node 2 to node 5 with capacity 14,\nan edge from node 2 to node 10 with capacity 14,\nan edge from node 3 to node 0 with capacity 3,\nan edge from node 3 to node 10 with capacity 15,\nan edge from node 4 to node 3 with capacity 17,\nan edge from node 4 to node 2 with capacity 1,\nan edge from node 4 to node 6 with capacity 5,\nan edge from node 5 to node 8 with capacity 8,\nan edge from node 5 to node 3 with capacity 7,\nan edge from node 5 to node 11 with capacity 14,\nan edge from node 5 to node 10 with capacity 18,\nan edge from node 6 to node 4 with capacity 6,\nan edge from node 6 to node 5 with capacity 8,\nan edge from node 6 to node 1 with capacity 1,\nan edge from node 7 to node 3 with capacity 19,\nan edge from node 7 to node 2 with capacity 18,\nan edge from node 8 to node 1 with capacity 16,\nan edge from node 8 to node 11 with capacity 2,\nan edge from node 8 to node 10 with capacity 18,\nan edge from node 9 to node 4 with capacity 5,\nan edge from node 9 to node 5 with capacity 7,\nan edge from node 9 to node 2 with capacity 12,\nan edge from node 9 to node 1 with capacity 12,\nan edge from node 9 to node 11 with capacity 2,\nan edge from node 10 to node 4 with capacity 17,\nan edge from node 10 to node 2 with capacity 15,\nan edge from node 10 to node 1 with capacity 11,\nan edge from node 11 to node 8 with capacity 2,\nan edge from node 11 to node 9 with capacity 18,\nan edge from node 11 to node 6 with capacity 16,\nan edge from node 11 to node 1 with capacity 17,\nan edge from node 11 to node 10 with capacity 18.\nQ: What is the maximum flow from node 11 to node 10?\nA:", "answer": "The maximum flow from node 11 to node 10 is 70.", "difficulty": "hard", "doc_id": "39"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 10 with capacity 1,\nan edge from node 0 to node 12 with capacity 7,\nan edge from node 2 to node 13 with capacity 10,\nan edge from node 2 to node 9 with capacity 4,\nan edge from node 2 to node 7 with capacity 2,\nan edge from node 3 to node 2 with capacity 10,\nan edge from node 3 to node 6 with capacity 19,\nan edge from node 3 to node 15 with capacity 1,\nan edge from node 3 to node 14 with capacity 6,\nan edge from node 4 to node 13 with capacity 19,\nan edge from node 4 to node 16 with capacity 1,\nan edge from node 4 to node 9 with capacity 8,\nan edge from node 4 to node 6 with capacity 7,\nan edge from node 4 to node 12 with capacity 16,\nan edge from node 5 to node 0 with capacity 8,\nan edge from node 5 to node 6 with capacity 11,\nan edge from node 5 to node 4 with capacity 1,\nan edge from node 6 to node 5 with capacity 3,\nan edge from node 7 to node 17 with capacity 8,\nan edge from node 7 to node 13 with capacity 11,\nan edge from node 7 to node 0 with capacity 11,\nan edge from node 7 to node 9 with capacity 1,\nan edge from node 7 to node 14 with capacity 3,\nan edge from node 7 to node 1 with capacity 19,\nan edge from node 8 to node 3 with capacity 16,\nan edge from node 8 to node 6 with capacity 10,\nan edge from node 8 to node 10 with capacity 16,\nan edge from node 8 to node 12 with capacity 3,\nan edge from node 8 to node 14 with capacity 2,\nan edge from node 9 to node 0 with capacity 19,\nan edge from node 9 to node 8 with capacity 11,\nan edge from node 9 to node 4 with capacity 3,\nan edge from node 9 to node 1 with capacity 13,\nan edge from node 10 to node 2 with capacity 7,\nan edge from node 10 to node 9 with capacity 17,\nan edge from node 10 to node 4 with capacity 9,\nan edge from node 11 to node 2 with capacity 19,\nan edge from node 11 to node 5 with capacity 16,\nan edge from node 11 to node 16 with capacity 2,\nan edge from node 11 to node 9 with capacity 4,\nan edge from node 12 to node 6 with capacity 7,\nan edge from node 12 to node 15 with capacity 18,\nan edge from node 13 to node 5 with capacity 3,\nan edge from node 13 to node 17 with capacity 4,\nan edge from node 13 to node 16 with capacity 4,\nan edge from node 13 to node 8 with capacity 19,\nan edge from node 13 to node 4 with capacity 9,\nan edge from node 14 to node 5 with capacity 10,\nan edge from node 14 to node 16 with capacity 13,\nan edge from node 14 to node 11 with capacity 15,\nan edge from node 15 to node 5 with capacity 14,\nan edge from node 15 to node 0 with capacity 9,\nan edge from node 15 to node 9 with capacity 19,\nan edge from node 15 to node 8 with capacity 20,\nan edge from node 15 to node 7 with capacity 2,\nan edge from node 15 to node 4 with capacity 19,\nan edge from node 16 to node 3 with capacity 11,\nan edge from node 16 to node 8 with capacity 4,\nan edge from node 16 to node 14 with capacity 13,\nan edge from node 17 to node 8 with capacity 2.\nQ: What is the maximum flow from node 5 to node 4?\nA:", "answer": "The maximum flow from node 5 to node 4 is 9.", "difficulty": "hard", "doc_id": "40"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 4 with capacity 4,\nan edge from node 1 to node 4 with capacity 8,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 2 to node 0 with capacity 5,\nan edge from node 3 to node 0 with capacity 9,\nan edge from node 4 to node 3 with capacity 5,\nan edge from node 4 to node 0 with capacity 7.\nQ: What is the maximum flow from node 2 to node 0?\nA:", "answer": "The maximum flow from node 2 to node 0 is 15.", "difficulty": "easy", "doc_id": "41"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 6 with capacity 10,\nan edge from node 1 to node 6 with capacity 2,\nan edge from node 1 to node 5 with capacity 5,\nan edge from node 2 to node 4 with capacity 1,\nan edge from node 2 to node 3 with capacity 3,\nan edge from node 2 to node 6 with capacity 9,\nan edge from node 3 to node 2 with capacity 10,\nan edge from node 4 to node 6 with capacity 5,\nan edge from node 4 to node 5 with capacity 10,\nan edge from node 6 to node 4 with capacity 9,\nan edge from node 6 to node 3 with capacity 10,\nan edge from node 6 to node 0 with capacity 1.\nQ: What is the maximum flow from node 2 to node 3?\nA:", "answer": "The maximum flow from node 2 to node 3 is 13.", "difficulty": "easy", "doc_id": "42"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 17 with capacity 13,\nan edge from node 0 to node 6 with capacity 19,\nan edge from node 0 to node 13 with capacity 19,\nan edge from node 1 to node 16 with capacity 11,\nan edge from node 1 to node 11 with capacity 1,\nan edge from node 1 to node 3 with capacity 18,\nan edge from node 1 to node 14 with capacity 16,\nan edge from node 2 to node 0 with capacity 8,\nan edge from node 2 to node 11 with capacity 11,\nan edge from node 2 to node 17 with capacity 10,\nan edge from node 2 to node 9 with capacity 18,\nan edge from node 2 to node 13 with capacity 20,\nan edge from node 2 to node 10 with capacity 17,\nan edge from node 2 to node 4 with capacity 20,\nan edge from node 3 to node 2 with capacity 15,\nan edge from node 3 to node 11 with capacity 13,\nan edge from node 3 to node 6 with capacity 15,\nan edge from node 3 to node 12 with capacity 16,\nan edge from node 3 to node 7 with capacity 9,\nan edge from node 4 to node 11 with capacity 14,\nan edge from node 4 to node 3 with capacity 4,\nan edge from node 4 to node 5 with capacity 16,\nan edge from node 4 to node 8 with capacity 2,\nan edge from node 4 to node 12 with capacity 17,\nan edge from node 4 to node 1 with capacity 12,\nan edge from node 4 to node 10 with capacity 3,\nan edge from node 5 to node 15 with capacity 7,\nan edge from node 5 to node 6 with capacity 7,\nan edge from node 5 to node 1 with capacity 2,\nan edge from node 6 to node 11 with capacity 14,\nan edge from node 6 to node 12 with capacity 9,\nan edge from node 6 to node 13 with capacity 12,\nan edge from node 6 to node 10 with capacity 18,\nan edge from node 6 to node 7 with capacity 16,\nan edge from node 7 to node 2 with capacity 13,\nan edge from node 7 to node 11 with capacity 3,\nan edge from node 7 to node 9 with capacity 1,\nan edge from node 8 to node 4 with capacity 4,\nan edge from node 9 to node 8 with capacity 18,\nan edge from node 9 to node 6 with capacity 15,\nan edge from node 9 to node 12 with capacity 7,\nan edge from node 9 to node 13 with capacity 2,\nan edge from node 10 to node 2 with capacity 16,\nan edge from node 10 to node 3 with capacity 16,\nan edge from node 11 to node 5 with capacity 19,\nan edge from node 11 to node 15 with capacity 5,\nan edge from node 11 to node 4 with capacity 13,\nan edge from node 12 to node 16 with capacity 11,\nan edge from node 12 to node 7 with capacity 5,\nan edge from node 13 to node 0 with capacity 11,\nan edge from node 13 to node 5 with capacity 4,\nan edge from node 13 to node 8 with capacity 16,\nan edge from node 13 to node 15 with capacity 10,\nan edge from node 13 to node 9 with capacity 1,\nan edge from node 13 to node 4 with capacity 11,\nan edge from node 14 to node 12 with capacity 2,\nan edge from node 14 to node 13 with capacity 4,\nan edge from node 15 to node 16 with capacity 4,\nan edge from node 15 to node 5 with capacity 20,\nan edge from node 15 to node 6 with capacity 8,\nan edge from node 15 to node 1 with capacity 11,\nan edge from node 15 to node 10 with capacity 20,\nan edge from node 15 to node 4 with capacity 11,\nan edge from node 16 to node 8 with capacity 20,\nan edge from node 16 to node 14 with capacity 2,\nan edge from node 16 to node 15 with capacity 8,\nan edge from node 16 to node 6 with capacity 9,\nan edge from node 17 to node 0 with capacity 18,\nan edge from node 17 to node 5 with capacity 4,\nan edge from node 17 to node 8 with capacity 8,\nan edge from node 17 to node 9 with capacity 10.\nQ: What is the maximum flow from node 17 to node 9?\nA:", "answer": "The maximum flow from node 17 to node 9 is 30.", "difficulty": "hard", "doc_id": "43"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 4 with capacity 9,\nan edge from node 0 to node 2 with capacity 10,\nan edge from node 1 to node 4 with capacity 9,\nan edge from node 1 to node 0 with capacity 7,\nan edge from node 1 to node 2 with capacity 10,\nan edge from node 4 to node 3 with capacity 1,\nan edge from node 4 to node 2 with capacity 1.\nQ: What is the maximum flow from node 1 to node 2?\nA:", "answer": "The maximum flow from node 1 to node 2 is 18.", "difficulty": "easy", "doc_id": "44"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 4 with capacity 3,\nan edge from node 0 to node 2 with capacity 12,\nan edge from node 0 to node 9 with capacity 4,\nan edge from node 0 to node 8 with capacity 11,\nan edge from node 1 to node 5 with capacity 14,\nan edge from node 1 to node 4 with capacity 20,\nan edge from node 1 to node 0 with capacity 4,\nan edge from node 1 to node 8 with capacity 7,\nan edge from node 2 to node 7 with capacity 19,\nan edge from node 2 to node 0 with capacity 9,\nan edge from node 2 to node 10 with capacity 8,\nan edge from node 2 to node 1 with capacity 19,\nan edge from node 3 to node 7 with capacity 7,\nan edge from node 3 to node 2 with capacity 18,\nan edge from node 3 to node 9 with capacity 17,\nan edge from node 4 to node 2 with capacity 13,\nan edge from node 5 to node 11 with capacity 5,\nan edge from node 5 to node 0 with capacity 12,\nan edge from node 5 to node 3 with capacity 6,\nan edge from node 6 to node 7 with capacity 18,\nan edge from node 6 to node 4 with capacity 1,\nan edge from node 6 to node 9 with capacity 19,\nan edge from node 6 to node 0 with capacity 19,\nan edge from node 7 to node 5 with capacity 5,\nan edge from node 7 to node 3 with capacity 15,\nan edge from node 8 to node 9 with capacity 6,\nan edge from node 8 to node 0 with capacity 9,\nan edge from node 8 to node 10 with capacity 1,\nan edge from node 9 to node 3 with capacity 15,\nan edge from node 10 to node 4 with capacity 10,\nan edge from node 10 to node 6 with capacity 15,\nan edge from node 10 to node 0 with capacity 12,\nan edge from node 10 to node 3 with capacity 16,\nan edge from node 10 to node 8 with capacity 17,\nan edge from node 10 to node 1 with capacity 6,\nan edge from node 11 to node 7 with capacity 9,\nan edge from node 11 to node 2 with capacity 15.\nQ: What is the maximum flow from node 7 to node 0?\nA:", "answer": "The maximum flow from node 7 to node 0 is 20.", "difficulty": "hard", "doc_id": "45"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 12 with capacity 3,\nan edge from node 0 to node 3 with capacity 16,\nan edge from node 1 to node 4 with capacity 20,\nan edge from node 1 to node 11 with capacity 5,\nan edge from node 2 to node 7 with capacity 20,\nan edge from node 3 to node 2 with capacity 10,\nan edge from node 3 to node 8 with capacity 11,\nan edge from node 4 to node 6 with capacity 7,\nan edge from node 4 to node 1 with capacity 12,\nan edge from node 4 to node 7 with capacity 4,\nan edge from node 4 to node 11 with capacity 17,\nan edge from node 4 to node 5 with capacity 18,\nan edge from node 4 to node 8 with capacity 4,\nan edge from node 5 to node 6 with capacity 10,\nan edge from node 5 to node 2 with capacity 5,\nan edge from node 6 to node 2 with capacity 9,\nan edge from node 6 to node 3 with capacity 1,\nan edge from node 6 to node 8 with capacity 9,\nan edge from node 7 to node 5 with capacity 2,\nan edge from node 8 to node 10 with capacity 12,\nan edge from node 8 to node 4 with capacity 16,\nan edge from node 9 to node 1 with capacity 6,\nan edge from node 9 to node 11 with capacity 17,\nan edge from node 10 to node 2 with capacity 12,\nan edge from node 10 to node 7 with capacity 13,\nan edge from node 10 to node 4 with capacity 19,\nan edge from node 10 to node 3 with capacity 6,\nan edge from node 10 to node 5 with capacity 11,\nan edge from node 11 to node 9 with capacity 17,\nan edge from node 11 to node 12 with capacity 17,\nan edge from node 11 to node 7 with capacity 1,\nan edge from node 11 to node 4 with capacity 11,\nan edge from node 11 to node 8 with capacity 12,\nan edge from node 12 to node 4 with capacity 19,\nan edge from node 12 to node 0 with capacity 19.\nQ: What is the maximum flow from node 10 to node 9?\nA:", "answer": "The maximum flow from node 10 to node 9 is 17.", "difficulty": "hard", "doc_id": "46"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 6 with capacity 17,\nan edge from node 0 to node 14 with capacity 7,\nan edge from node 0 to node 15 with capacity 8,\nan edge from node 0 to node 5 with capacity 3,\nan edge from node 1 to node 6 with capacity 14,\nan edge from node 1 to node 14 with capacity 6,\nan edge from node 1 to node 4 with capacity 4,\nan edge from node 1 to node 0 with capacity 2,\nan edge from node 1 to node 3 with capacity 4,\nan edge from node 1 to node 2 with capacity 17,\nan edge from node 1 to node 9 with capacity 4,\nan edge from node 1 to node 12 with capacity 6,\nan edge from node 1 to node 5 with capacity 13,\nan edge from node 2 to node 6 with capacity 14,\nan edge from node 2 to node 0 with capacity 6,\nan edge from node 2 to node 7 with capacity 17,\nan edge from node 2 to node 8 with capacity 16,\nan edge from node 2 to node 5 with capacity 12,\nan edge from node 3 to node 14 with capacity 17,\nan edge from node 3 to node 10 with capacity 7,\nan edge from node 3 to node 16 with capacity 10,\nan edge from node 3 to node 0 with capacity 14,\nan edge from node 3 to node 9 with capacity 8,\nan edge from node 3 to node 11 with capacity 6,\nan edge from node 4 to node 6 with capacity 15,\nan edge from node 4 to node 10 with capacity 14,\nan edge from node 4 to node 16 with capacity 13,\nan edge from node 5 to node 6 with capacity 16,\nan edge from node 5 to node 16 with capacity 12,\nan edge from node 5 to node 2 with capacity 3,\nan edge from node 5 to node 13 with capacity 6,\nan edge from node 6 to node 7 with capacity 10,\nan edge from node 6 to node 12 with capacity 4,\nan edge from node 6 to node 5 with capacity 7,\nan edge from node 7 to node 6 with capacity 20,\nan edge from node 7 to node 1 with capacity 17,\nan edge from node 7 to node 2 with capacity 9,\nan edge from node 7 to node 11 with capacity 10,\nan edge from node 8 to node 14 with capacity 11,\nan edge from node 8 to node 16 with capacity 17,\nan edge from node 8 to node 7 with capacity 11,\nan edge from node 8 to node 11 with capacity 1,\nan edge from node 8 to node 15 with capacity 3,\nan edge from node 8 to node 5 with capacity 15,\nan edge from node 9 to node 1 with capacity 19,\nan edge from node 9 to node 11 with capacity 19,\nan edge from node 9 to node 15 with capacity 13,\nan edge from node 10 to node 9 with capacity 20,\nan edge from node 10 to node 13 with capacity 14,\nan edge from node 11 to node 3 with capacity 5,\nan edge from node 11 to node 2 with capacity 7,\nan edge from node 11 to node 9 with capacity 19,\nan edge from node 11 to node 12 with capacity 7,\nan edge from node 12 to node 6 with capacity 10,\nan edge from node 12 to node 16 with capacity 7,\nan edge from node 12 to node 3 with capacity 11,\nan edge from node 12 to node 13 with capacity 18,\nan edge from node 12 to node 8 with capacity 18,\nan edge from node 13 to node 10 with capacity 6,\nan edge from node 13 to node 3 with capacity 19,\nan edge from node 13 to node 12 with capacity 8,\nan edge from node 13 to node 5 with capacity 3,\nan edge from node 14 to node 7 with capacity 19,\nan edge from node 14 to node 12 with capacity 14,\nan edge from node 15 to node 6 with capacity 4,\nan edge from node 15 to node 14 with capacity 8,\nan edge from node 15 to node 10 with capacity 1,\nan edge from node 15 to node 3 with capacity 2,\nan edge from node 15 to node 9 with capacity 18,\nan edge from node 16 to node 14 with capacity 20,\nan edge from node 16 to node 10 with capacity 20,\nan edge from node 16 to node 4 with capacity 6,\nan edge from node 16 to node 1 with capacity 14,\nan edge from node 16 to node 2 with capacity 9,\nan edge from node 16 to node 12 with capacity 5,\nan edge from node 16 to node 15 with capacity 17,\nan edge from node 16 to node 5 with capacity 8.\nQ: What is the maximum flow from node 15 to node 4?\nA:", "answer": "The maximum flow from node 15 to node 4 is 10.", "difficulty": "hard", "doc_id": "47"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 7,\nan edge from node 0 to node 2 with capacity 4,\nan edge from node 2 to node 3 with capacity 4,\nan edge from node 2 to node 6 with capacity 9,\nan edge from node 3 to node 2 with capacity 6,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 5 to node 2 with capacity 3,\nan edge from node 5 to node 6 with capacity 3,\nan edge from node 6 to node 0 with capacity 9.\nQ: What is the maximum flow from node 5 to node 2?\nA:", "answer": "The maximum flow from node 5 to node 2 is 6.", "difficulty": "easy", "doc_id": "48"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 2 with capacity 13,\nan edge from node 0 to node 7 with capacity 17,\nan edge from node 1 to node 2 with capacity 11,\nan edge from node 1 to node 3 with capacity 4,\nan edge from node 1 to node 5 with capacity 14,\nan edge from node 1 to node 9 with capacity 17,\nan edge from node 2 to node 6 with capacity 8,\nan edge from node 4 to node 3 with capacity 15,\nan edge from node 5 to node 8 with capacity 6,\nan edge from node 5 to node 9 with capacity 11,\nan edge from node 5 to node 6 with capacity 2,\nan edge from node 6 to node 2 with capacity 12,\nan edge from node 6 to node 4 with capacity 12,\nan edge from node 6 to node 9 with capacity 14,\nan edge from node 7 to node 0 with capacity 14,\nan edge from node 7 to node 6 with capacity 16,\nan edge from node 7 to node 1 with capacity 15,\nan edge from node 8 to node 10 with capacity 18,\nan edge from node 8 to node 5 with capacity 15,\nan edge from node 8 to node 9 with capacity 10,\nan edge from node 9 to node 8 with capacity 11,\nan edge from node 9 to node 5 with capacity 19,\nan edge from node 9 to node 7 with capacity 8,\nan edge from node 10 to node 5 with capacity 18,\nan edge from node 10 to node 7 with capacity 19,\nan edge from node 10 to node 1 with capacity 10.\nQ: What is the maximum flow from node 6 to node 10?\nA:", "answer": "The maximum flow from node 6 to node 10 is 14.", "difficulty": "hard", "doc_id": "49"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 5 with capacity 16,\nan edge from node 0 to node 7 with capacity 18,\nan edge from node 0 to node 11 with capacity 7,\nan edge from node 0 to node 1 with capacity 19,\nan edge from node 0 to node 10 with capacity 19,\nan edge from node 0 to node 8 with capacity 5,\nan edge from node 0 to node 18 with capacity 8,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 1 to node 7 with capacity 4,\nan edge from node 1 to node 17 with capacity 15,\nan edge from node 1 to node 12 with capacity 14,\nan edge from node 1 to node 14 with capacity 7,\nan edge from node 1 to node 18 with capacity 11,\nan edge from node 2 to node 4 with capacity 4,\nan edge from node 2 to node 9 with capacity 3,\nan edge from node 2 to node 3 with capacity 5,\nan edge from node 3 to node 1 with capacity 18,\nan edge from node 3 to node 6 with capacity 15,\nan edge from node 3 to node 0 with capacity 11,\nan edge from node 4 to node 5 with capacity 9,\nan edge from node 4 to node 1 with capacity 16,\nan edge from node 4 to node 9 with capacity 3,\nan edge from node 4 to node 2 with capacity 17,\nan edge from node 4 to node 8 with capacity 10,\nan edge from node 4 to node 3 with capacity 1,\nan edge from node 5 to node 7 with capacity 4,\nan edge from node 5 to node 17 with capacity 9,\nan edge from node 5 to node 12 with capacity 2,\nan edge from node 5 to node 2 with capacity 5,\nan edge from node 5 to node 6 with capacity 13,\nan edge from node 5 to node 15 with capacity 10,\nan edge from node 5 to node 18 with capacity 14,\nan edge from node 5 to node 16 with capacity 16,\nan edge from node 6 to node 4 with capacity 5,\nan edge from node 6 to node 7 with capacity 10,\nan edge from node 6 to node 11 with capacity 18,\nan edge from node 6 to node 8 with capacity 9,\nan edge from node 6 to node 0 with capacity 13,\nan edge from node 6 to node 18 with capacity 8,\nan edge from node 6 to node 3 with capacity 2,\nan edge from node 7 to node 5 with capacity 6,\nan edge from node 7 to node 17 with capacity 6,\nan edge from node 7 to node 12 with capacity 18,\nan edge from node 7 to node 9 with capacity 11,\nan edge from node 7 to node 13 with capacity 15,\nan edge from node 7 to node 14 with capacity 13,\nan edge from node 7 to node 15 with capacity 18,\nan edge from node 7 to node 3 with capacity 19,\nan edge from node 8 to node 4 with capacity 17,\nan edge from node 8 to node 12 with capacity 14,\nan edge from node 8 to node 11 with capacity 19,\nan edge from node 8 to node 1 with capacity 15,\nan edge from node 8 to node 2 with capacity 11,\nan edge from node 8 to node 15 with capacity 7,\nan edge from node 8 to node 0 with capacity 17,\nan edge from node 9 to node 14 with capacity 2,\nan edge from node 9 to node 15 with capacity 4,\nan edge from node 9 to node 16 with capacity 10,\nan edge from node 9 to node 3 with capacity 2,\nan edge from node 10 to node 7 with capacity 20,\nan edge from node 10 to node 17 with capacity 17,\nan edge from node 10 to node 12 with capacity 11,\nan edge from node 10 to node 8 with capacity 1,\nan edge from node 10 to node 6 with capacity 8,\nan edge from node 10 to node 3 with capacity 2,\nan edge from node 11 to node 8 with capacity 19,\nan edge from node 11 to node 15 with capacity 2,\nan edge from node 12 to node 13 with capacity 9,\nan edge from node 12 to node 15 with capacity 8,\nan edge from node 12 to node 0 with capacity 16,\nan edge from node 13 to node 4 with capacity 3,\nan edge from node 13 to node 5 with capacity 20,\nan edge from node 13 to node 11 with capacity 10,\nan edge from node 13 to node 2 with capacity 14,\nan edge from node 13 to node 8 with capacity 20,\nan edge from node 13 to node 6 with capacity 16,\nan edge from node 13 to node 0 with capacity 10,\nan edge from node 13 to node 16 with capacity 16,\nan edge from node 14 to node 10 with capacity 8,\nan edge from node 14 to node 15 with capacity 3,\nan edge from node 14 to node 18 with capacity 6,\nan edge from node 15 to node 4 with capacity 6,\nan edge from node 15 to node 12 with capacity 3,\nan edge from node 15 to node 11 with capacity 9,\nan edge from node 15 to node 9 with capacity 18,\nan edge from node 15 to node 13 with capacity 20,\nan edge from node 16 to node 7 with capacity 5,\nan edge from node 16 to node 8 with capacity 6,\nan edge from node 16 to node 6 with capacity 8,\nan edge from node 16 to node 15 with capacity 16,\nan edge from node 16 to node 3 with capacity 19,\nan edge from node 17 to node 7 with capacity 13,\nan edge from node 17 to node 12 with capacity 9,\nan edge from node 17 to node 8 with capacity 11,\nan edge from node 17 to node 6 with capacity 15,\nan edge from node 17 to node 0 with capacity 5,\nan edge from node 18 to node 4 with capacity 14,\nan edge from node 18 to node 13 with capacity 20,\nan edge from node 18 to node 15 with capacity 14.\nQ: What is the maximum flow from node 9 to node 7?\nA:", "answer": "The maximum flow from node 9 to node 7 is 18.", "difficulty": "hard", "doc_id": "50"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 8 with capacity 8,\nan edge from node 1 to node 3 with capacity 9,\nan edge from node 1 to node 2 with capacity 3,\nan edge from node 2 to node 6 with capacity 5,\nan edge from node 2 to node 8 with capacity 8,\nan edge from node 3 to node 8 with capacity 7,\nan edge from node 4 to node 3 with capacity 3,\nan edge from node 5 to node 7 with capacity 2,\nan edge from node 5 to node 6 with capacity 1,\nan edge from node 5 to node 2 with capacity 8,\nan edge from node 6 to node 8 with capacity 8,\nan edge from node 7 to node 8 with capacity 2,\nan edge from node 7 to node 5 with capacity 5,\nan edge from node 8 to node 7 with capacity 10.\nQ: What is the maximum flow from node 5 to node 8?\nA:", "answer": "The maximum flow from node 5 to node 8 is 11.", "difficulty": "easy", "doc_id": "51"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 1 with capacity 14,\nan edge from node 0 to node 2 with capacity 4,\nan edge from node 1 to node 6 with capacity 20,\nan edge from node 1 to node 7 with capacity 14,\nan edge from node 1 to node 0 with capacity 13,\nan edge from node 1 to node 10 with capacity 13,\nan edge from node 2 to node 7 with capacity 8,\nan edge from node 2 to node 1 with capacity 7,\nan edge from node 2 to node 3 with capacity 5,\nan edge from node 2 to node 10 with capacity 20,\nan edge from node 3 to node 4 with capacity 9,\nan edge from node 3 to node 6 with capacity 14,\nan edge from node 4 to node 9 with capacity 11,\nan edge from node 4 to node 7 with capacity 10,\nan edge from node 4 to node 0 with capacity 13,\nan edge from node 4 to node 2 with capacity 17,\nan edge from node 5 to node 4 with capacity 3,\nan edge from node 5 to node 7 with capacity 4,\nan edge from node 5 to node 0 with capacity 14,\nan edge from node 6 to node 9 with capacity 2,\nan edge from node 6 to node 1 with capacity 4,\nan edge from node 6 to node 2 with capacity 18,\nan edge from node 6 to node 3 with capacity 15,\nan edge from node 6 to node 5 with capacity 7,\nan edge from node 7 to node 4 with capacity 12,\nan edge from node 7 to node 3 with capacity 2,\nan edge from node 7 to node 10 with capacity 14,\nan edge from node 7 to node 5 with capacity 1,\nan edge from node 8 to node 9 with capacity 16,\nan edge from node 8 to node 0 with capacity 15,\nan edge from node 8 to node 2 with capacity 19,\nan edge from node 8 to node 10 with capacity 15,\nan edge from node 9 to node 2 with capacity 13,\nan edge from node 9 to node 3 with capacity 12,\nan edge from node 10 to node 9 with capacity 14,\nan edge from node 10 to node 8 with capacity 15.\nQ: What is the maximum flow from node 5 to node 7?\nA:", "answer": "The maximum flow from node 5 to node 7 is 21.", "difficulty": "hard", "doc_id": "52"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 19 with capacity 2,\nan edge from node 0 to node 7 with capacity 1,\nan edge from node 0 to node 1 with capacity 3,\nan edge from node 0 to node 5 with capacity 15,\nan edge from node 0 to node 16 with capacity 17,\nan edge from node 1 to node 17 with capacity 5,\nan edge from node 1 to node 9 with capacity 18,\nan edge from node 2 to node 12 with capacity 6,\nan edge from node 2 to node 5 with capacity 20,\nan edge from node 2 to node 0 with capacity 11,\nan edge from node 2 to node 9 with capacity 3,\nan edge from node 2 to node 15 with capacity 1,\nan edge from node 3 to node 19 with capacity 16,\nan edge from node 3 to node 0 with capacity 12,\nan edge from node 3 to node 15 with capacity 7,\nan edge from node 4 to node 19 with capacity 19,\nan edge from node 4 to node 1 with capacity 3,\nan edge from node 4 to node 16 with capacity 20,\nan edge from node 4 to node 9 with capacity 10,\nan edge from node 4 to node 6 with capacity 6,\nan edge from node 4 to node 14 with capacity 20,\nan edge from node 5 to node 10 with capacity 17,\nan edge from node 5 to node 13 with capacity 8,\nan edge from node 5 to node 1 with capacity 14,\nan edge from node 5 to node 17 with capacity 18,\nan edge from node 5 to node 15 with capacity 6,\nan edge from node 5 to node 8 with capacity 12,\nan edge from node 6 to node 10 with capacity 11,\nan edge from node 6 to node 1 with capacity 14,\nan edge from node 6 to node 9 with capacity 15,\nan edge from node 6 to node 3 with capacity 18,\nan edge from node 6 to node 14 with capacity 4,\nan edge from node 7 to node 12 with capacity 10,\nan edge from node 7 to node 10 with capacity 9,\nan edge from node 7 to node 1 with capacity 17,\nan edge from node 7 to node 6 with capacity 2,\nan edge from node 8 to node 12 with capacity 4,\nan edge from node 8 to node 7 with capacity 20,\nan edge from node 8 to node 5 with capacity 16,\nan edge from node 8 to node 0 with capacity 11,\nan edge from node 8 to node 16 with capacity 14,\nan edge from node 8 to node 4 with capacity 17,\nan edge from node 8 to node 2 with capacity 3,\nan edge from node 9 to node 1 with capacity 20,\nan edge from node 9 to node 5 with capacity 19,\nan edge from node 9 to node 15 with capacity 8,\nan edge from node 9 to node 6 with capacity 10,\nan edge from node 10 to node 11 with capacity 14,\nan edge from node 10 to node 6 with capacity 11,\nan edge from node 11 to node 10 with capacity 13,\nan edge from node 11 to node 13 with capacity 16,\nan edge from node 11 to node 7 with capacity 16,\nan edge from node 11 to node 14 with capacity 2,\nan edge from node 12 to node 18 with capacity 4,\nan edge from node 12 to node 13 with capacity 20,\nan edge from node 12 to node 3 with capacity 18,\nan edge from node 13 to node 19 with capacity 8,\nan edge from node 13 to node 0 with capacity 14,\nan edge from node 13 to node 3 with capacity 4,\nan edge from node 14 to node 5 with capacity 17,\nan edge from node 14 to node 3 with capacity 17,\nan edge from node 15 to node 17 with capacity 2,\nan edge from node 15 to node 9 with capacity 12,\nan edge from node 15 to node 3 with capacity 17,\nan edge from node 15 to node 14 with capacity 19,\nan edge from node 16 to node 19 with capacity 20,\nan edge from node 16 to node 0 with capacity 5,\nan edge from node 16 to node 4 with capacity 16,\nan edge from node 16 to node 17 with capacity 9,\nan edge from node 16 to node 15 with capacity 9,\nan edge from node 16 to node 14 with capacity 10,\nan edge from node 17 to node 18 with capacity 4,\nan edge from node 17 to node 5 with capacity 8,\nan edge from node 17 to node 9 with capacity 11,\nan edge from node 17 to node 11 with capacity 3,\nan edge from node 17 to node 6 with capacity 11,\nan edge from node 18 to node 0 with capacity 19,\nan edge from node 18 to node 6 with capacity 17,\nan edge from node 19 to node 18 with capacity 9,\nan edge from node 19 to node 9 with capacity 15,\nan edge from node 19 to node 14 with capacity 20.\nQ: What is the maximum flow from node 19 to node 13?\nA:", "answer": "The maximum flow from node 19 to node 13 is 38.", "difficulty": "hard", "doc_id": "53"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 12 with capacity 20,\nan edge from node 0 to node 11 with capacity 11,\nan edge from node 1 to node 4 with capacity 16,\nan edge from node 1 to node 11 with capacity 7,\nan edge from node 2 to node 7 with capacity 14,\nan edge from node 2 to node 11 with capacity 6,\nan edge from node 3 to node 9 with capacity 20,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 11 with capacity 16,\nan edge from node 4 to node 8 with capacity 7,\nan edge from node 4 to node 5 with capacity 8,\nan edge from node 4 to node 0 with capacity 6,\nan edge from node 5 to node 8 with capacity 13,\nan edge from node 5 to node 9 with capacity 11,\nan edge from node 5 to node 3 with capacity 17,\nan edge from node 5 to node 4 with capacity 3,\nan edge from node 6 to node 10 with capacity 7,\nan edge from node 6 to node 7 with capacity 11,\nan edge from node 6 to node 2 with capacity 12,\nan edge from node 6 to node 5 with capacity 10,\nan edge from node 6 to node 13 with capacity 6,\nan edge from node 6 to node 4 with capacity 1,\nan edge from node 6 to node 0 with capacity 12,\nan edge from node 7 to node 6 with capacity 11,\nan edge from node 8 to node 10 with capacity 1,\nan edge from node 8 to node 13 with capacity 6,\nan edge from node 9 to node 8 with capacity 17,\nan edge from node 9 to node 1 with capacity 14,\nan edge from node 9 to node 13 with capacity 4,\nan edge from node 9 to node 0 with capacity 1,\nan edge from node 10 to node 12 with capacity 19,\nan edge from node 10 to node 1 with capacity 8,\nan edge from node 10 to node 2 with capacity 15,\nan edge from node 10 to node 0 with capacity 4,\nan edge from node 11 to node 6 with capacity 18,\nan edge from node 11 to node 8 with capacity 14,\nan edge from node 11 to node 9 with capacity 4,\nan edge from node 11 to node 10 with capacity 15,\nan edge from node 11 to node 1 with capacity 8,\nan edge from node 11 to node 2 with capacity 4,\nan edge from node 12 to node 6 with capacity 11,\nan edge from node 12 to node 8 with capacity 1,\nan edge from node 12 to node 7 with capacity 15,\nan edge from node 12 to node 13 with capacity 9,\nan edge from node 12 to node 4 with capacity 18,\nan edge from node 13 to node 10 with capacity 19,\nan edge from node 13 to node 5 with capacity 12,\nan edge from node 13 to node 11 with capacity 5,\nan edge from node 13 to node 0 with capacity 5.\nQ: What is the maximum flow from node 2 to node 13?\nA:", "answer": "The maximum flow from node 2 to node 13 is 17.", "difficulty": "hard", "doc_id": "54"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 17 with capacity 19,\nan edge from node 0 to node 11 with capacity 10,\nan edge from node 0 to node 7 with capacity 8,\nan edge from node 0 to node 10 with capacity 9,\nan edge from node 0 to node 9 with capacity 2,\nan edge from node 0 to node 1 with capacity 11,\nan edge from node 1 to node 11 with capacity 19,\nan edge from node 1 to node 10 with capacity 1,\nan edge from node 1 to node 16 with capacity 16,\nan edge from node 1 to node 6 with capacity 10,\nan edge from node 2 to node 17 with capacity 2,\nan edge from node 2 to node 15 with capacity 17,\nan edge from node 2 to node 19 with capacity 13,\nan edge from node 2 to node 7 with capacity 15,\nan edge from node 2 to node 4 with capacity 9,\nan edge from node 3 to node 17 with capacity 2,\nan edge from node 3 to node 5 with capacity 2,\nan edge from node 3 to node 9 with capacity 3,\nan edge from node 4 to node 0 with capacity 2,\nan edge from node 4 to node 17 with capacity 7,\nan edge from node 4 to node 7 with capacity 20,\nan edge from node 4 to node 10 with capacity 5,\nan edge from node 4 to node 2 with capacity 16,\nan edge from node 4 to node 1 with capacity 10,\nan edge from node 4 to node 16 with capacity 14,\nan edge from node 5 to node 3 with capacity 3,\nan edge from node 5 to node 17 with capacity 13,\nan edge from node 5 to node 14 with capacity 13,\nan edge from node 5 to node 11 with capacity 16,\nan edge from node 5 to node 8 with capacity 6,\nan edge from node 5 to node 1 with capacity 13,\nan edge from node 6 to node 0 with capacity 12,\nan edge from node 6 to node 17 with capacity 11,\nan edge from node 6 to node 16 with capacity 9,\nan edge from node 7 to node 10 with capacity 11,\nan edge from node 7 to node 5 with capacity 10,\nan edge from node 7 to node 2 with capacity 16,\nan edge from node 7 to node 1 with capacity 7,\nan edge from node 7 to node 16 with capacity 14,\nan edge from node 7 to node 6 with capacity 18,\nan edge from node 8 to node 17 with capacity 7,\nan edge from node 8 to node 7 with capacity 18,\nan edge from node 8 to node 5 with capacity 8,\nan edge from node 8 to node 16 with capacity 4,\nan edge from node 9 to node 15 with capacity 12,\nan edge from node 9 to node 11 with capacity 18,\nan edge from node 9 to node 13 with capacity 17,\nan edge from node 9 to node 4 with capacity 9,\nan edge from node 9 to node 12 with capacity 11,\nan edge from node 10 to node 14 with capacity 9,\nan edge from node 10 to node 9 with capacity 8,\nan edge from node 10 to node 2 with capacity 7,\nan edge from node 11 to node 15 with capacity 1,\nan edge from node 11 to node 12 with capacity 4,\nan edge from node 12 to node 14 with capacity 17,\nan edge from node 12 to node 7 with capacity 10,\nan edge from node 12 to node 8 with capacity 15,\nan edge from node 12 to node 13 with capacity 17,\nan edge from node 12 to node 4 with capacity 20,\nan edge from node 12 to node 5 with capacity 18,\nan edge from node 12 to node 9 with capacity 8,\nan edge from node 12 to node 2 with capacity 6,\nan edge from node 12 to node 1 with capacity 19,\nan edge from node 13 to node 3 with capacity 7,\nan edge from node 13 to node 18 with capacity 8,\nan edge from node 13 to node 4 with capacity 19,\nan edge from node 13 to node 9 with capacity 6,\nan edge from node 14 to node 13 with capacity 13,\nan edge from node 14 to node 5 with capacity 12,\nan edge from node 14 to node 1 with capacity 12,\nan edge from node 15 to node 3 with capacity 10,\nan edge from node 15 to node 7 with capacity 5,\nan edge from node 15 to node 8 with capacity 12,\nan edge from node 15 to node 10 with capacity 8,\nan edge from node 15 to node 5 with capacity 19,\nan edge from node 15 to node 6 with capacity 17,\nan edge from node 16 to node 17 with capacity 12,\nan edge from node 16 to node 11 with capacity 1,\nan edge from node 16 to node 8 with capacity 7,\nan edge from node 16 to node 12 with capacity 8,\nan edge from node 17 to node 15 with capacity 19,\nan edge from node 17 to node 11 with capacity 7,\nan edge from node 17 to node 8 with capacity 9,\nan edge from node 17 to node 12 with capacity 12,\nan edge from node 17 to node 2 with capacity 3,\nan edge from node 17 to node 6 with capacity 6,\nan edge from node 18 to node 19 with capacity 5,\nan edge from node 19 to node 0 with capacity 15,\nan edge from node 19 to node 17 with capacity 15,\nan edge from node 19 to node 15 with capacity 19,\nan edge from node 19 to node 18 with capacity 11,\nan edge from node 19 to node 11 with capacity 9,\nan edge from node 19 to node 9 with capacity 5,\nan edge from node 19 to node 2 with capacity 7,\nan edge from node 19 to node 1 with capacity 4.\nQ: What is the maximum flow from node 13 to node 16?\nA:", "answer": "The maximum flow from node 13 to node 16 is 37.", "difficulty": "hard", "doc_id": "55"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 4 with capacity 15,\nan edge from node 0 to node 13 with capacity 3,\nan edge from node 0 to node 7 with capacity 15,\nan edge from node 0 to node 1 with capacity 18,\nan edge from node 1 to node 14 with capacity 2,\nan edge from node 1 to node 11 with capacity 4,\nan edge from node 1 to node 9 with capacity 11,\nan edge from node 1 to node 6 with capacity 3,\nan edge from node 2 to node 4 with capacity 12,\nan edge from node 2 to node 14 with capacity 7,\nan edge from node 2 to node 10 with capacity 18,\nan edge from node 2 to node 6 with capacity 20,\nan edge from node 2 to node 8 with capacity 20,\nan edge from node 3 to node 2 with capacity 2,\nan edge from node 3 to node 1 with capacity 5,\nan edge from node 4 to node 14 with capacity 2,\nan edge from node 4 to node 11 with capacity 20,\nan edge from node 4 to node 1 with capacity 12,\nan edge from node 5 to node 4 with capacity 18,\nan edge from node 5 to node 14 with capacity 6,\nan edge from node 5 to node 10 with capacity 10,\nan edge from node 5 to node 8 with capacity 1,\nan edge from node 6 to node 4 with capacity 8,\nan edge from node 6 to node 11 with capacity 18,\nan edge from node 6 to node 12 with capacity 13,\nan edge from node 6 to node 10 with capacity 16,\nan edge from node 7 to node 4 with capacity 19,\nan edge from node 7 to node 3 with capacity 8,\nan edge from node 7 to node 12 with capacity 14,\nan edge from node 7 to node 1 with capacity 9,\nan edge from node 8 to node 9 with capacity 8,\nan edge from node 8 to node 0 with capacity 20,\nan edge from node 8 to node 12 with capacity 6,\nan edge from node 8 to node 7 with capacity 6,\nan edge from node 8 to node 6 with capacity 19,\nan edge from node 8 to node 2 with capacity 18,\nan edge from node 9 to node 4 with capacity 10,\nan edge from node 9 to node 14 with capacity 4,\nan edge from node 9 to node 5 with capacity 3,\nan edge from node 9 to node 13 with capacity 1,\nan edge from node 9 to node 7 with capacity 14,\nan edge from node 9 to node 10 with capacity 12,\nan edge from node 10 to node 4 with capacity 4,\nan edge from node 10 to node 2 with capacity 11,\nan edge from node 11 to node 14 with capacity 8,\nan edge from node 12 to node 4 with capacity 17,\nan edge from node 12 to node 5 with capacity 3,\nan edge from node 12 to node 9 with capacity 7,\nan edge from node 12 to node 13 with capacity 9,\nan edge from node 13 to node 12 with capacity 19,\nan edge from node 13 to node 10 with capacity 10,\nan edge from node 13 to node 8 with capacity 16,\nan edge from node 14 to node 11 with capacity 5,\nan edge from node 14 to node 12 with capacity 4,\nan edge from node 14 to node 8 with capacity 8.\nQ: What is the maximum flow from node 3 to node 14?\nA:", "answer": "The maximum flow from node 3 to node 14 is 7.", "difficulty": "hard", "doc_id": "56"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 5 with capacity 2,\nan edge from node 0 to node 4 with capacity 4,\nan edge from node 1 to node 0 with capacity 8,\nan edge from node 2 to node 3 with capacity 3,\nan edge from node 3 to node 4 with capacity 8,\nan edge from node 4 to node 2 with capacity 5,\nan edge from node 4 to node 0 with capacity 6,\nan edge from node 4 to node 3 with capacity 8,\nan edge from node 5 to node 4 with capacity 5,\nan edge from node 5 to node 3 with capacity 8,\nan edge from node 6 to node 1 with capacity 10.\nQ: What is the maximum flow from node 0 to node 3?\nA:", "answer": "The maximum flow from node 0 to node 3 is 6.", "difficulty": "easy", "doc_id": "57"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 4,\nan edge from node 1 to node 0 with capacity 9,\nan edge from node 2 to node 0 with capacity 3,\nan edge from node 3 to node 0 with capacity 1,\nan edge from node 4 to node 0 with capacity 8,\nan edge from node 4 to node 2 with capacity 8,\nan edge from node 4 to node 3 with capacity 5.\nQ: What is the maximum flow from node 4 to node 1?\nA:", "answer": "The maximum flow from node 4 to node 1 is 4.", "difficulty": "easy", "doc_id": "58"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 3 with capacity 7,\nan edge from node 0 to node 2 with capacity 9,\nan edge from node 1 to node 7 with capacity 10,\nan edge from node 1 to node 6 with capacity 8,\nan edge from node 1 to node 3 with capacity 4,\nan edge from node 2 to node 1 with capacity 3,\nan edge from node 2 to node 5 with capacity 9,\nan edge from node 3 to node 5 with capacity 2,\nan edge from node 3 to node 6 with capacity 4,\nan edge from node 4 to node 8 with capacity 4,\nan edge from node 4 to node 0 with capacity 9,\nan edge from node 5 to node 6 with capacity 1,\nan edge from node 7 to node 1 with capacity 1,\nan edge from node 7 to node 8 with capacity 8,\nan edge from node 7 to node 0 with capacity 6,\nan edge from node 7 to node 2 with capacity 9,\nan edge from node 8 to node 3 with capacity 9,\nan edge from node 8 to node 2 with capacity 8.\nQ: What is the maximum flow from node 0 to node 6?\nA:", "answer": "The maximum flow from node 0 to node 6 is 8.", "difficulty": "easy", "doc_id": "59"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 10 with capacity 5,\nan edge from node 0 to node 7 with capacity 13,\nan edge from node 0 to node 4 with capacity 20,\nan edge from node 0 to node 3 with capacity 2,\nan edge from node 0 to node 9 with capacity 18,\nan edge from node 1 to node 5 with capacity 8,\nan edge from node 1 to node 4 with capacity 19,\nan edge from node 1 to node 13 with capacity 1,\nan edge from node 1 to node 9 with capacity 2,\nan edge from node 2 to node 8 with capacity 7,\nan edge from node 2 to node 9 with capacity 18,\nan edge from node 3 to node 0 with capacity 16,\nan edge from node 3 to node 2 with capacity 14,\nan edge from node 3 to node 7 with capacity 12,\nan edge from node 4 to node 0 with capacity 16,\nan edge from node 4 to node 12 with capacity 15,\nan edge from node 4 to node 15 with capacity 17,\nan edge from node 4 to node 3 with capacity 2,\nan edge from node 4 to node 6 with capacity 3,\nan edge from node 5 to node 0 with capacity 9,\nan edge from node 5 to node 2 with capacity 2,\nan edge from node 5 to node 7 with capacity 7,\nan edge from node 5 to node 11 with capacity 12,\nan edge from node 6 to node 11 with capacity 10,\nan edge from node 6 to node 13 with capacity 6,\nan edge from node 7 to node 10 with capacity 1,\nan edge from node 7 to node 12 with capacity 10,\nan edge from node 7 to node 8 with capacity 14,\nan edge from node 7 to node 13 with capacity 7,\nan edge from node 7 to node 3 with capacity 19,\nan edge from node 8 to node 2 with capacity 2,\nan edge from node 8 to node 10 with capacity 8,\nan edge from node 8 to node 3 with capacity 13,\nan edge from node 9 to node 8 with capacity 16,\nan edge from node 9 to node 6 with capacity 13,\nan edge from node 10 to node 2 with capacity 15,\nan edge from node 10 to node 7 with capacity 2,\nan edge from node 10 to node 12 with capacity 9,\nan edge from node 10 to node 8 with capacity 2,\nan edge from node 10 to node 5 with capacity 9,\nan edge from node 10 to node 3 with capacity 11,\nan edge from node 10 to node 9 with capacity 6,\nan edge from node 10 to node 6 with capacity 16,\nan edge from node 11 to node 0 with capacity 17,\nan edge from node 11 to node 14 with capacity 17,\nan edge from node 11 to node 4 with capacity 20,\nan edge from node 12 to node 7 with capacity 11,\nan edge from node 12 to node 8 with capacity 20,\nan edge from node 12 to node 9 with capacity 13,\nan edge from node 12 to node 6 with capacity 4,\nan edge from node 13 to node 2 with capacity 19,\nan edge from node 13 to node 11 with capacity 3,\nan edge from node 14 to node 10 with capacity 11,\nan edge from node 14 to node 7 with capacity 2,\nan edge from node 14 to node 11 with capacity 9,\nan edge from node 14 to node 4 with capacity 17,\nan edge from node 14 to node 3 with capacity 20,\nan edge from node 14 to node 6 with capacity 7,\nan edge from node 15 to node 7 with capacity 6,\nan edge from node 15 to node 12 with capacity 20,\nan edge from node 15 to node 1 with capacity 7,\nan edge from node 15 to node 4 with capacity 18.\nQ: What is the maximum flow from node 9 to node 1?\nA:", "answer": "The maximum flow from node 9 to node 1 is 7.", "difficulty": "hard", "doc_id": "60"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 2 with capacity 5,\nan edge from node 0 to node 1 with capacity 6,\nan edge from node 0 to node 6 with capacity 6,\nan edge from node 1 to node 8 with capacity 10,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 2 to node 4 with capacity 7,\nan edge from node 3 to node 1 with capacity 1,\nan edge from node 3 to node 7 with capacity 2,\nan edge from node 3 to node 0 with capacity 10,\nan edge from node 4 to node 1 with capacity 8,\nan edge from node 5 to node 3 with capacity 7,\nan edge from node 5 to node 7 with capacity 1,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 6 to node 3 with capacity 10,\nan edge from node 6 to node 4 with capacity 9,\nan edge from node 7 to node 5 with capacity 5,\nan edge from node 7 to node 8 with capacity 6,\nan edge from node 7 to node 4 with capacity 3,\nan edge from node 8 to node 1 with capacity 3.\nQ: What is the maximum flow from node 7 to node 8?\nA:", "answer": "The maximum flow from node 7 to node 8 is 14.", "difficulty": "easy", "doc_id": "61"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 1 with capacity 15,\nan edge from node 0 to node 13 with capacity 14,\nan edge from node 0 to node 17 with capacity 7,\nan edge from node 0 to node 2 with capacity 17,\nan edge from node 0 to node 8 with capacity 4,\nan edge from node 0 to node 9 with capacity 5,\nan edge from node 0 to node 11 with capacity 11,\nan edge from node 0 to node 5 with capacity 4,\nan edge from node 1 to node 14 with capacity 16,\nan edge from node 1 to node 13 with capacity 10,\nan edge from node 1 to node 9 with capacity 2,\nan edge from node 1 to node 18 with capacity 4,\nan edge from node 2 to node 3 with capacity 3,\nan edge from node 2 to node 0 with capacity 9,\nan edge from node 2 to node 14 with capacity 6,\nan edge from node 2 to node 8 with capacity 8,\nan edge from node 2 to node 12 with capacity 18,\nan edge from node 2 to node 11 with capacity 5,\nan edge from node 3 to node 8 with capacity 17,\nan edge from node 3 to node 11 with capacity 19,\nan edge from node 4 to node 3 with capacity 17,\nan edge from node 4 to node 14 with capacity 11,\nan edge from node 4 to node 13 with capacity 15,\nan edge from node 4 to node 18 with capacity 5,\nan edge from node 4 to node 5 with capacity 20,\nan edge from node 5 to node 6 with capacity 18,\nan edge from node 5 to node 9 with capacity 14,\nan edge from node 6 to node 3 with capacity 14,\nan edge from node 6 to node 0 with capacity 4,\nan edge from node 6 to node 1 with capacity 5,\nan edge from node 6 to node 10 with capacity 19,\nan edge from node 6 to node 2 with capacity 15,\nan edge from node 6 to node 16 with capacity 12,\nan edge from node 6 to node 7 with capacity 12,\nan edge from node 6 to node 18 with capacity 18,\nan edge from node 6 to node 4 with capacity 20,\nan edge from node 7 to node 3 with capacity 8,\nan edge from node 7 to node 14 with capacity 14,\nan edge from node 7 to node 1 with capacity 16,\nan edge from node 7 to node 10 with capacity 9,\nan edge from node 7 to node 6 with capacity 10,\nan edge from node 7 to node 9 with capacity 18,\nan edge from node 7 to node 18 with capacity 13,\nan edge from node 7 to node 4 with capacity 13,\nan edge from node 8 to node 0 with capacity 16,\nan edge from node 8 to node 14 with capacity 13,\nan edge from node 8 to node 1 with capacity 3,\nan edge from node 8 to node 6 with capacity 15,\nan edge from node 8 to node 4 with capacity 9,\nan edge from node 9 to node 0 with capacity 10,\nan edge from node 9 to node 17 with capacity 6,\nan edge from node 9 to node 6 with capacity 10,\nan edge from node 9 to node 11 with capacity 6,\nan edge from node 10 to node 1 with capacity 3,\nan edge from node 10 to node 17 with capacity 3,\nan edge from node 10 to node 18 with capacity 2,\nan edge from node 11 to node 0 with capacity 15,\nan edge from node 11 to node 14 with capacity 5,\nan edge from node 11 to node 16 with capacity 4,\nan edge from node 11 to node 6 with capacity 8,\nan edge from node 11 to node 9 with capacity 12,\nan edge from node 11 to node 18 with capacity 6,\nan edge from node 11 to node 5 with capacity 8,\nan edge from node 12 to node 1 with capacity 4,\nan edge from node 12 to node 17 with capacity 4,\nan edge from node 12 to node 8 with capacity 18,\nan edge from node 12 to node 15 with capacity 11,\nan edge from node 13 to node 14 with capacity 3,\nan edge from node 13 to node 1 with capacity 18,\nan edge from node 13 to node 6 with capacity 10,\nan edge from node 14 to node 13 with capacity 20,\nan edge from node 14 to node 2 with capacity 18,\nan edge from node 14 to node 7 with capacity 14,\nan edge from node 14 to node 6 with capacity 19,\nan edge from node 14 to node 15 with capacity 14,\nan edge from node 14 to node 18 with capacity 10,\nan edge from node 15 to node 13 with capacity 11,\nan edge from node 15 to node 10 with capacity 10,\nan edge from node 15 to node 8 with capacity 11,\nan edge from node 15 to node 18 with capacity 19,\nan edge from node 15 to node 4 with capacity 5,\nan edge from node 16 to node 0 with capacity 8,\nan edge from node 16 to node 18 with capacity 3,\nan edge from node 16 to node 5 with capacity 5,\nan edge from node 17 to node 3 with capacity 10,\nan edge from node 17 to node 13 with capacity 13,\nan edge from node 17 to node 16 with capacity 18,\nan edge from node 17 to node 12 with capacity 15,\nan edge from node 17 to node 11 with capacity 17,\nan edge from node 17 to node 4 with capacity 3,\nan edge from node 17 to node 5 with capacity 4,\nan edge from node 18 to node 3 with capacity 14,\nan edge from node 18 to node 13 with capacity 7,\nan edge from node 18 to node 17 with capacity 16,\nan edge from node 18 to node 12 with capacity 1,\nan edge from node 18 to node 15 with capacity 3.\nQ: What is the maximum flow from node 1 to node 12?\nA:", "answer": "The maximum flow from node 1 to node 12 is 32.", "difficulty": "hard", "doc_id": "62"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 6 with capacity 14,\nan edge from node 0 to node 4 with capacity 4,\nan edge from node 0 to node 1 with capacity 14,\nan edge from node 0 to node 12 with capacity 12,\nan edge from node 1 to node 7 with capacity 10,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 9 with capacity 8,\nan edge from node 2 to node 0 with capacity 5,\nan edge from node 2 to node 4 with capacity 13,\nan edge from node 3 to node 13 with capacity 5,\nan edge from node 4 to node 12 with capacity 17,\nan edge from node 4 to node 11 with capacity 1,\nan edge from node 5 to node 7 with capacity 5,\nan edge from node 5 to node 6 with capacity 4,\nan edge from node 5 to node 3 with capacity 13,\nan edge from node 6 to node 2 with capacity 9,\nan edge from node 6 to node 1 with capacity 6,\nan edge from node 6 to node 13 with capacity 17,\nan edge from node 7 to node 8 with capacity 15,\nan edge from node 7 to node 6 with capacity 15,\nan edge from node 7 to node 1 with capacity 12,\nan edge from node 8 to node 4 with capacity 9,\nan edge from node 9 to node 5 with capacity 16,\nan edge from node 10 to node 6 with capacity 12,\nan edge from node 10 to node 1 with capacity 2,\nan edge from node 10 to node 13 with capacity 11,\nan edge from node 10 to node 11 with capacity 5,\nan edge from node 11 to node 5 with capacity 19,\nan edge from node 11 to node 12 with capacity 15,\nan edge from node 12 to node 7 with capacity 12,\nan edge from node 12 to node 4 with capacity 13,\nan edge from node 12 to node 3 with capacity 19,\nan edge from node 13 to node 10 with capacity 13,\nan edge from node 13 to node 7 with capacity 8,\nan edge from node 13 to node 8 with capacity 1,\nan edge from node 13 to node 5 with capacity 9,\nan edge from node 13 to node 9 with capacity 2,\nan edge from node 13 to node 6 with capacity 19.\nQ: What is the maximum flow from node 3 to node 0?\nA:", "answer": "The maximum flow from node 3 to node 0 is 5.", "difficulty": "hard", "doc_id": "63"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 7 with capacity 17,\nan edge from node 0 to node 13 with capacity 9,\nan edge from node 1 to node 9 with capacity 3,\nan edge from node 2 to node 4 with capacity 8,\nan edge from node 2 to node 15 with capacity 20,\nan edge from node 2 to node 5 with capacity 13,\nan edge from node 2 to node 10 with capacity 9,\nan edge from node 2 to node 0 with capacity 5,\nan edge from node 3 to node 15 with capacity 10,\nan edge from node 4 to node 7 with capacity 6,\nan edge from node 4 to node 15 with capacity 11,\nan edge from node 4 to node 1 with capacity 3,\nan edge from node 4 to node 5 with capacity 11,\nan edge from node 4 to node 17 with capacity 4,\nan edge from node 4 to node 12 with capacity 4,\nan edge from node 4 to node 2 with capacity 2,\nan edge from node 5 to node 1 with capacity 19,\nan edge from node 5 to node 17 with capacity 2,\nan edge from node 5 to node 3 with capacity 7,\nan edge from node 5 to node 12 with capacity 8,\nan edge from node 5 to node 11 with capacity 10,\nan edge from node 6 to node 8 with capacity 1,\nan edge from node 6 to node 11 with capacity 12,\nan edge from node 6 to node 16 with capacity 3,\nan edge from node 6 to node 14 with capacity 19,\nan edge from node 7 to node 1 with capacity 16,\nan edge from node 7 to node 6 with capacity 3,\nan edge from node 8 to node 15 with capacity 18,\nan edge from node 9 to node 15 with capacity 12,\nan edge from node 9 to node 3 with capacity 2,\nan edge from node 9 to node 8 with capacity 16,\nan edge from node 9 to node 10 with capacity 1,\nan edge from node 9 to node 16 with capacity 14,\nan edge from node 10 to node 7 with capacity 17,\nan edge from node 10 to node 13 with capacity 15,\nan edge from node 10 to node 6 with capacity 18,\nan edge from node 11 to node 4 with capacity 17,\nan edge from node 11 to node 5 with capacity 18,\nan edge from node 11 to node 8 with capacity 10,\nan edge from node 11 to node 13 with capacity 12,\nan edge from node 11 to node 0 with capacity 7,\nan edge from node 12 to node 4 with capacity 2,\nan edge from node 12 to node 15 with capacity 2,\nan edge from node 12 to node 13 with capacity 2,\nan edge from node 13 to node 7 with capacity 10,\nan edge from node 13 to node 3 with capacity 18,\nan edge from node 13 to node 12 with capacity 5,\nan edge from node 13 to node 16 with capacity 6,\nan edge from node 14 to node 7 with capacity 8,\nan edge from node 14 to node 4 with capacity 12,\nan edge from node 14 to node 17 with capacity 17,\nan edge from node 14 to node 3 with capacity 1,\nan edge from node 14 to node 0 with capacity 5,\nan edge from node 15 to node 12 with capacity 4,\nan edge from node 15 to node 6 with capacity 1,\nan edge from node 16 to node 15 with capacity 16,\nan edge from node 16 to node 5 with capacity 20,\nan edge from node 17 to node 7 with capacity 19,\nan edge from node 17 to node 4 with capacity 7,\nan edge from node 17 to node 15 with capacity 12,\nan edge from node 17 to node 0 with capacity 1.\nQ: What is the maximum flow from node 13 to node 10?\nA:", "answer": "The maximum flow from node 13 to node 10 is 3.", "difficulty": "hard", "doc_id": "64"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 6 with capacity 14,\nan edge from node 0 to node 10 with capacity 6,\nan edge from node 0 to node 7 with capacity 1,\nan edge from node 0 to node 8 with capacity 9,\nan edge from node 1 to node 7 with capacity 17,\nan edge from node 1 to node 8 with capacity 7,\nan edge from node 3 to node 4 with capacity 15,\nan edge from node 3 to node 9 with capacity 6,\nan edge from node 3 to node 5 with capacity 14,\nan edge from node 4 to node 1 with capacity 13,\nan edge from node 4 to node 7 with capacity 7,\nan edge from node 5 to node 1 with capacity 7,\nan edge from node 5 to node 7 with capacity 4,\nan edge from node 6 to node 8 with capacity 17,\nan edge from node 7 to node 4 with capacity 1,\nan edge from node 7 to node 3 with capacity 6,\nan edge from node 8 to node 6 with capacity 5,\nan edge from node 8 to node 9 with capacity 9,\nan edge from node 8 to node 5 with capacity 3,\nan edge from node 8 to node 0 with capacity 17,\nan edge from node 8 to node 2 with capacity 2,\nan edge from node 9 to node 1 with capacity 12,\nan edge from node 9 to node 10 with capacity 12,\nan edge from node 9 to node 7 with capacity 20,\nan edge from node 10 to node 4 with capacity 17,\nan edge from node 10 to node 9 with capacity 5,\nan edge from node 10 to node 5 with capacity 13,\nan edge from node 10 to node 0 with capacity 7.\nQ: What is the maximum flow from node 4 to node 3?\nA:", "answer": "The maximum flow from node 4 to node 3 is 6.", "difficulty": "hard", "doc_id": "65"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 9 with capacity 17,\nan edge from node 0 to node 8 with capacity 17,\nan edge from node 0 to node 17 with capacity 6,\nan edge from node 0 to node 3 with capacity 2,\nan edge from node 0 to node 4 with capacity 19,\nan edge from node 0 to node 10 with capacity 2,\nan edge from node 1 to node 9 with capacity 7,\nan edge from node 1 to node 16 with capacity 5,\nan edge from node 1 to node 4 with capacity 18,\nan edge from node 1 to node 13 with capacity 2,\nan edge from node 1 to node 6 with capacity 5,\nan edge from node 1 to node 15 with capacity 12,\nan edge from node 2 to node 1 with capacity 12,\nan edge from node 2 to node 17 with capacity 11,\nan edge from node 2 to node 4 with capacity 18,\nan edge from node 2 to node 13 with capacity 15,\nan edge from node 2 to node 5 with capacity 12,\nan edge from node 3 to node 4 with capacity 3,\nan edge from node 3 to node 0 with capacity 6,\nan edge from node 3 to node 6 with capacity 2,\nan edge from node 3 to node 15 with capacity 1,\nan edge from node 3 to node 10 with capacity 4,\nan edge from node 4 to node 17 with capacity 20,\nan edge from node 4 to node 13 with capacity 13,\nan edge from node 4 to node 7 with capacity 12,\nan edge from node 4 to node 12 with capacity 12,\nan edge from node 5 to node 11 with capacity 10,\nan edge from node 6 to node 9 with capacity 19,\nan edge from node 6 to node 2 with capacity 8,\nan edge from node 6 to node 5 with capacity 17,\nan edge from node 6 to node 12 with capacity 10,\nan edge from node 7 to node 9 with capacity 12,\nan edge from node 7 to node 16 with capacity 16,\nan edge from node 7 to node 4 with capacity 9,\nan edge from node 7 to node 2 with capacity 11,\nan edge from node 7 to node 11 with capacity 14,\nan edge from node 8 to node 9 with capacity 15,\nan edge from node 8 to node 1 with capacity 8,\nan edge from node 8 to node 3 with capacity 3,\nan edge from node 8 to node 4 with capacity 7,\nan edge from node 8 to node 11 with capacity 15,\nan edge from node 9 to node 13 with capacity 12,\nan edge from node 9 to node 2 with capacity 3,\nan edge from node 9 to node 5 with capacity 2,\nan edge from node 9 to node 10 with capacity 1,\nan edge from node 10 to node 9 with capacity 6,\nan edge from node 10 to node 1 with capacity 20,\nan edge from node 10 to node 3 with capacity 7,\nan edge from node 10 to node 4 with capacity 9,\nan edge from node 10 to node 6 with capacity 17,\nan edge from node 10 to node 11 with capacity 3,\nan edge from node 10 to node 7 with capacity 17,\nan edge from node 11 to node 9 with capacity 14,\nan edge from node 11 to node 3 with capacity 18,\nan edge from node 11 to node 16 with capacity 11,\nan edge from node 11 to node 4 with capacity 16,\nan edge from node 11 to node 0 with capacity 6,\nan edge from node 12 to node 3 with capacity 5,\nan edge from node 12 to node 16 with capacity 11,\nan edge from node 12 to node 6 with capacity 3,\nan edge from node 13 to node 14 with capacity 19,\nan edge from node 13 to node 1 with capacity 18,\nan edge from node 13 to node 3 with capacity 17,\nan edge from node 13 to node 6 with capacity 6,\nan edge from node 14 to node 4 with capacity 3,\nan edge from node 14 to node 2 with capacity 19,\nan edge from node 14 to node 10 with capacity 11,\nan edge from node 15 to node 14 with capacity 8,\nan edge from node 15 to node 9 with capacity 19,\nan edge from node 15 to node 8 with capacity 17,\nan edge from node 15 to node 5 with capacity 6,\nan edge from node 15 to node 11 with capacity 9,\nan edge from node 15 to node 7 with capacity 9,\nan edge from node 16 to node 9 with capacity 7,\nan edge from node 16 to node 0 with capacity 20,\nan edge from node 16 to node 5 with capacity 14,\nan edge from node 16 to node 12 with capacity 8,\nan edge from node 16 to node 10 with capacity 12,\nan edge from node 17 to node 9 with capacity 14,\nan edge from node 17 to node 15 with capacity 18,\nan edge from node 17 to node 2 with capacity 20,\nan edge from node 17 to node 7 with capacity 18,\nan edge from node 17 to node 10 with capacity 15.\nQ: What is the maximum flow from node 14 to node 4?\nA:", "answer": "The maximum flow from node 14 to node 4 is 33.", "difficulty": "hard", "doc_id": "66"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 5 with capacity 4,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 0 to node 8 with capacity 3,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 3 to node 5 with capacity 4,\nan edge from node 3 to node 0 with capacity 3,\nan edge from node 3 to node 8 with capacity 7,\nan edge from node 5 to node 6 with capacity 6,\nan edge from node 8 to node 5 with capacity 1,\nan edge from node 8 to node 3 with capacity 3,\nan edge from node 8 to node 7 with capacity 2.\nQ: What is the maximum flow from node 8 to node 6?\nA:", "answer": "The maximum flow from node 8 to node 6 is 4.", "difficulty": "easy", "doc_id": "67"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 4 with capacity 5,\nan edge from node 0 to node 3 with capacity 2,\nan edge from node 1 to node 9 with capacity 3,\nan edge from node 1 to node 2 with capacity 7,\nan edge from node 1 to node 7 with capacity 1,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 8 with capacity 8,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 3 to node 7 with capacity 2,\nan edge from node 5 to node 9 with capacity 1,\nan edge from node 5 to node 6 with capacity 4,\nan edge from node 6 to node 0 with capacity 8,\nan edge from node 7 to node 4 with capacity 7,\nan edge from node 7 to node 5 with capacity 1,\nan edge from node 7 to node 2 with capacity 9,\nan edge from node 8 to node 1 with capacity 2,\nan edge from node 8 to node 2 with capacity 2,\nan edge from node 8 to node 6 with capacity 1,\nan edge from node 9 to node 5 with capacity 6,\nan edge from node 9 to node 8 with capacity 8.\nQ: What is the maximum flow from node 8 to node 7?\nA:", "answer": "The maximum flow from node 8 to node 7 is 3.", "difficulty": "easy", "doc_id": "68"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 5 with capacity 10,\nan edge from node 0 to node 1 with capacity 8,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 2 to node 0 with capacity 9,\nan edge from node 2 to node 3 with capacity 3,\nan edge from node 3 to node 6 with capacity 8,\nan edge from node 4 to node 1 with capacity 9,\nan edge from node 5 to node 2 with capacity 4,\nan edge from node 5 to node 3 with capacity 6,\nan edge from node 6 to node 0 with capacity 3,\nan edge from node 8 to node 5 with capacity 2,\nan edge from node 8 to node 1 with capacity 1.\nQ: What is the maximum flow from node 0 to node 6?\nA:", "answer": "The maximum flow from node 0 to node 6 is 8.", "difficulty": "easy", "doc_id": "69"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 5 with capacity 5,\nan edge from node 0 to node 2 with capacity 1,\nan edge from node 1 to node 3 with capacity 5,\nan edge from node 2 to node 3 with capacity 10,\nan edge from node 3 to node 1 with capacity 9,\nan edge from node 4 to node 3 with capacity 4,\nan edge from node 4 to node 1 with capacity 1,\nan edge from node 4 to node 2 with capacity 4,\nan edge from node 5 to node 2 with capacity 1,\nan edge from node 5 to node 0 with capacity 8.\nQ: What is the maximum flow from node 4 to node 1?\nA:", "answer": "The maximum flow from node 4 to node 1 is 9.", "difficulty": "easy", "doc_id": "70"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 1 with capacity 3,\nan edge from node 0 to node 5 with capacity 1,\nan edge from node 1 to node 5 with capacity 9,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 2 to node 3 with capacity 10,\nan edge from node 2 to node 0 with capacity 5,\nan edge from node 3 to node 0 with capacity 2,\nan edge from node 4 to node 3 with capacity 8,\nan edge from node 4 to node 5 with capacity 8,\nan edge from node 4 to node 6 with capacity 1,\nan edge from node 5 to node 0 with capacity 5,\nan edge from node 6 to node 4 with capacity 9,\nan edge from node 6 to node 5 with capacity 8,\nan edge from node 6 to node 0 with capacity 6.\nQ: What is the maximum flow from node 6 to node 1?\nA:", "answer": "The maximum flow from node 6 to node 1 is 3.", "difficulty": "easy", "doc_id": "71"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 14 with capacity 11,\nan edge from node 0 to node 8 with capacity 18,\nan edge from node 0 to node 10 with capacity 1,\nan edge from node 0 to node 18 with capacity 18,\nan edge from node 0 to node 2 with capacity 19,\nan edge from node 1 to node 11 with capacity 7,\nan edge from node 1 to node 14 with capacity 16,\nan edge from node 1 to node 17 with capacity 14,\nan edge from node 1 to node 9 with capacity 16,\nan edge from node 1 to node 0 with capacity 5,\nan edge from node 1 to node 3 with capacity 4,\nan edge from node 1 to node 15 with capacity 5,\nan edge from node 1 to node 13 with capacity 14,\nan edge from node 2 to node 5 with capacity 17,\nan edge from node 2 to node 14 with capacity 18,\nan edge from node 2 to node 17 with capacity 19,\nan edge from node 2 to node 15 with capacity 18,\nan edge from node 2 to node 18 with capacity 8,\nan edge from node 2 to node 19 with capacity 17,\nan edge from node 2 to node 4 with capacity 4,\nan edge from node 2 to node 1 with capacity 15,\nan edge from node 2 to node 13 with capacity 19,\nan edge from node 3 to node 5 with capacity 1,\nan edge from node 3 to node 7 with capacity 10,\nan edge from node 4 to node 5 with capacity 17,\nan edge from node 4 to node 17 with capacity 13,\nan edge from node 5 to node 12 with capacity 18,\nan edge from node 5 to node 16 with capacity 19,\nan edge from node 5 to node 0 with capacity 10,\nan edge from node 5 to node 8 with capacity 1,\nan edge from node 5 to node 19 with capacity 18,\nan edge from node 5 to node 1 with capacity 12,\nan edge from node 6 to node 11 with capacity 7,\nan edge from node 6 to node 0 with capacity 8,\nan edge from node 6 to node 15 with capacity 6,\nan edge from node 6 to node 18 with capacity 3,\nan edge from node 6 to node 2 with capacity 8,\nan edge from node 6 to node 7 with capacity 17,\nan edge from node 7 to node 5 with capacity 2,\nan edge from node 7 to node 16 with capacity 14,\nan edge from node 7 to node 17 with capacity 7,\nan edge from node 7 to node 9 with capacity 5,\nan edge from node 7 to node 0 with capacity 9,\nan edge from node 7 to node 3 with capacity 16,\nan edge from node 7 to node 15 with capacity 12,\nan edge from node 7 to node 2 with capacity 11,\nan edge from node 7 to node 1 with capacity 10,\nan edge from node 7 to node 13 with capacity 5,\nan edge from node 8 to node 12 with capacity 5,\nan edge from node 8 to node 16 with capacity 10,\nan edge from node 8 to node 0 with capacity 14,\nan edge from node 8 to node 18 with capacity 16,\nan edge from node 8 to node 4 with capacity 13,\nan edge from node 9 to node 12 with capacity 6,\nan edge from node 9 to node 17 with capacity 11,\nan edge from node 9 to node 19 with capacity 1,\nan edge from node 9 to node 7 with capacity 4,\nan edge from node 10 to node 9 with capacity 13,\nan edge from node 10 to node 3 with capacity 1,\nan edge from node 10 to node 8 with capacity 5,\nan edge from node 10 to node 6 with capacity 2,\nan edge from node 11 to node 5 with capacity 7,\nan edge from node 11 to node 17 with capacity 3,\nan edge from node 11 to node 9 with capacity 11,\nan edge from node 11 to node 15 with capacity 4,\nan edge from node 11 to node 18 with capacity 15,\nan edge from node 11 to node 4 with capacity 18,\nan edge from node 12 to node 11 with capacity 8,\nan edge from node 12 to node 5 with capacity 5,\nan edge from node 12 to node 0 with capacity 2,\nan edge from node 12 to node 18 with capacity 8,\nan edge from node 12 to node 19 with capacity 14,\nan edge from node 12 to node 13 with capacity 5,\nan edge from node 13 to node 5 with capacity 13,\nan edge from node 13 to node 10 with capacity 16,\nan edge from node 13 to node 15 with capacity 4,\nan edge from node 13 to node 4 with capacity 17,\nan edge from node 14 to node 11 with capacity 20,\nan edge from node 14 to node 12 with capacity 18,\nan edge from node 14 to node 10 with capacity 14,\nan edge from node 14 to node 15 with capacity 1,\nan edge from node 14 to node 4 with capacity 3,\nan edge from node 15 to node 16 with capacity 9,\nan edge from node 15 to node 10 with capacity 10,\nan edge from node 15 to node 18 with capacity 3,\nan edge from node 15 to node 2 with capacity 12,\nan edge from node 16 to node 11 with capacity 14,\nan edge from node 16 to node 5 with capacity 12,\nan edge from node 16 to node 12 with capacity 12,\nan edge from node 16 to node 6 with capacity 19,\nan edge from node 16 to node 19 with capacity 8,\nan edge from node 16 to node 4 with capacity 13,\nan edge from node 17 to node 11 with capacity 4,\nan edge from node 17 to node 12 with capacity 18,\nan edge from node 17 to node 19 with capacity 13,\nan edge from node 17 to node 13 with capacity 19,\nan edge from node 18 to node 14 with capacity 2,\nan edge from node 18 to node 12 with capacity 13,\nan edge from node 18 to node 17 with capacity 4,\nan edge from node 18 to node 1 with capacity 6,\nan edge from node 18 to node 13 with capacity 5,\nan edge from node 19 to node 12 with capacity 12,\nan edge from node 19 to node 3 with capacity 5,\nan edge from node 19 to node 15 with capacity 12,\nan edge from node 19 to node 6 with capacity 6,\nan edge from node 19 to node 7 with capacity 5,\nan edge from node 19 to node 13 with capacity 7.\nQ: What is the maximum flow from node 11 to node 16?\nA:", "answer": "The maximum flow from node 11 to node 16 is 52.", "difficulty": "hard", "doc_id": "72"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 1 to node 5 with capacity 5,\nan edge from node 2 to node 1 with capacity 8,\nan edge from node 2 to node 4 with capacity 8,\nan edge from node 3 to node 1 with capacity 2,\nan edge from node 3 to node 2 with capacity 9,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 4 to node 5 with capacity 4,\nan edge from node 5 to node 2 with capacity 6.\nQ: What is the maximum flow from node 3 to node 0?\nA:", "answer": "The maximum flow from node 3 to node 0 is 6.", "difficulty": "easy", "doc_id": "73"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 2 to node 4 with capacity 2,\nan edge from node 2 to node 3 with capacity 3,\nan edge from node 3 to node 1 with capacity 6,\nan edge from node 3 to node 0 with capacity 6,\nan edge from node 4 to node 1 with capacity 1,\nan edge from node 4 to node 3 with capacity 3,\nan edge from node 4 to node 0 with capacity 6.\nQ: What is the maximum flow from node 2 to node 0?\nA:", "answer": "The maximum flow from node 2 to node 0 is 5.", "difficulty": "easy", "doc_id": "74"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 5 with capacity 8,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 3 to node 4 with capacity 3,\nan edge from node 3 to node 0 with capacity 5,\nan edge from node 4 to node 6 with capacity 3,\nan edge from node 4 to node 2 with capacity 1,\nan edge from node 5 to node 3 with capacity 2,\nan edge from node 5 to node 0 with capacity 5,\nan edge from node 6 to node 4 with capacity 2.\nQ: What is the maximum flow from node 1 to node 2?\nA:", "answer": "The maximum flow from node 1 to node 2 is 7.", "difficulty": "easy", "doc_id": "75"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 5 with capacity 9,\nan edge from node 0 to node 8 with capacity 14,\nan edge from node 1 to node 11 with capacity 18,\nan edge from node 1 to node 9 with capacity 12,\nan edge from node 1 to node 5 with capacity 14,\nan edge from node 1 to node 4 with capacity 12,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 6 with capacity 19,\nan edge from node 2 to node 1 with capacity 11,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 3 to node 6 with capacity 13,\nan edge from node 4 to node 7 with capacity 11,\nan edge from node 5 to node 4 with capacity 5,\nan edge from node 6 to node 7 with capacity 6,\nan edge from node 6 to node 10 with capacity 13,\nan edge from node 6 to node 2 with capacity 3,\nan edge from node 7 to node 11 with capacity 12,\nan edge from node 7 to node 10 with capacity 19,\nan edge from node 7 to node 5 with capacity 18,\nan edge from node 7 to node 8 with capacity 3,\nan edge from node 7 to node 1 with capacity 4,\nan edge from node 8 to node 7 with capacity 4,\nan edge from node 10 to node 12 with capacity 15,\nan edge from node 10 to node 6 with capacity 7,\nan edge from node 10 to node 2 with capacity 15,\nan edge from node 11 to node 8 with capacity 2,\nan edge from node 11 to node 1 with capacity 9.\nQ: What is the maximum flow from node 8 to node 11?\nA:", "answer": "The maximum flow from node 8 to node 11 is 4.", "difficulty": "hard", "doc_id": "76"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 1 to node 3 with capacity 9,\nan edge from node 1 to node 8 with capacity 8,\nan edge from node 1 to node 6 with capacity 7,\nan edge from node 2 to node 1 with capacity 8,\nan edge from node 2 to node 7 with capacity 4,\nan edge from node 3 to node 8 with capacity 10,\nan edge from node 4 to node 1 with capacity 3,\nan edge from node 4 to node 5 with capacity 3,\nan edge from node 4 to node 6 with capacity 8,\nan edge from node 5 to node 4 with capacity 4,\nan edge from node 5 to node 8 with capacity 5,\nan edge from node 6 to node 7 with capacity 3,\nan edge from node 6 to node 5 with capacity 10,\nan edge from node 6 to node 8 with capacity 4,\nan edge from node 7 to node 3 with capacity 9,\nan edge from node 8 to node 0 with capacity 2,\nan edge from node 8 to node 5 with capacity 6,\nan edge from node 8 to node 3 with capacity 9,\nan edge from node 8 to node 4 with capacity 1.\nQ: What is the maximum flow from node 4 to node 0?\nA:", "answer": "The maximum flow from node 4 to node 0 is 2.", "difficulty": "easy", "doc_id": "77"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 9 with capacity 12,\nan edge from node 0 to node 11 with capacity 18,\nan edge from node 0 to node 10 with capacity 9,\nan edge from node 0 to node 12 with capacity 9,\nan edge from node 1 to node 11 with capacity 11,\nan edge from node 1 to node 10 with capacity 12,\nan edge from node 1 to node 12 with capacity 12,\nan edge from node 2 to node 9 with capacity 10,\nan edge from node 2 to node 6 with capacity 19,\nan edge from node 2 to node 14 with capacity 11,\nan edge from node 3 to node 4 with capacity 13,\nan edge from node 3 to node 5 with capacity 6,\nan edge from node 3 to node 2 with capacity 12,\nan edge from node 3 to node 12 with capacity 15,\nan edge from node 4 to node 6 with capacity 20,\nan edge from node 4 to node 3 with capacity 1,\nan edge from node 5 to node 9 with capacity 17,\nan edge from node 5 to node 0 with capacity 20,\nan edge from node 6 to node 7 with capacity 3,\nan edge from node 6 to node 12 with capacity 11,\nan edge from node 6 to node 13 with capacity 4,\nan edge from node 7 to node 8 with capacity 13,\nan edge from node 7 to node 0 with capacity 18,\nan edge from node 7 to node 12 with capacity 17,\nan edge from node 7 to node 13 with capacity 10,\nan edge from node 8 to node 7 with capacity 7,\nan edge from node 8 to node 10 with capacity 5,\nan edge from node 9 to node 7 with capacity 3,\nan edge from node 9 to node 11 with capacity 17,\nan edge from node 9 to node 6 with capacity 5,\nan edge from node 9 to node 10 with capacity 12,\nan edge from node 9 to node 15 with capacity 1,\nan edge from node 9 to node 3 with capacity 16,\nan edge from node 9 to node 13 with capacity 4,\nan edge from node 10 to node 9 with capacity 13,\nan edge from node 10 to node 11 with capacity 9,\nan edge from node 11 to node 7 with capacity 3,\nan edge from node 11 to node 1 with capacity 3,\nan edge from node 11 to node 5 with capacity 17,\nan edge from node 11 to node 8 with capacity 13,\nan edge from node 11 to node 14 with capacity 2,\nan edge from node 12 to node 11 with capacity 16,\nan edge from node 12 to node 10 with capacity 18,\nan edge from node 13 to node 7 with capacity 12,\nan edge from node 13 to node 1 with capacity 5,\nan edge from node 13 to node 14 with capacity 3,\nan edge from node 13 to node 12 with capacity 14,\nan edge from node 14 to node 6 with capacity 10,\nan edge from node 14 to node 8 with capacity 14,\nan edge from node 14 to node 2 with capacity 5,\nan edge from node 14 to node 13 with capacity 16,\nan edge from node 15 to node 10 with capacity 3,\nan edge from node 15 to node 8 with capacity 17,\nan edge from node 15 to node 2 with capacity 18,\nan edge from node 15 to node 14 with capacity 10.\nQ: What is the maximum flow from node 5 to node 8?\nA:", "answer": "The maximum flow from node 5 to node 8 is 37.", "difficulty": "hard", "doc_id": "78"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 16 with capacity 9,\nan edge from node 0 to node 15 with capacity 5,\nan edge from node 1 to node 14 with capacity 9,\nan edge from node 1 to node 4 with capacity 17,\nan edge from node 1 to node 13 with capacity 11,\nan edge from node 2 to node 9 with capacity 11,\nan edge from node 2 to node 15 with capacity 9,\nan edge from node 2 to node 0 with capacity 4,\nan edge from node 3 to node 16 with capacity 13,\nan edge from node 3 to node 2 with capacity 11,\nan edge from node 3 to node 12 with capacity 13,\nan edge from node 4 to node 9 with capacity 14,\nan edge from node 4 to node 7 with capacity 20,\nan edge from node 4 to node 12 with capacity 18,\nan edge from node 4 to node 15 with capacity 4,\nan edge from node 4 to node 5 with capacity 12,\nan edge from node 5 to node 16 with capacity 19,\nan edge from node 5 to node 6 with capacity 9,\nan edge from node 6 to node 9 with capacity 17,\nan edge from node 6 to node 7 with capacity 12,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 6 to node 17 with capacity 15,\nan edge from node 7 to node 11 with capacity 5,\nan edge from node 7 to node 6 with capacity 14,\nan edge from node 7 to node 3 with capacity 3,\nan edge from node 7 to node 15 with capacity 14,\nan edge from node 8 to node 2 with capacity 2,\nan edge from node 8 to node 7 with capacity 15,\nan edge from node 8 to node 12 with capacity 16,\nan edge from node 8 to node 17 with capacity 13,\nan edge from node 8 to node 0 with capacity 6,\nan edge from node 9 to node 6 with capacity 4,\nan edge from node 9 to node 7 with capacity 8,\nan edge from node 9 to node 0 with capacity 3,\nan edge from node 10 to node 11 with capacity 9,\nan edge from node 10 to node 14 with capacity 17,\nan edge from node 10 to node 1 with capacity 19,\nan edge from node 10 to node 9 with capacity 18,\nan edge from node 10 to node 3 with capacity 3,\nan edge from node 10 to node 4 with capacity 9,\nan edge from node 10 to node 5 with capacity 3,\nan edge from node 10 to node 17 with capacity 13,\nan edge from node 11 to node 16 with capacity 8,\nan edge from node 11 to node 2 with capacity 6,\nan edge from node 11 to node 8 with capacity 18,\nan edge from node 11 to node 4 with capacity 13,\nan edge from node 12 to node 11 with capacity 19,\nan edge from node 12 to node 1 with capacity 17,\nan edge from node 12 to node 8 with capacity 16,\nan edge from node 12 to node 15 with capacity 13,\nan edge from node 13 to node 16 with capacity 20,\nan edge from node 13 to node 14 with capacity 9,\nan edge from node 13 to node 6 with capacity 14,\nan edge from node 13 to node 9 with capacity 16,\nan edge from node 13 to node 8 with capacity 7,\nan edge from node 13 to node 7 with capacity 1,\nan edge from node 13 to node 5 with capacity 7,\nan edge from node 13 to node 17 with capacity 14,\nan edge from node 14 to node 8 with capacity 14,\nan edge from node 14 to node 12 with capacity 8,\nan edge from node 14 to node 13 with capacity 4,\nan edge from node 14 to node 0 with capacity 12,\nan edge from node 15 to node 10 with capacity 13,\nan edge from node 15 to node 2 with capacity 14,\nan edge from node 15 to node 12 with capacity 14,\nan edge from node 15 to node 17 with capacity 5,\nan edge from node 16 to node 14 with capacity 20,\nan edge from node 16 to node 7 with capacity 17,\nan edge from node 16 to node 5 with capacity 16,\nan edge from node 16 to node 13 with capacity 1,\nan edge from node 17 to node 10 with capacity 20,\nan edge from node 17 to node 1 with capacity 17,\nan edge from node 17 to node 12 with capacity 3,\nan edge from node 17 to node 15 with capacity 9.\nQ: What is the maximum flow from node 1 to node 13?\nA:", "answer": "The maximum flow from node 1 to node 13 is 16.", "difficulty": "hard", "doc_id": "79"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 5 with capacity 7,\nan edge from node 0 to node 1 with capacity 4,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 0 to node 2 with capacity 4,\nan edge from node 1 to node 5 with capacity 7,\nan edge from node 1 to node 2 with capacity 8,\nan edge from node 2 to node 3 with capacity 8,\nan edge from node 3 to node 5 with capacity 2,\nan edge from node 3 to node 1 with capacity 2,\nan edge from node 3 to node 4 with capacity 10,\nan edge from node 3 to node 2 with capacity 7,\nan edge from node 4 to node 0 with capacity 6,\nan edge from node 4 to node 5 with capacity 1,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 2 with capacity 2,\nan edge from node 5 to node 4 with capacity 10.\nQ: What is the maximum flow from node 3 to node 0?\nA:", "answer": "The maximum flow from node 3 to node 0 is 6.", "difficulty": "easy", "doc_id": "80"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 18 with capacity 1,\nan edge from node 0 to node 16 with capacity 4,\nan edge from node 0 to node 3 with capacity 6,\nan edge from node 0 to node 4 with capacity 6,\nan edge from node 0 to node 11 with capacity 11,\nan edge from node 0 to node 12 with capacity 20,\nan edge from node 1 to node 9 with capacity 13,\nan edge from node 1 to node 10 with capacity 17,\nan edge from node 1 to node 12 with capacity 3,\nan edge from node 2 to node 9 with capacity 10,\nan edge from node 2 to node 0 with capacity 11,\nan edge from node 2 to node 10 with capacity 16,\nan edge from node 2 to node 4 with capacity 17,\nan edge from node 2 to node 14 with capacity 13,\nan edge from node 2 to node 1 with capacity 15,\nan edge from node 3 to node 4 with capacity 17,\nan edge from node 3 to node 7 with capacity 6,\nan edge from node 3 to node 11 with capacity 15,\nan edge from node 4 to node 9 with capacity 10,\nan edge from node 4 to node 18 with capacity 1,\nan edge from node 4 to node 16 with capacity 10,\nan edge from node 4 to node 6 with capacity 14,\nan edge from node 4 to node 12 with capacity 7,\nan edge from node 5 to node 10 with capacity 15,\nan edge from node 6 to node 9 with capacity 15,\nan edge from node 6 to node 18 with capacity 8,\nan edge from node 6 to node 16 with capacity 19,\nan edge from node 6 to node 8 with capacity 1,\nan edge from node 6 to node 2 with capacity 1,\nan edge from node 6 to node 1 with capacity 10,\nan edge from node 7 to node 18 with capacity 16,\nan edge from node 7 to node 16 with capacity 14,\nan edge from node 7 to node 8 with capacity 3,\nan edge from node 7 to node 6 with capacity 14,\nan edge from node 7 to node 17 with capacity 10,\nan edge from node 7 to node 12 with capacity 8,\nan edge from node 8 to node 18 with capacity 2,\nan edge from node 8 to node 0 with capacity 12,\nan edge from node 8 to node 16 with capacity 13,\nan edge from node 8 to node 7 with capacity 1,\nan edge from node 8 to node 11 with capacity 14,\nan edge from node 9 to node 13 with capacity 1,\nan edge from node 9 to node 14 with capacity 19,\nan edge from node 9 to node 17 with capacity 1,\nan edge from node 9 to node 1 with capacity 10,\nan edge from node 10 to node 0 with capacity 2,\nan edge from node 10 to node 8 with capacity 11,\nan edge from node 10 to node 2 with capacity 7,\nan edge from node 10 to node 12 with capacity 13,\nan edge from node 11 to node 10 with capacity 4,\nan edge from node 11 to node 15 with capacity 3,\nan edge from node 11 to node 1 with capacity 16,\nan edge from node 11 to node 12 with capacity 2,\nan edge from node 12 to node 18 with capacity 9,\nan edge from node 12 to node 0 with capacity 12,\nan edge from node 12 to node 10 with capacity 6,\nan edge from node 12 to node 5 with capacity 18,\nan edge from node 13 to node 15 with capacity 19,\nan edge from node 13 to node 14 with capacity 5,\nan edge from node 13 to node 1 with capacity 8,\nan edge from node 14 to node 16 with capacity 5,\nan edge from node 14 to node 2 with capacity 13,\nan edge from node 14 to node 7 with capacity 4,\nan edge from node 15 to node 9 with capacity 8,\nan edge from node 15 to node 5 with capacity 15,\nan edge from node 15 to node 16 with capacity 15,\nan edge from node 15 to node 17 with capacity 11,\nan edge from node 15 to node 2 with capacity 18,\nan edge from node 15 to node 11 with capacity 17,\nan edge from node 16 to node 5 with capacity 1,\nan edge from node 17 to node 3 with capacity 9,\nan edge from node 17 to node 1 with capacity 1,\nan edge from node 18 to node 13 with capacity 2,\nan edge from node 18 to node 0 with capacity 5,\nan edge from node 18 to node 16 with capacity 13,\nan edge from node 18 to node 14 with capacity 7,\nan edge from node 18 to node 7 with capacity 11,\nan edge from node 18 to node 11 with capacity 5.\nQ: What is the maximum flow from node 0 to node 1?\nA:", "answer": "The maximum flow from node 0 to node 1 is 45.", "difficulty": "hard", "doc_id": "81"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 7 with capacity 9,\nan edge from node 0 to node 9 with capacity 10,\nan edge from node 1 to node 6 with capacity 8,\nan edge from node 1 to node 8 with capacity 2,\nan edge from node 1 to node 4 with capacity 3,\nan edge from node 2 to node 7 with capacity 4,\nan edge from node 2 to node 9 with capacity 10,\nan edge from node 2 to node 8 with capacity 2,\nan edge from node 3 to node 7 with capacity 10,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 3 to node 1 with capacity 3,\nan edge from node 3 to node 8 with capacity 2,\nan edge from node 4 to node 2 with capacity 3,\nan edge from node 4 to node 9 with capacity 10,\nan edge from node 4 to node 3 with capacity 1,\nan edge from node 5 to node 7 with capacity 7,\nan edge from node 6 to node 1 with capacity 1,\nan edge from node 7 to node 6 with capacity 4,\nan edge from node 7 to node 1 with capacity 7,\nan edge from node 8 to node 2 with capacity 9,\nan edge from node 9 to node 6 with capacity 8,\nan edge from node 9 to node 2 with capacity 4,\nan edge from node 9 to node 0 with capacity 3,\nan edge from node 9 to node 1 with capacity 3,\nan edge from node 9 to node 8 with capacity 10.\nQ: What is the maximum flow from node 4 to node 2?\nA:", "answer": "The maximum flow from node 4 to node 2 is 14.", "difficulty": "easy", "doc_id": "82"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 3 with capacity 7,\nan edge from node 1 to node 7 with capacity 9,\nan edge from node 2 to node 5 with capacity 3,\nan edge from node 3 to node 5 with capacity 1,\nan edge from node 3 to node 4 with capacity 7,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 0 with capacity 2,\nan edge from node 5 to node 1 with capacity 7,\nan edge from node 5 to node 6 with capacity 1,\nan edge from node 6 to node 7 with capacity 2.\nQ: What is the maximum flow from node 0 to node 7?\nA:", "answer": "The maximum flow from node 0 to node 7 is 7.", "difficulty": "easy", "doc_id": "83"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 9 with capacity 1,\nan edge from node 1 to node 8 with capacity 16,\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 2 to node 8 with capacity 17,\nan edge from node 2 to node 4 with capacity 13,\nan edge from node 2 to node 0 with capacity 19,\nan edge from node 2 to node 3 with capacity 11,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 3 to node 8 with capacity 15,\nan edge from node 3 to node 1 with capacity 13,\nan edge from node 3 to node 7 with capacity 13,\nan edge from node 3 to node 5 with capacity 11,\nan edge from node 4 to node 8 with capacity 12,\nan edge from node 4 to node 2 with capacity 7,\nan edge from node 5 to node 9 with capacity 14,\nan edge from node 5 to node 1 with capacity 1,\nan edge from node 6 to node 8 with capacity 7,\nan edge from node 6 to node 7 with capacity 9,\nan edge from node 6 to node 5 with capacity 15,\nan edge from node 7 to node 5 with capacity 18,\nan edge from node 7 to node 2 with capacity 1,\nan edge from node 8 to node 6 with capacity 17,\nan edge from node 8 to node 2 with capacity 12,\nan edge from node 9 to node 7 with capacity 17,\nan edge from node 9 to node 5 with capacity 17,\nan edge from node 10 to node 9 with capacity 1,\nan edge from node 10 to node 4 with capacity 5,\nan edge from node 10 to node 2 with capacity 7.\nQ: What is the maximum flow from node 2 to node 6?\nA:", "answer": "The maximum flow from node 2 to node 6 is 17.", "difficulty": "hard", "doc_id": "84"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 8 with capacity 1,\nan edge from node 0 to node 12 with capacity 18,\nan edge from node 0 to node 4 with capacity 4,\nan edge from node 0 to node 5 with capacity 16,\nan edge from node 0 to node 10 with capacity 11,\nan edge from node 0 to node 9 with capacity 2,\nan edge from node 0 to node 2 with capacity 5,\nan edge from node 0 to node 7 with capacity 15,\nan edge from node 1 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 5,\nan edge from node 1 to node 11 with capacity 15,\nan edge from node 2 to node 8 with capacity 14,\nan edge from node 2 to node 14 with capacity 14,\nan edge from node 2 to node 5 with capacity 18,\nan edge from node 2 to node 9 with capacity 9,\nan edge from node 2 to node 0 with capacity 13,\nan edge from node 3 to node 13 with capacity 13,\nan edge from node 3 to node 15 with capacity 14,\nan edge from node 3 to node 11 with capacity 16,\nan edge from node 3 to node 9 with capacity 8,\nan edge from node 4 to node 13 with capacity 17,\nan edge from node 4 to node 5 with capacity 18,\nan edge from node 4 to node 3 with capacity 4,\nan edge from node 5 to node 12 with capacity 17,\nan edge from node 5 to node 1 with capacity 13,\nan edge from node 5 to node 16 with capacity 7,\nan edge from node 5 to node 6 with capacity 3,\nan edge from node 6 to node 4 with capacity 6,\nan edge from node 6 to node 14 with capacity 6,\nan edge from node 6 to node 7 with capacity 9,\nan edge from node 7 to node 10 with capacity 13,\nan edge from node 7 to node 0 with capacity 11,\nan edge from node 8 to node 1 with capacity 13,\nan edge from node 8 to node 13 with capacity 13,\nan edge from node 8 to node 6 with capacity 13,\nan edge from node 8 to node 7 with capacity 3,\nan edge from node 9 to node 1 with capacity 6,\nan edge from node 9 to node 16 with capacity 15,\nan edge from node 10 to node 13 with capacity 13,\nan edge from node 10 to node 3 with capacity 10,\nan edge from node 10 to node 15 with capacity 5,\nan edge from node 10 to node 9 with capacity 20,\nan edge from node 11 to node 12 with capacity 4,\nan edge from node 11 to node 4 with capacity 18,\nan edge from node 11 to node 10 with capacity 2,\nan edge from node 11 to node 2 with capacity 3,\nan edge from node 12 to node 5 with capacity 20,\nan edge from node 12 to node 6 with capacity 9,\nan edge from node 13 to node 4 with capacity 11,\nan edge from node 13 to node 1 with capacity 14,\nan edge from node 13 to node 15 with capacity 3,\nan edge from node 13 to node 2 with capacity 3,\nan edge from node 14 to node 13 with capacity 3,\nan edge from node 14 to node 3 with capacity 16,\nan edge from node 14 to node 15 with capacity 10,\nan edge from node 14 to node 11 with capacity 11,\nan edge from node 14 to node 6 with capacity 3,\nan edge from node 15 to node 16 with capacity 17,\nan edge from node 16 to node 2 with capacity 5,\nan edge from node 16 to node 7 with capacity 12.\nQ: What is the maximum flow from node 1 to node 16?\nA:", "answer": "The maximum flow from node 1 to node 16 is 30.", "difficulty": "hard", "doc_id": "85"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 6 with capacity 8,\nan edge from node 1 to node 6 with capacity 1,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 3 to node 7 with capacity 8,\nan edge from node 4 to node 5 with capacity 8,\nan edge from node 5 to node 3 with capacity 4,\nan edge from node 5 to node 1 with capacity 2,\nan edge from node 5 to node 4 with capacity 4,\nan edge from node 6 to node 4 with capacity 6,\nan edge from node 7 to node 2 with capacity 7.\nQ: What is the maximum flow from node 4 to node 1?\nA:", "answer": "The maximum flow from node 4 to node 1 is 6.", "difficulty": "easy", "doc_id": "86"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 9 with capacity 17,\nan edge from node 0 to node 7 with capacity 7,\nan edge from node 0 to node 11 with capacity 19,\nan edge from node 1 to node 9 with capacity 20,\nan edge from node 1 to node 0 with capacity 20,\nan edge from node 1 to node 10 with capacity 12,\nan edge from node 2 to node 4 with capacity 20,\nan edge from node 2 to node 9 with capacity 10,\nan edge from node 2 to node 5 with capacity 11,\nan edge from node 2 to node 3 with capacity 8,\nan edge from node 2 to node 11 with capacity 20,\nan edge from node 3 to node 4 with capacity 13,\nan edge from node 3 to node 9 with capacity 20,\nan edge from node 3 to node 7 with capacity 19,\nan edge from node 3 to node 8 with capacity 7,\nan edge from node 4 to node 9 with capacity 16,\nan edge from node 4 to node 7 with capacity 18,\nan edge from node 4 to node 8 with capacity 20,\nan edge from node 4 to node 12 with capacity 18,\nan edge from node 5 to node 2 with capacity 7,\nan edge from node 5 to node 9 with capacity 15,\nan edge from node 5 to node 3 with capacity 5,\nan edge from node 5 to node 11 with capacity 15,\nan edge from node 5 to node 12 with capacity 12,\nan edge from node 6 to node 5 with capacity 11,\nan edge from node 6 to node 0 with capacity 19,\nan edge from node 6 to node 11 with capacity 8,\nan edge from node 6 to node 12 with capacity 17,\nan edge from node 7 to node 2 with capacity 17,\nan edge from node 7 to node 0 with capacity 1,\nan edge from node 7 to node 3 with capacity 8,\nan edge from node 8 to node 4 with capacity 16,\nan edge from node 8 to node 1 with capacity 13,\nan edge from node 8 to node 9 with capacity 13,\nan edge from node 8 to node 11 with capacity 17,\nan edge from node 8 to node 6 with capacity 10,\nan edge from node 9 to node 4 with capacity 16,\nan edge from node 9 to node 6 with capacity 4,\nan edge from node 10 to node 2 with capacity 8,\nan edge from node 10 to node 11 with capacity 6,\nan edge from node 10 to node 12 with capacity 9,\nan edge from node 11 to node 5 with capacity 5,\nan edge from node 11 to node 3 with capacity 19,\nan edge from node 11 to node 8 with capacity 9,\nan edge from node 11 to node 12 with capacity 4,\nan edge from node 12 to node 5 with capacity 11,\nan edge from node 12 to node 0 with capacity 4,\nan edge from node 12 to node 3 with capacity 19,\nan edge from node 12 to node 8 with capacity 6.\nQ: What is the maximum flow from node 6 to node 4?\nA:", "answer": "The maximum flow from node 6 to node 4 is 55.", "difficulty": "hard", "doc_id": "87"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 13 with capacity 2,\nan edge from node 0 to node 6 with capacity 2,\nan edge from node 0 to node 16 with capacity 13,\nan edge from node 0 to node 7 with capacity 10,\nan edge from node 0 to node 17 with capacity 2,\nan edge from node 0 to node 14 with capacity 11,\nan edge from node 1 to node 16 with capacity 19,\nan edge from node 1 to node 17 with capacity 5,\nan edge from node 2 to node 10 with capacity 9,\nan edge from node 2 to node 11 with capacity 4,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 2 to node 1 with capacity 13,\nan edge from node 3 to node 15 with capacity 18,\nan edge from node 3 to node 5 with capacity 14,\nan edge from node 4 to node 16 with capacity 2,\nan edge from node 4 to node 15 with capacity 7,\nan edge from node 4 to node 14 with capacity 13,\nan edge from node 5 to node 8 with capacity 6,\nan edge from node 5 to node 3 with capacity 20,\nan edge from node 5 to node 15 with capacity 13,\nan edge from node 5 to node 14 with capacity 14,\nan edge from node 6 to node 8 with capacity 20,\nan edge from node 6 to node 10 with capacity 18,\nan edge from node 6 to node 11 with capacity 6,\nan edge from node 6 to node 5 with capacity 17,\nan edge from node 6 to node 0 with capacity 2,\nan edge from node 6 to node 9 with capacity 8,\nan edge from node 6 to node 12 with capacity 15,\nan edge from node 7 to node 8 with capacity 12,\nan edge from node 7 to node 6 with capacity 9,\nan edge from node 7 to node 10 with capacity 10,\nan edge from node 7 to node 15 with capacity 15,\nan edge from node 7 to node 17 with capacity 4,\nan edge from node 8 to node 2 with capacity 10,\nan edge from node 8 to node 0 with capacity 3,\nan edge from node 8 to node 7 with capacity 6,\nan edge from node 8 to node 12 with capacity 5,\nan edge from node 8 to node 1 with capacity 10,\nan edge from node 8 to node 14 with capacity 12,\nan edge from node 9 to node 13 with capacity 10,\nan edge from node 9 to node 3 with capacity 19,\nan edge from node 9 to node 4 with capacity 17,\nan edge from node 9 to node 16 with capacity 15,\nan edge from node 9 to node 11 with capacity 10,\nan edge from node 9 to node 5 with capacity 3,\nan edge from node 9 to node 0 with capacity 15,\nan edge from node 10 to node 16 with capacity 1,\nan edge from node 10 to node 15 with capacity 17,\nan edge from node 10 to node 5 with capacity 3,\nan edge from node 10 to node 7 with capacity 16,\nan edge from node 11 to node 15 with capacity 7,\nan edge from node 11 to node 5 with capacity 13,\nan edge from node 11 to node 9 with capacity 12,\nan edge from node 11 to node 17 with capacity 3,\nan edge from node 11 to node 14 with capacity 7,\nan edge from node 12 to node 13 with capacity 20,\nan edge from node 12 to node 5 with capacity 4,\nan edge from node 12 to node 17 with capacity 2,\nan edge from node 13 to node 3 with capacity 14,\nan edge from node 13 to node 4 with capacity 1,\nan edge from node 14 to node 8 with capacity 10,\nan edge from node 14 to node 6 with capacity 19,\nan edge from node 14 to node 5 with capacity 16,\nan edge from node 14 to node 7 with capacity 19,\nan edge from node 14 to node 17 with capacity 13,\nan edge from node 15 to node 8 with capacity 6,\nan edge from node 15 to node 16 with capacity 20,\nan edge from node 16 to node 10 with capacity 9,\nan edge from node 16 to node 9 with capacity 18,\nan edge from node 17 to node 13 with capacity 13,\nan edge from node 17 to node 8 with capacity 4,\nan edge from node 17 to node 5 with capacity 19,\nan edge from node 17 to node 12 with capacity 2,\nan edge from node 17 to node 14 with capacity 16.\nQ: What is the maximum flow from node 8 to node 9?\nA:", "answer": "The maximum flow from node 8 to node 9 is 36.", "difficulty": "hard", "doc_id": "88"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 5 with capacity 3,\nan edge from node 1 to node 4 with capacity 2,\nan edge from node 1 to node 7 with capacity 3,\nan edge from node 2 to node 1 with capacity 8,\nan edge from node 3 to node 6 with capacity 5,\nan edge from node 3 to node 4 with capacity 2,\nan edge from node 3 to node 5 with capacity 5,\nan edge from node 4 to node 6 with capacity 7,\nan edge from node 4 to node 2 with capacity 3,\nan edge from node 5 to node 6 with capacity 2,\nan edge from node 5 to node 2 with capacity 2,\nan edge from node 7 to node 6 with capacity 2,\nan edge from node 7 to node 0 with capacity 2,\nan edge from node 7 to node 3 with capacity 2.\nQ: What is the maximum flow from node 5 to node 6?\nA:", "answer": "The maximum flow from node 5 to node 6 is 4.", "difficulty": "easy", "doc_id": "89"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 3,\nan edge from node 0 to node 1 with capacity 6,\nan edge from node 0 to node 2 with capacity 3,\nan edge from node 1 to node 2 with capacity 4,\nan edge from node 1 to node 0 with capacity 4,\nan edge from node 2 to node 4 with capacity 9,\nan edge from node 2 to node 3 with capacity 7,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 0 with capacity 5,\nan edge from node 4 to node 1 with capacity 4,\nan edge from node 4 to node 2 with capacity 4,\nan edge from node 4 to node 0 with capacity 2,\nan edge from node 5 to node 0 with capacity 3.\nQ: What is the maximum flow from node 2 to node 4?\nA:", "answer": "The maximum flow from node 2 to node 4 is 16.", "difficulty": "easy", "doc_id": "90"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 2 with capacity 13,\nan edge from node 0 to node 10 with capacity 15,\nan edge from node 0 to node 12 with capacity 18,\nan edge from node 1 to node 2 with capacity 9,\nan edge from node 1 to node 5 with capacity 6,\nan edge from node 1 to node 8 with capacity 7,\nan edge from node 1 to node 9 with capacity 19,\nan edge from node 2 to node 10 with capacity 7,\nan edge from node 2 to node 14 with capacity 2,\nan edge from node 2 to node 9 with capacity 1,\nan edge from node 2 to node 1 with capacity 9,\nan edge from node 2 to node 12 with capacity 3,\nan edge from node 2 to node 4 with capacity 8,\nan edge from node 3 to node 10 with capacity 7,\nan edge from node 4 to node 13 with capacity 19,\nan edge from node 4 to node 1 with capacity 11,\nan edge from node 4 to node 0 with capacity 9,\nan edge from node 5 to node 11 with capacity 14,\nan edge from node 5 to node 1 with capacity 17,\nan edge from node 6 to node 2 with capacity 1,\nan edge from node 6 to node 9 with capacity 8,\nan edge from node 7 to node 13 with capacity 3,\nan edge from node 8 to node 10 with capacity 5,\nan edge from node 8 to node 13 with capacity 7,\nan edge from node 8 to node 11 with capacity 9,\nan edge from node 8 to node 1 with capacity 4,\nan edge from node 9 to node 2 with capacity 18,\nan edge from node 9 to node 6 with capacity 13,\nan edge from node 9 to node 7 with capacity 17,\nan edge from node 9 to node 0 with capacity 1,\nan edge from node 9 to node 15 with capacity 20,\nan edge from node 10 to node 11 with capacity 3,\nan edge from node 10 to node 7 with capacity 16,\nan edge from node 10 to node 0 with capacity 6,\nan edge from node 10 to node 12 with capacity 12,\nan edge from node 11 to node 10 with capacity 16,\nan edge from node 11 to node 1 with capacity 5,\nan edge from node 11 to node 12 with capacity 3,\nan edge from node 11 to node 4 with capacity 8,\nan edge from node 12 to node 2 with capacity 8,\nan edge from node 12 to node 1 with capacity 14,\nan edge from node 12 to node 4 with capacity 1,\nan edge from node 13 to node 2 with capacity 11,\nan edge from node 13 to node 10 with capacity 8,\nan edge from node 13 to node 8 with capacity 19,\nan edge from node 13 to node 12 with capacity 1,\nan edge from node 13 to node 4 with capacity 17,\nan edge from node 14 to node 10 with capacity 15,\nan edge from node 14 to node 8 with capacity 2,\nan edge from node 15 to node 10 with capacity 18,\nan edge from node 15 to node 14 with capacity 6,\nan edge from node 15 to node 8 with capacity 16,\nan edge from node 15 to node 13 with capacity 9,\nan edge from node 15 to node 11 with capacity 10,\nan edge from node 15 to node 3 with capacity 15,\nan edge from node 15 to node 4 with capacity 15.\nQ: What is the maximum flow from node 15 to node 13?\nA:", "answer": "The maximum flow from node 15 to node 13 is 38.", "difficulty": "hard", "doc_id": "91"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 4 with capacity 1,\nan edge from node 0 to node 1 with capacity 3,\nan edge from node 0 to node 7 with capacity 2,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 1 to node 9 with capacity 4,\nan edge from node 1 to node 5 with capacity 2,\nan edge from node 2 to node 1 with capacity 4,\nan edge from node 2 to node 8 with capacity 2,\nan edge from node 2 to node 7 with capacity 6,\nan edge from node 3 to node 0 with capacity 7,\nan edge from node 3 to node 7 with capacity 2,\nan edge from node 4 to node 0 with capacity 2,\nan edge from node 4 to node 6 with capacity 4,\nan edge from node 6 to node 5 with capacity 10,\nan edge from node 6 to node 8 with capacity 5,\nan edge from node 8 to node 6 with capacity 9,\nan edge from node 9 to node 4 with capacity 5.\nQ: What is the maximum flow from node 2 to node 5?\nA:", "answer": "The maximum flow from node 2 to node 5 is 6.", "difficulty": "easy", "doc_id": "92"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 9,\nan edge from node 0 to node 1 with capacity 3,\nan edge from node 1 to node 3 with capacity 1,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 4 to node 1 with capacity 3.\nQ: What is the maximum flow from node 4 to node 3?\nA:", "answer": "The maximum flow from node 4 to node 3 is 10.", "difficulty": "easy", "doc_id": "93"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 1 with capacity 6,\nan edge from node 0 to node 2 with capacity 3,\nan edge from node 1 to node 5 with capacity 10,\nan edge from node 2 to node 5 with capacity 8,\nan edge from node 2 to node 1 with capacity 9,\nan edge from node 3 to node 4 with capacity 10,\nan edge from node 3 to node 5 with capacity 4,\nan edge from node 4 to node 2 with capacity 1,\nan edge from node 5 to node 0 with capacity 1.\nQ: What is the maximum flow from node 5 to node 1?\nA:", "answer": "The maximum flow from node 5 to node 1 is 1.", "difficulty": "easy", "doc_id": "94"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 4 with capacity 7,\nan edge from node 1 to node 0 with capacity 4,\nan edge from node 1 to node 4 with capacity 1,\nan edge from node 2 to node 0 with capacity 1,\nan edge from node 2 to node 4 with capacity 1,\nan edge from node 3 to node 0 with capacity 9,\nan edge from node 4 to node 1 with capacity 5,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 4 to node 2 with capacity 9.\nQ: What is the maximum flow from node 2 to node 0?\nA:", "answer": "The maximum flow from node 2 to node 0 is 2.", "difficulty": "easy", "doc_id": "95"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 4 with capacity 3,\nan edge from node 1 to node 6 with capacity 1,\nan edge from node 3 to node 0 with capacity 7,\nan edge from node 3 to node 6 with capacity 6,\nan edge from node 3 to node 1 with capacity 1,\nan edge from node 4 to node 3 with capacity 7,\nan edge from node 5 to node 3 with capacity 2,\nan edge from node 5 to node 6 with capacity 2,\nan edge from node 5 to node 1 with capacity 5,\nan edge from node 6 to node 5 with capacity 3.\nQ: What is the maximum flow from node 5 to node 6?\nA:", "answer": "The maximum flow from node 5 to node 6 is 5.", "difficulty": "easy", "doc_id": "96"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 10 with capacity 8,\nan edge from node 0 to node 11 with capacity 13,\nan edge from node 0 to node 8 with capacity 2,\nan edge from node 0 to node 7 with capacity 13,\nan edge from node 0 to node 4 with capacity 16,\nan edge from node 1 to node 2 with capacity 20,\nan edge from node 1 to node 5 with capacity 7,\nan edge from node 2 to node 0 with capacity 5,\nan edge from node 2 to node 12 with capacity 4,\nan edge from node 3 to node 10 with capacity 11,\nan edge from node 3 to node 0 with capacity 17,\nan edge from node 3 to node 2 with capacity 2,\nan edge from node 3 to node 9 with capacity 15,\nan edge from node 3 to node 13 with capacity 9,\nan edge from node 4 to node 10 with capacity 6,\nan edge from node 4 to node 8 with capacity 1,\nan edge from node 4 to node 12 with capacity 15,\nan edge from node 4 to node 5 with capacity 3,\nan edge from node 4 to node 13 with capacity 8,\nan edge from node 5 to node 10 with capacity 10,\nan edge from node 5 to node 3 with capacity 16,\nan edge from node 5 to node 8 with capacity 7,\nan edge from node 5 to node 12 with capacity 16,\nan edge from node 6 to node 7 with capacity 9,\nan edge from node 6 to node 9 with capacity 4,\nan edge from node 7 to node 0 with capacity 20,\nan edge from node 7 to node 2 with capacity 5,\nan edge from node 7 to node 1 with capacity 11,\nan edge from node 7 to node 9 with capacity 1,\nan edge from node 8 to node 10 with capacity 7,\nan edge from node 8 to node 3 with capacity 2,\nan edge from node 8 to node 11 with capacity 11,\nan edge from node 8 to node 4 with capacity 10,\nan edge from node 8 to node 6 with capacity 19,\nan edge from node 9 to node 0 with capacity 17,\nan edge from node 9 to node 7 with capacity 16,\nan edge from node 9 to node 5 with capacity 16,\nan edge from node 10 to node 12 with capacity 17,\nan edge from node 10 to node 7 with capacity 5,\nan edge from node 10 to node 4 with capacity 8,\nan edge from node 11 to node 8 with capacity 2,\nan edge from node 11 to node 2 with capacity 12,\nan edge from node 12 to node 9 with capacity 9,\nan edge from node 12 to node 13 with capacity 1,\nan edge from node 13 to node 10 with capacity 5,\nan edge from node 13 to node 0 with capacity 9,\nan edge from node 13 to node 11 with capacity 15,\nan edge from node 13 to node 1 with capacity 2,\nan edge from node 13 to node 4 with capacity 14,\nan edge from node 13 to node 9 with capacity 2.\nQ: What is the maximum flow from node 3 to node 9?\nA:", "answer": "The maximum flow from node 3 to node 9 is 31.", "difficulty": "hard", "doc_id": "97"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 7 with capacity 3,\nan edge from node 0 to node 1 with capacity 3,\nan edge from node 2 to node 8 with capacity 6,\nan edge from node 4 to node 7 with capacity 3,\nan edge from node 4 to node 5 with capacity 7,\nan edge from node 4 to node 3 with capacity 7,\nan edge from node 5 to node 7 with capacity 6,\nan edge from node 6 to node 1 with capacity 7,\nan edge from node 7 to node 2 with capacity 10,\nan edge from node 9 to node 8 with capacity 2,\nan edge from node 9 to node 5 with capacity 5,\nan edge from node 9 to node 4 with capacity 3,\nan edge from node 9 to node 3 with capacity 9.\nQ: What is the maximum flow from node 9 to node 8?\nA:", "answer": "The maximum flow from node 9 to node 8 is 8.", "difficulty": "easy", "doc_id": "98"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 1 to node 3 with capacity 9,\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 2 to node 4 with capacity 3,\nan edge from node 2 to node 0 with capacity 7,\nan edge from node 3 to node 5 with capacity 3,\nan edge from node 4 to node 6 with capacity 8,\nan edge from node 5 to node 6 with capacity 8,\nan edge from node 5 to node 0 with capacity 4,\nan edge from node 6 to node 4 with capacity 7,\nan edge from node 6 to node 2 with capacity 2,\nan edge from node 7 to node 3 with capacity 8.\nQ: What is the maximum flow from node 1 to node 0?\nA:", "answer": "The maximum flow from node 1 to node 0 is 9.", "difficulty": "easy", "doc_id": "99"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 1,\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 2 to node 0 with capacity 1,\nan edge from node 2 to node 3 with capacity 7,\nan edge from node 3 to node 2 with capacity 8,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 2 with capacity 6,\nan edge from node 4 to node 3 with capacity 7.\nQ: What is the maximum flow from node 4 to node 1?\nA:", "answer": "The maximum flow from node 4 to node 1 is 3.", "difficulty": "easy", "doc_id": "100"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 1 to node 0 with capacity 7,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 2 with capacity 1,\nan edge from node 1 to node 4 with capacity 9,\nan edge from node 2 to node 0 with capacity 9,\nan edge from node 2 to node 3 with capacity 7,\nan edge from node 3 to node 0 with capacity 9,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 8.\nQ: What is the maximum flow from node 1 to node 3?\nA:", "answer": "The maximum flow from node 1 to node 3 is 12.", "difficulty": "easy", "doc_id": "101"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 2 with capacity 9,\nan edge from node 0 to node 8 with capacity 1,\nan edge from node 1 to node 8 with capacity 5,\nan edge from node 2 to node 8 with capacity 9,\nan edge from node 2 to node 3 with capacity 9,\nan edge from node 2 to node 7 with capacity 2,\nan edge from node 4 to node 6 with capacity 10,\nan edge from node 4 to node 8 with capacity 5,\nan edge from node 5 to node 6 with capacity 7,\nan edge from node 5 to node 8 with capacity 2,\nan edge from node 6 to node 8 with capacity 3,\nan edge from node 7 to node 3 with capacity 7,\nan edge from node 8 to node 2 with capacity 5,\nan edge from node 8 to node 3 with capacity 3.\nQ: What is the maximum flow from node 8 to node 3?\nA:", "answer": "The maximum flow from node 8 to node 3 is 8.", "difficulty": "easy", "doc_id": "102"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 8,\nan edge from node 0 to node 1 with capacity 9,\nan edge from node 1 to node 5 with capacity 5,\nan edge from node 1 to node 0 with capacity 8,\nan edge from node 2 to node 6 with capacity 7,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 3 to node 0 with capacity 2,\nan edge from node 3 to node 1 with capacity 8,\nan edge from node 4 to node 5 with capacity 8,\nan edge from node 4 to node 0 with capacity 7,\nan edge from node 4 to node 1 with capacity 8,\nan edge from node 5 to node 3 with capacity 2,\nan edge from node 5 to node 2 with capacity 9,\nan edge from node 6 to node 5 with capacity 10,\nan edge from node 6 to node 0 with capacity 7.\nQ: What is the maximum flow from node 4 to node 0?\nA:", "answer": "The maximum flow from node 4 to node 0 is 23.", "difficulty": "easy", "doc_id": "103"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 6 with capacity 3,\nan edge from node 0 to node 1 with capacity 7,\nan edge from node 1 to node 0 with capacity 7,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 0 with capacity 8,\nan edge from node 3 to node 5 with capacity 2,\nan edge from node 4 to node 6 with capacity 3,\nan edge from node 5 to node 1 with capacity 6,\nan edge from node 5 to node 2 with capacity 6,\nan edge from node 5 to node 4 with capacity 1,\nan edge from node 5 to node 0 with capacity 4.\nQ: What is the maximum flow from node 3 to node 0?\nA:", "answer": "The maximum flow from node 3 to node 0 is 2.", "difficulty": "easy", "doc_id": "104"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 4 with capacity 18,\nan edge from node 0 to node 3 with capacity 4,\nan edge from node 1 to node 0 with capacity 1,\nan edge from node 1 to node 7 with capacity 5,\nan edge from node 2 to node 15 with capacity 4,\nan edge from node 2 to node 0 with capacity 15,\nan edge from node 2 to node 12 with capacity 8,\nan edge from node 2 to node 5 with capacity 6,\nan edge from node 2 to node 8 with capacity 13,\nan edge from node 2 to node 7 with capacity 11,\nan edge from node 3 to node 4 with capacity 6,\nan edge from node 3 to node 15 with capacity 14,\nan edge from node 3 to node 5 with capacity 17,\nan edge from node 3 to node 8 with capacity 9,\nan edge from node 3 to node 11 with capacity 8,\nan edge from node 4 to node 14 with capacity 19,\nan edge from node 4 to node 12 with capacity 5,\nan edge from node 4 to node 5 with capacity 17,\nan edge from node 4 to node 6 with capacity 2,\nan edge from node 5 to node 0 with capacity 6,\nan edge from node 5 to node 12 with capacity 5,\nan edge from node 5 to node 1 with capacity 8,\nan edge from node 5 to node 7 with capacity 18,\nan edge from node 5 to node 6 with capacity 16,\nan edge from node 6 to node 3 with capacity 15,\nan edge from node 6 to node 0 with capacity 6,\nan edge from node 7 to node 4 with capacity 1,\nan edge from node 7 to node 13 with capacity 15,\nan edge from node 7 to node 0 with capacity 8,\nan edge from node 7 to node 16 with capacity 20,\nan edge from node 7 to node 12 with capacity 18,\nan edge from node 7 to node 8 with capacity 16,\nan edge from node 8 to node 4 with capacity 3,\nan edge from node 8 to node 2 with capacity 18,\nan edge from node 8 to node 10 with capacity 7,\nan edge from node 8 to node 16 with capacity 5,\nan edge from node 8 to node 6 with capacity 10,\nan edge from node 9 to node 3 with capacity 3,\nan edge from node 9 to node 10 with capacity 7,\nan edge from node 9 to node 14 with capacity 8,\nan edge from node 9 to node 7 with capacity 9,\nan edge from node 9 to node 6 with capacity 10,\nan edge from node 10 to node 15 with capacity 15,\nan edge from node 10 to node 16 with capacity 3,\nan edge from node 10 to node 1 with capacity 6,\nan edge from node 10 to node 6 with capacity 14,\nan edge from node 10 to node 11 with capacity 11,\nan edge from node 11 to node 4 with capacity 2,\nan edge from node 11 to node 10 with capacity 12,\nan edge from node 11 to node 7 with capacity 17,\nan edge from node 12 to node 2 with capacity 14,\nan edge from node 12 to node 13 with capacity 11,\nan edge from node 12 to node 10 with capacity 3,\nan edge from node 12 to node 1 with capacity 7,\nan edge from node 12 to node 6 with capacity 4,\nan edge from node 13 to node 4 with capacity 6,\nan edge from node 13 to node 8 with capacity 5,\nan edge from node 13 to node 7 with capacity 15,\nan edge from node 14 to node 10 with capacity 16,\nan edge from node 14 to node 8 with capacity 3,\nan edge from node 14 to node 6 with capacity 19,\nan edge from node 14 to node 11 with capacity 15,\nan edge from node 15 to node 13 with capacity 10,\nan edge from node 15 to node 0 with capacity 18,\nan edge from node 15 to node 10 with capacity 9,\nan edge from node 15 to node 8 with capacity 14,\nan edge from node 15 to node 9 with capacity 20,\nan edge from node 16 to node 5 with capacity 5,\nan edge from node 16 to node 9 with capacity 12.\nQ: What is the maximum flow from node 8 to node 7?\nA:", "answer": "The maximum flow from node 8 to node 7 is 43.", "difficulty": "hard", "doc_id": "105"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 16 with capacity 12,\nan edge from node 0 to node 4 with capacity 13,\nan edge from node 0 to node 17 with capacity 5,\nan edge from node 1 to node 10 with capacity 13,\nan edge from node 1 to node 16 with capacity 7,\nan edge from node 1 to node 4 with capacity 1,\nan edge from node 1 to node 0 with capacity 1,\nan edge from node 1 to node 9 with capacity 6,\nan edge from node 1 to node 5 with capacity 8,\nan edge from node 1 to node 13 with capacity 5,\nan edge from node 1 to node 14 with capacity 3,\nan edge from node 2 to node 11 with capacity 16,\nan edge from node 2 to node 10 with capacity 5,\nan edge from node 2 to node 19 with capacity 1,\nan edge from node 2 to node 7 with capacity 5,\nan edge from node 2 to node 6 with capacity 8,\nan edge from node 3 to node 16 with capacity 15,\nan edge from node 3 to node 0 with capacity 12,\nan edge from node 3 to node 6 with capacity 17,\nan edge from node 4 to node 2 with capacity 14,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 4 to node 3 with capacity 12,\nan edge from node 4 to node 13 with capacity 3,\nan edge from node 4 to node 6 with capacity 6,\nan edge from node 5 to node 11 with capacity 8,\nan edge from node 5 to node 19 with capacity 3,\nan edge from node 5 to node 6 with capacity 17,\nan edge from node 6 to node 0 with capacity 4,\nan edge from node 6 to node 9 with capacity 9,\nan edge from node 6 to node 17 with capacity 18,\nan edge from node 6 to node 13 with capacity 13,\nan edge from node 6 to node 12 with capacity 9,\nan edge from node 6 to node 7 with capacity 1,\nan edge from node 6 to node 14 with capacity 13,\nan edge from node 6 to node 18 with capacity 19,\nan edge from node 7 to node 8 with capacity 16,\nan edge from node 7 to node 11 with capacity 3,\nan edge from node 7 to node 15 with capacity 6,\nan edge from node 7 to node 4 with capacity 15,\nan edge from node 7 to node 9 with capacity 5,\nan edge from node 7 to node 14 with capacity 16,\nan edge from node 8 to node 0 with capacity 12,\nan edge from node 8 to node 1 with capacity 1,\nan edge from node 8 to node 9 with capacity 14,\nan edge from node 8 to node 7 with capacity 13,\nan edge from node 8 to node 6 with capacity 19,\nan edge from node 8 to node 14 with capacity 8,\nan edge from node 8 to node 18 with capacity 7,\nan edge from node 9 to node 10 with capacity 10,\nan edge from node 9 to node 16 with capacity 16,\nan edge from node 9 to node 1 with capacity 17,\nan edge from node 9 to node 3 with capacity 6,\nan edge from node 9 to node 19 with capacity 18,\nan edge from node 9 to node 5 with capacity 18,\nan edge from node 9 to node 18 with capacity 12,\nan edge from node 10 to node 8 with capacity 6,\nan edge from node 10 to node 11 with capacity 19,\nan edge from node 10 to node 15 with capacity 19,\nan edge from node 11 to node 16 with capacity 14,\nan edge from node 11 to node 4 with capacity 2,\nan edge from node 11 to node 0 with capacity 13,\nan edge from node 11 to node 7 with capacity 19,\nan edge from node 12 to node 4 with capacity 18,\nan edge from node 12 to node 1 with capacity 10,\nan edge from node 12 to node 17 with capacity 18,\nan edge from node 12 to node 3 with capacity 19,\nan edge from node 12 to node 13 with capacity 10,\nan edge from node 13 to node 4 with capacity 9,\nan edge from node 13 to node 18 with capacity 7,\nan edge from node 14 to node 8 with capacity 4,\nan edge from node 14 to node 2 with capacity 8,\nan edge from node 14 to node 1 with capacity 16,\nan edge from node 14 to node 17 with capacity 8,\nan edge from node 14 to node 19 with capacity 5,\nan edge from node 14 to node 13 with capacity 18,\nan edge from node 14 to node 12 with capacity 4,\nan edge from node 15 to node 8 with capacity 14,\nan edge from node 15 to node 16 with capacity 9,\nan edge from node 15 to node 4 with capacity 10,\nan edge from node 15 to node 9 with capacity 16,\nan edge from node 15 to node 12 with capacity 9,\nan edge from node 16 to node 9 with capacity 20,\nan edge from node 16 to node 3 with capacity 1,\nan edge from node 18 to node 16 with capacity 6,\nan edge from node 18 to node 3 with capacity 6,\nan edge from node 18 to node 12 with capacity 3,\nan edge from node 18 to node 6 with capacity 14,\nan edge from node 18 to node 14 with capacity 6,\nan edge from node 19 to node 10 with capacity 14,\nan edge from node 19 to node 16 with capacity 1,\nan edge from node 19 to node 4 with capacity 20,\nan edge from node 19 to node 0 with capacity 8,\nan edge from node 19 to node 5 with capacity 16,\nan edge from node 19 to node 7 with capacity 9.\nQ: What is the maximum flow from node 5 to node 17?\nA:", "answer": "The maximum flow from node 5 to node 17 is 28.", "difficulty": "hard", "doc_id": "106"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 4 with capacity 17,\nan edge from node 0 to node 3 with capacity 19,\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 0 to node 5 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 1 to node 7 with capacity 12,\nan edge from node 1 to node 5 with capacity 8,\nan edge from node 2 to node 11 with capacity 2,\nan edge from node 3 to node 9 with capacity 2,\nan edge from node 3 to node 1 with capacity 11,\nan edge from node 4 to node 0 with capacity 15,\nan edge from node 4 to node 10 with capacity 5,\nan edge from node 4 to node 1 with capacity 18,\nan edge from node 5 to node 9 with capacity 17,\nan edge from node 5 to node 3 with capacity 17,\nan edge from node 6 to node 9 with capacity 10,\nan edge from node 6 to node 7 with capacity 19,\nan edge from node 7 to node 8 with capacity 9,\nan edge from node 7 to node 1 with capacity 13,\nan edge from node 7 to node 5 with capacity 9,\nan edge from node 8 to node 0 with capacity 14,\nan edge from node 8 to node 2 with capacity 9,\nan edge from node 8 to node 4 with capacity 19,\nan edge from node 8 to node 9 with capacity 16,\nan edge from node 8 to node 11 with capacity 11,\nan edge from node 8 to node 3 with capacity 11,\nan edge from node 9 to node 4 with capacity 19,\nan edge from node 9 to node 10 with capacity 19,\nan edge from node 9 to node 11 with capacity 12,\nan edge from node 9 to node 3 with capacity 18,\nan edge from node 10 to node 4 with capacity 1,\nan edge from node 10 to node 8 with capacity 11,\nan edge from node 10 to node 7 with capacity 6,\nan edge from node 11 to node 12 with capacity 7,\nan edge from node 11 to node 7 with capacity 10,\nan edge from node 12 to node 4 with capacity 14,\nan edge from node 12 to node 10 with capacity 19,\nan edge from node 12 to node 1 with capacity 1,\nan edge from node 12 to node 7 with capacity 19,\nan edge from node 12 to node 5 with capacity 10.\nQ: What is the maximum flow from node 5 to node 2?\nA:", "answer": "The maximum flow from node 5 to node 2 is 9.", "difficulty": "hard", "doc_id": "107"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 5 with capacity 6,\nan edge from node 0 to node 7 with capacity 12,\nan edge from node 1 to node 8 with capacity 2,\nan edge from node 1 to node 3 with capacity 1,\nan edge from node 2 to node 9 with capacity 13,\nan edge from node 2 to node 10 with capacity 10,\nan edge from node 2 to node 6 with capacity 11,\nan edge from node 3 to node 11 with capacity 14,\nan edge from node 3 to node 5 with capacity 16,\nan edge from node 3 to node 10 with capacity 12,\nan edge from node 4 to node 1 with capacity 12,\nan edge from node 4 to node 10 with capacity 9,\nan edge from node 5 to node 11 with capacity 18,\nan edge from node 5 to node 4 with capacity 12,\nan edge from node 5 to node 7 with capacity 10,\nan edge from node 6 to node 5 with capacity 12,\nan edge from node 7 to node 3 with capacity 7,\nan edge from node 7 to node 6 with capacity 12,\nan edge from node 8 to node 11 with capacity 6,\nan edge from node 8 to node 9 with capacity 13,\nan edge from node 9 to node 3 with capacity 20,\nan edge from node 10 to node 5 with capacity 3,\nan edge from node 10 to node 3 with capacity 9,\nan edge from node 10 to node 4 with capacity 14,\nan edge from node 11 to node 10 with capacity 20.\nQ: What is the maximum flow from node 5 to node 4?\nA:", "answer": "The maximum flow from node 5 to node 4 is 26.", "difficulty": "hard", "doc_id": "108"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 1 with capacity 5,\nan edge from node 0 to node 15 with capacity 3,\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 1 to node 8 with capacity 3,\nan edge from node 1 to node 6 with capacity 13,\nan edge from node 2 to node 0 with capacity 7,\nan edge from node 2 to node 13 with capacity 18,\nan edge from node 2 to node 1 with capacity 20,\nan edge from node 2 to node 3 with capacity 4,\nan edge from node 3 to node 13 with capacity 11,\nan edge from node 3 to node 10 with capacity 20,\nan edge from node 3 to node 1 with capacity 4,\nan edge from node 3 to node 2 with capacity 15,\nan edge from node 3 to node 7 with capacity 1,\nan edge from node 3 to node 15 with capacity 8,\nan edge from node 4 to node 1 with capacity 10,\nan edge from node 4 to node 8 with capacity 13,\nan edge from node 4 to node 9 with capacity 4,\nan edge from node 4 to node 7 with capacity 19,\nan edge from node 4 to node 15 with capacity 11,\nan edge from node 5 to node 2 with capacity 6,\nan edge from node 5 to node 3 with capacity 7,\nan edge from node 5 to node 15 with capacity 17,\nan edge from node 6 to node 0 with capacity 3,\nan edge from node 6 to node 11 with capacity 20,\nan edge from node 6 to node 10 with capacity 13,\nan edge from node 6 to node 3 with capacity 17,\nan edge from node 7 to node 6 with capacity 17,\nan edge from node 7 to node 5 with capacity 15,\nan edge from node 8 to node 0 with capacity 11,\nan edge from node 8 to node 11 with capacity 8,\nan edge from node 8 to node 14 with capacity 15,\nan edge from node 8 to node 2 with capacity 10,\nan edge from node 9 to node 1 with capacity 8,\nan edge from node 9 to node 14 with capacity 2,\nan edge from node 9 to node 6 with capacity 8,\nan edge from node 9 to node 7 with capacity 1,\nan edge from node 10 to node 1 with capacity 17,\nan edge from node 11 to node 13 with capacity 17,\nan edge from node 11 to node 8 with capacity 18,\nan edge from node 11 to node 5 with capacity 18,\nan edge from node 11 to node 15 with capacity 6,\nan edge from node 12 to node 0 with capacity 2,\nan edge from node 12 to node 14 with capacity 3,\nan edge from node 12 to node 5 with capacity 15,\nan edge from node 13 to node 10 with capacity 15,\nan edge from node 14 to node 0 with capacity 7,\nan edge from node 15 to node 2 with capacity 13,\nan edge from node 15 to node 6 with capacity 15,\nan edge from node 15 to node 7 with capacity 10.\nQ: What is the maximum flow from node 6 to node 14?\nA:", "answer": "The maximum flow from node 6 to node 14 is 15.", "difficulty": "hard", "doc_id": "109"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 11 with capacity 17,\nan edge from node 0 to node 2 with capacity 16,\nan edge from node 1 to node 9 with capacity 11,\nan edge from node 1 to node 0 with capacity 15,\nan edge from node 1 to node 2 with capacity 9,\nan edge from node 2 to node 3 with capacity 15,\nan edge from node 3 to node 8 with capacity 7,\nan edge from node 4 to node 0 with capacity 15,\nan edge from node 4 to node 1 with capacity 4,\nan edge from node 4 to node 3 with capacity 19,\nan edge from node 4 to node 7 with capacity 16,\nan edge from node 5 to node 9 with capacity 10,\nan edge from node 5 to node 2 with capacity 13,\nan edge from node 6 to node 11 with capacity 3,\nan edge from node 7 to node 10 with capacity 15,\nan edge from node 7 to node 11 with capacity 15,\nan edge from node 7 to node 6 with capacity 10,\nan edge from node 7 to node 0 with capacity 10,\nan edge from node 7 to node 8 with capacity 5,\nan edge from node 7 to node 3 with capacity 16,\nan edge from node 7 to node 2 with capacity 10,\nan edge from node 8 to node 4 with capacity 2,\nan edge from node 8 to node 11 with capacity 15,\nan edge from node 8 to node 6 with capacity 8,\nan edge from node 8 to node 3 with capacity 6,\nan edge from node 8 to node 2 with capacity 16,\nan edge from node 9 to node 1 with capacity 8,\nan edge from node 9 to node 3 with capacity 18,\nan edge from node 9 to node 2 with capacity 6,\nan edge from node 10 to node 4 with capacity 17,\nan edge from node 10 to node 5 with capacity 13,\nan edge from node 10 to node 1 with capacity 10,\nan edge from node 10 to node 3 with capacity 5,\nan edge from node 10 to node 2 with capacity 1,\nan edge from node 11 to node 4 with capacity 3,\nan edge from node 11 to node 6 with capacity 3,\nan edge from node 11 to node 9 with capacity 9,\nan edge from node 11 to node 8 with capacity 16,\nan edge from node 11 to node 1 with capacity 3,\nan edge from node 11 to node 3 with capacity 7.\nQ: What is the maximum flow from node 4 to node 3?\nA:", "answer": "The maximum flow from node 4 to node 3 is 54.", "difficulty": "hard", "doc_id": "110"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 5 with capacity 6,\nan edge from node 0 to node 6 with capacity 7,\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 2 to node 9 with capacity 3,\nan edge from node 4 to node 3 with capacity 7,\nan edge from node 5 to node 2 with capacity 9,\nan edge from node 5 to node 1 with capacity 2,\nan edge from node 6 to node 8 with capacity 6,\nan edge from node 6 to node 7 with capacity 10,\nan edge from node 7 to node 3 with capacity 6,\nan edge from node 8 to node 3 with capacity 8,\nan edge from node 8 to node 1 with capacity 7,\nan edge from node 9 to node 0 with capacity 5.\nQ: What is the maximum flow from node 9 to node 3?\nA:", "answer": "The maximum flow from node 9 to node 3 is 5.", "difficulty": "easy", "doc_id": "111"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 0 to node 1 with capacity 7,\nan edge from node 1 to node 0 with capacity 9,\nan edge from node 1 to node 7 with capacity 2,\nan edge from node 2 to node 1 with capacity 3,\nan edge from node 2 to node 7 with capacity 10,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 3 to node 8 with capacity 2,\nan edge from node 3 to node 7 with capacity 6,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 5 to node 6 with capacity 7,\nan edge from node 5 to node 1 with capacity 6,\nan edge from node 5 to node 3 with capacity 1,\nan edge from node 7 to node 2 with capacity 9,\nan edge from node 7 to node 1 with capacity 6,\nan edge from node 8 to node 6 with capacity 1,\nan edge from node 8 to node 0 with capacity 7,\nan edge from node 8 to node 9 with capacity 9,\nan edge from node 9 to node 6 with capacity 5,\nan edge from node 9 to node 3 with capacity 5,\nan edge from node 9 to node 5 with capacity 10.\nQ: What is the maximum flow from node 0 to node 1?\nA:", "answer": "The maximum flow from node 0 to node 1 is 13.", "difficulty": "easy", "doc_id": "112"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 16 with capacity 19,\nan edge from node 0 to node 9 with capacity 6,\nan edge from node 0 to node 7 with capacity 1,\nan edge from node 1 to node 4 with capacity 1,\nan edge from node 1 to node 10 with capacity 16,\nan edge from node 1 to node 6 with capacity 10,\nan edge from node 1 to node 5 with capacity 2,\nan edge from node 1 to node 19 with capacity 5,\nan edge from node 1 to node 2 with capacity 11,\nan edge from node 2 to node 3 with capacity 5,\nan edge from node 2 to node 7 with capacity 11,\nan edge from node 2 to node 13 with capacity 9,\nan edge from node 3 to node 9 with capacity 3,\nan edge from node 3 to node 5 with capacity 19,\nan edge from node 3 to node 19 with capacity 10,\nan edge from node 3 to node 13 with capacity 12,\nan edge from node 3 to node 17 with capacity 15,\nan edge from node 4 to node 16 with capacity 6,\nan edge from node 4 to node 10 with capacity 10,\nan edge from node 4 to node 7 with capacity 11,\nan edge from node 4 to node 18 with capacity 14,\nan edge from node 4 to node 13 with capacity 14,\nan edge from node 5 to node 4 with capacity 17,\nan edge from node 5 to node 9 with capacity 19,\nan edge from node 5 to node 11 with capacity 19,\nan edge from node 5 to node 7 with capacity 16,\nan edge from node 5 to node 1 with capacity 18,\nan edge from node 6 to node 16 with capacity 11,\nan edge from node 6 to node 14 with capacity 18,\nan edge from node 6 to node 5 with capacity 15,\nan edge from node 6 to node 7 with capacity 11,\nan edge from node 6 to node 18 with capacity 6,\nan edge from node 6 to node 1 with capacity 11,\nan edge from node 6 to node 2 with capacity 4,\nan edge from node 7 to node 0 with capacity 18,\nan edge from node 7 to node 14 with capacity 9,\nan edge from node 7 to node 9 with capacity 15,\nan edge from node 7 to node 3 with capacity 17,\nan edge from node 7 to node 15 with capacity 7,\nan edge from node 7 to node 19 with capacity 5,\nan edge from node 7 to node 8 with capacity 17,\nan edge from node 8 to node 14 with capacity 16,\nan edge from node 8 to node 6 with capacity 12,\nan edge from node 8 to node 15 with capacity 8,\nan edge from node 8 to node 13 with capacity 11,\nan edge from node 9 to node 16 with capacity 5,\nan edge from node 9 to node 0 with capacity 16,\nan edge from node 9 to node 4 with capacity 16,\nan edge from node 10 to node 4 with capacity 20,\nan edge from node 10 to node 9 with capacity 18,\nan edge from node 10 to node 5 with capacity 19,\nan edge from node 11 to node 7 with capacity 3,\nan edge from node 11 to node 18 with capacity 8,\nan edge from node 11 to node 8 with capacity 12,\nan edge from node 12 to node 4 with capacity 11,\nan edge from node 12 to node 11 with capacity 13,\nan edge from node 12 to node 2 with capacity 4,\nan edge from node 13 to node 0 with capacity 6,\nan edge from node 13 to node 4 with capacity 16,\nan edge from node 13 to node 10 with capacity 11,\nan edge from node 13 to node 6 with capacity 4,\nan edge from node 13 to node 3 with capacity 13,\nan edge from node 13 to node 7 with capacity 7,\nan edge from node 13 to node 17 with capacity 18,\nan edge from node 14 to node 10 with capacity 5,\nan edge from node 14 to node 5 with capacity 8,\nan edge from node 14 to node 8 with capacity 17,\nan edge from node 15 to node 14 with capacity 1,\nan edge from node 15 to node 9 with capacity 11,\nan edge from node 15 to node 7 with capacity 2,\nan edge from node 15 to node 19 with capacity 3,\nan edge from node 15 to node 17 with capacity 16,\nan edge from node 16 to node 4 with capacity 16,\nan edge from node 16 to node 10 with capacity 1,\nan edge from node 16 to node 9 with capacity 3,\nan edge from node 16 to node 5 with capacity 13,\nan edge from node 16 to node 7 with capacity 4,\nan edge from node 16 to node 13 with capacity 2,\nan edge from node 17 to node 14 with capacity 11,\nan edge from node 17 to node 10 with capacity 18,\nan edge from node 17 to node 9 with capacity 20,\nan edge from node 17 to node 11 with capacity 14,\nan edge from node 17 to node 15 with capacity 10,\nan edge from node 17 to node 13 with capacity 11,\nan edge from node 17 to node 8 with capacity 20,\nan edge from node 17 to node 2 with capacity 6,\nan edge from node 18 to node 0 with capacity 4,\nan edge from node 18 to node 11 with capacity 12,\nan edge from node 19 to node 16 with capacity 1,\nan edge from node 19 to node 0 with capacity 9,\nan edge from node 19 to node 4 with capacity 10,\nan edge from node 19 to node 9 with capacity 4,\nan edge from node 19 to node 3 with capacity 8,\nan edge from node 19 to node 7 with capacity 1,\nan edge from node 19 to node 1 with capacity 2,\nan edge from node 19 to node 13 with capacity 11.\nQ: What is the maximum flow from node 6 to node 3?\nA:", "answer": "The maximum flow from node 6 to node 3 is 43.", "difficulty": "hard", "doc_id": "113"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 3,\nan edge from node 1 to node 4 with capacity 4,\nan edge from node 1 to node 3 with capacity 9,\nan edge from node 2 to node 4 with capacity 2,\nan edge from node 2 to node 3 with capacity 9,\nan edge from node 4 to node 2 with capacity 1,\nan edge from node 4 to node 3 with capacity 6,\nan edge from node 4 to node 0 with capacity 2,\nan edge from node 5 to node 4 with capacity 9,\nan edge from node 5 to node 3 with capacity 7.\nQ: What is the maximum flow from node 1 to node 3?\nA:", "answer": "The maximum flow from node 1 to node 3 is 13.", "difficulty": "easy", "doc_id": "114"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 8 with capacity 6,\nan edge from node 0 to node 12 with capacity 5,\nan edge from node 0 to node 2 with capacity 11,\nan edge from node 0 to node 9 with capacity 1,\nan edge from node 0 to node 11 with capacity 17,\nan edge from node 0 to node 1 with capacity 11,\nan edge from node 1 to node 3 with capacity 19,\nan edge from node 1 to node 9 with capacity 10,\nan edge from node 1 to node 11 with capacity 19,\nan edge from node 1 to node 6 with capacity 19,\nan edge from node 2 to node 3 with capacity 1,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 7 with capacity 5,\nan edge from node 3 to node 8 with capacity 2,\nan edge from node 3 to node 12 with capacity 18,\nan edge from node 3 to node 2 with capacity 2,\nan edge from node 3 to node 9 with capacity 15,\nan edge from node 4 to node 13 with capacity 6,\nan edge from node 4 to node 8 with capacity 8,\nan edge from node 4 to node 2 with capacity 19,\nan edge from node 4 to node 0 with capacity 6,\nan edge from node 5 to node 1 with capacity 18,\nan edge from node 6 to node 3 with capacity 15,\nan edge from node 6 to node 10 with capacity 3,\nan edge from node 6 to node 1 with capacity 11,\nan edge from node 6 to node 5 with capacity 4,\nan edge from node 7 to node 9 with capacity 3,\nan edge from node 8 to node 1 with capacity 4,\nan edge from node 8 to node 0 with capacity 5,\nan edge from node 9 to node 13 with capacity 5,\nan edge from node 9 to node 7 with capacity 3,\nan edge from node 9 to node 8 with capacity 20,\nan edge from node 9 to node 6 with capacity 10,\nan edge from node 9 to node 1 with capacity 9,\nan edge from node 10 to node 13 with capacity 14,\nan edge from node 10 to node 9 with capacity 17,\nan edge from node 10 to node 4 with capacity 20,\nan edge from node 10 to node 1 with capacity 12,\nan edge from node 11 to node 2 with capacity 15,\nan edge from node 11 to node 4 with capacity 7,\nan edge from node 12 to node 2 with capacity 18,\nan edge from node 12 to node 9 with capacity 6,\nan edge from node 12 to node 11 with capacity 18,\nan edge from node 12 to node 10 with capacity 10,\nan edge from node 12 to node 6 with capacity 1,\nan edge from node 12 to node 1 with capacity 2,\nan edge from node 13 to node 8 with capacity 20,\nan edge from node 13 to node 12 with capacity 15,\nan edge from node 13 to node 11 with capacity 8,\nan edge from node 13 to node 10 with capacity 16,\nan edge from node 13 to node 1 with capacity 14.\nQ: What is the maximum flow from node 5 to node 3?\nA:", "answer": "The maximum flow from node 5 to node 3 is 18.", "difficulty": "hard", "doc_id": "115"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 16 with capacity 12,\nan edge from node 0 to node 10 with capacity 3,\nan edge from node 0 to node 1 with capacity 16,\nan edge from node 0 to node 3 with capacity 2,\nan edge from node 0 to node 9 with capacity 1,\nan edge from node 1 to node 17 with capacity 4,\nan edge from node 1 to node 5 with capacity 20,\nan edge from node 1 to node 12 with capacity 3,\nan edge from node 1 to node 14 with capacity 7,\nan edge from node 1 to node 11 with capacity 7,\nan edge from node 2 to node 16 with capacity 12,\nan edge from node 2 to node 10 with capacity 8,\nan edge from node 2 to node 12 with capacity 2,\nan edge from node 2 to node 9 with capacity 20,\nan edge from node 3 to node 16 with capacity 19,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 3 to node 13 with capacity 18,\nan edge from node 4 to node 17 with capacity 15,\nan edge from node 4 to node 10 with capacity 2,\nan edge from node 4 to node 6 with capacity 17,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 5 to node 7 with capacity 2,\nan edge from node 5 to node 6 with capacity 15,\nan edge from node 5 to node 13 with capacity 13,\nan edge from node 5 to node 9 with capacity 8,\nan edge from node 6 to node 7 with capacity 13,\nan edge from node 6 to node 13 with capacity 7,\nan edge from node 6 to node 12 with capacity 6,\nan edge from node 6 to node 9 with capacity 7,\nan edge from node 7 to node 5 with capacity 7,\nan edge from node 7 to node 2 with capacity 13,\nan edge from node 7 to node 9 with capacity 19,\nan edge from node 8 to node 10 with capacity 3,\nan edge from node 8 to node 4 with capacity 6,\nan edge from node 8 to node 13 with capacity 6,\nan edge from node 8 to node 12 with capacity 9,\nan edge from node 9 to node 2 with capacity 13,\nan edge from node 9 to node 1 with capacity 14,\nan edge from node 9 to node 0 with capacity 15,\nan edge from node 9 to node 11 with capacity 11,\nan edge from node 10 to node 5 with capacity 15,\nan edge from node 10 to node 2 with capacity 15,\nan edge from node 10 to node 6 with capacity 1,\nan edge from node 10 to node 4 with capacity 18,\nan edge from node 10 to node 11 with capacity 8,\nan edge from node 11 to node 6 with capacity 18,\nan edge from node 11 to node 15 with capacity 9,\nan edge from node 12 to node 0 with capacity 14,\nan edge from node 12 to node 13 with capacity 8,\nan edge from node 13 to node 5 with capacity 16,\nan edge from node 13 to node 10 with capacity 14,\nan edge from node 13 to node 2 with capacity 1,\nan edge from node 13 to node 4 with capacity 10,\nan edge from node 13 to node 15 with capacity 20,\nan edge from node 13 to node 14 with capacity 18,\nan edge from node 14 to node 5 with capacity 3,\nan edge from node 14 to node 16 with capacity 5,\nan edge from node 14 to node 10 with capacity 16,\nan edge from node 14 to node 1 with capacity 7,\nan edge from node 14 to node 0 with capacity 19,\nan edge from node 14 to node 11 with capacity 1,\nan edge from node 14 to node 3 with capacity 20,\nan edge from node 15 to node 7 with capacity 7,\nan edge from node 15 to node 8 with capacity 7,\nan edge from node 15 to node 14 with capacity 11,\nan edge from node 16 to node 8 with capacity 12,\nan edge from node 16 to node 13 with capacity 13,\nan edge from node 16 to node 12 with capacity 19,\nan edge from node 17 to node 2 with capacity 19,\nan edge from node 17 to node 1 with capacity 4,\nan edge from node 17 to node 0 with capacity 18,\nan edge from node 17 to node 15 with capacity 2,\nan edge from node 17 to node 12 with capacity 19,\nan edge from node 17 to node 9 with capacity 12.\nQ: What is the maximum flow from node 15 to node 16?\nA:", "answer": "The maximum flow from node 15 to node 16 is 25.", "difficulty": "hard", "doc_id": "116"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 5 with capacity 19,\nan edge from node 0 to node 13 with capacity 19,\nan edge from node 0 to node 10 with capacity 4,\nan edge from node 0 to node 15 with capacity 4,\nan edge from node 0 to node 4 with capacity 17,\nan edge from node 1 to node 5 with capacity 6,\nan edge from node 1 to node 7 with capacity 1,\nan edge from node 1 to node 8 with capacity 7,\nan edge from node 1 to node 3 with capacity 1,\nan edge from node 1 to node 6 with capacity 14,\nan edge from node 2 to node 5 with capacity 5,\nan edge from node 2 to node 11 with capacity 1,\nan edge from node 2 to node 3 with capacity 4,\nan edge from node 2 to node 1 with capacity 18,\nan edge from node 3 to node 8 with capacity 12,\nan edge from node 3 to node 18 with capacity 18,\nan edge from node 3 to node 14 with capacity 1,\nan edge from node 3 to node 16 with capacity 14,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 17 with capacity 5,\nan edge from node 3 to node 0 with capacity 19,\nan edge from node 4 to node 11 with capacity 1,\nan edge from node 4 to node 3 with capacity 7,\nan edge from node 4 to node 12 with capacity 6,\nan edge from node 4 to node 13 with capacity 12,\nan edge from node 4 to node 10 with capacity 13,\nan edge from node 4 to node 1 with capacity 6,\nan edge from node 4 to node 16 with capacity 1,\nan edge from node 5 to node 11 with capacity 11,\nan edge from node 5 to node 7 with capacity 20,\nan edge from node 5 to node 1 with capacity 17,\nan edge from node 5 to node 16 with capacity 20,\nan edge from node 6 to node 11 with capacity 20,\nan edge from node 6 to node 3 with capacity 6,\nan edge from node 6 to node 18 with capacity 6,\nan edge from node 7 to node 9 with capacity 16,\nan edge from node 7 to node 2 with capacity 15,\nan edge from node 7 to node 3 with capacity 17,\nan edge from node 7 to node 14 with capacity 11,\nan edge from node 7 to node 15 with capacity 6,\nan edge from node 7 to node 17 with capacity 12,\nan edge from node 8 to node 7 with capacity 19,\nan edge from node 8 to node 6 with capacity 20,\nan edge from node 8 to node 1 with capacity 7,\nan edge from node 8 to node 16 with capacity 3,\nan edge from node 8 to node 4 with capacity 2,\nan edge from node 8 to node 17 with capacity 12,\nan edge from node 9 to node 2 with capacity 20,\nan edge from node 9 to node 8 with capacity 2,\nan edge from node 9 to node 6 with capacity 17,\nan edge from node 9 to node 13 with capacity 2,\nan edge from node 9 to node 1 with capacity 20,\nan edge from node 9 to node 4 with capacity 13,\nan edge from node 9 to node 17 with capacity 9,\nan edge from node 10 to node 9 with capacity 18,\nan edge from node 10 to node 5 with capacity 18,\nan edge from node 10 to node 11 with capacity 16,\nan edge from node 11 to node 9 with capacity 20,\nan edge from node 11 to node 5 with capacity 10,\nan edge from node 11 to node 7 with capacity 18,\nan edge from node 11 to node 10 with capacity 12,\nan edge from node 11 to node 18 with capacity 6,\nan edge from node 11 to node 16 with capacity 10,\nan edge from node 12 to node 9 with capacity 18,\nan edge from node 12 to node 3 with capacity 7,\nan edge from node 12 to node 1 with capacity 1,\nan edge from node 13 to node 9 with capacity 4,\nan edge from node 13 to node 2 with capacity 4,\nan edge from node 13 to node 5 with capacity 17,\nan edge from node 13 to node 14 with capacity 12,\nan edge from node 13 to node 17 with capacity 8,\nan edge from node 14 to node 11 with capacity 10,\nan edge from node 14 to node 3 with capacity 5,\nan edge from node 14 to node 18 with capacity 5,\nan edge from node 15 to node 3 with capacity 2,\nan edge from node 15 to node 1 with capacity 9,\nan edge from node 15 to node 0 with capacity 13,\nan edge from node 16 to node 11 with capacity 5,\nan edge from node 16 to node 8 with capacity 4,\nan edge from node 16 to node 18 with capacity 16,\nan edge from node 16 to node 0 with capacity 15,\nan edge from node 17 to node 7 with capacity 8,\nan edge from node 17 to node 18 with capacity 12,\nan edge from node 17 to node 14 with capacity 13,\nan edge from node 18 to node 8 with capacity 14,\nan edge from node 18 to node 12 with capacity 12,\nan edge from node 18 to node 14 with capacity 13,\nan edge from node 18 to node 4 with capacity 15,\nan edge from node 18 to node 0 with capacity 6.\nQ: What is the maximum flow from node 6 to node 13?\nA:", "answer": "The maximum flow from node 6 to node 13 is 32.", "difficulty": "hard", "doc_id": "117"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 13 with capacity 5,\nan edge from node 0 to node 3 with capacity 15,\nan edge from node 0 to node 5 with capacity 1,\nan edge from node 0 to node 1 with capacity 16,\nan edge from node 0 to node 8 with capacity 18,\nan edge from node 0 to node 11 with capacity 5,\nan edge from node 1 to node 13 with capacity 3,\nan edge from node 1 to node 5 with capacity 2,\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 1 to node 6 with capacity 11,\nan edge from node 2 to node 3 with capacity 18,\nan edge from node 2 to node 15 with capacity 8,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 2 to node 5 with capacity 6,\nan edge from node 2 to node 10 with capacity 12,\nan edge from node 2 to node 0 with capacity 12,\nan edge from node 3 to node 16 with capacity 20,\nan edge from node 3 to node 14 with capacity 7,\nan edge from node 3 to node 12 with capacity 1,\nan edge from node 3 to node 6 with capacity 17,\nan edge from node 4 to node 15 with capacity 15,\nan edge from node 4 to node 9 with capacity 20,\nan edge from node 4 to node 16 with capacity 4,\nan edge from node 4 to node 10 with capacity 2,\nan edge from node 4 to node 12 with capacity 2,\nan edge from node 4 to node 0 with capacity 17,\nan edge from node 5 to node 13 with capacity 3,\nan edge from node 5 to node 3 with capacity 4,\nan edge from node 5 to node 7 with capacity 11,\nan edge from node 5 to node 9 with capacity 17,\nan edge from node 5 to node 16 with capacity 17,\nan edge from node 5 to node 17 with capacity 14,\nan edge from node 5 to node 14 with capacity 12,\nan edge from node 5 to node 6 with capacity 20,\nan edge from node 6 to node 7 with capacity 5,\nan edge from node 6 to node 18 with capacity 2,\nan edge from node 6 to node 10 with capacity 13,\nan edge from node 6 to node 12 with capacity 18,\nan edge from node 6 to node 11 with capacity 14,\nan edge from node 6 to node 2 with capacity 5,\nan edge from node 7 to node 4 with capacity 2,\nan edge from node 7 to node 9 with capacity 10,\nan edge from node 7 to node 16 with capacity 10,\nan edge from node 7 to node 0 with capacity 19,\nan edge from node 7 to node 6 with capacity 5,\nan edge from node 8 to node 4 with capacity 16,\nan edge from node 8 to node 9 with capacity 9,\nan edge from node 8 to node 10 with capacity 11,\nan edge from node 8 to node 14 with capacity 3,\nan edge from node 8 to node 12 with capacity 17,\nan edge from node 9 to node 13 with capacity 12,\nan edge from node 9 to node 10 with capacity 9,\nan edge from node 9 to node 2 with capacity 10,\nan edge from node 9 to node 19 with capacity 3,\nan edge from node 9 to node 6 with capacity 13,\nan edge from node 10 to node 13 with capacity 18,\nan edge from node 10 to node 3 with capacity 19,\nan edge from node 10 to node 15 with capacity 18,\nan edge from node 10 to node 5 with capacity 20,\nan edge from node 10 to node 16 with capacity 12,\nan edge from node 10 to node 0 with capacity 2,\nan edge from node 10 to node 6 with capacity 1,\nan edge from node 11 to node 17 with capacity 6,\nan edge from node 11 to node 18 with capacity 14,\nan edge from node 11 to node 10 with capacity 13,\nan edge from node 11 to node 14 with capacity 3,\nan edge from node 12 to node 18 with capacity 19,\nan edge from node 12 to node 14 with capacity 11,\nan edge from node 12 to node 8 with capacity 13,\nan edge from node 12 to node 11 with capacity 20,\nan edge from node 13 to node 3 with capacity 17,\nan edge from node 13 to node 15 with capacity 9,\nan edge from node 13 to node 5 with capacity 4,\nan edge from node 13 to node 14 with capacity 5,\nan edge from node 13 to node 12 with capacity 20,\nan edge from node 13 to node 11 with capacity 9,\nan edge from node 13 to node 2 with capacity 14,\nan edge from node 13 to node 19 with capacity 18,\nan edge from node 14 to node 3 with capacity 4,\nan edge from node 14 to node 4 with capacity 15,\nan edge from node 14 to node 18 with capacity 15,\nan edge from node 15 to node 7 with capacity 7,\nan edge from node 15 to node 5 with capacity 20,\nan edge from node 15 to node 1 with capacity 2,\nan edge from node 15 to node 14 with capacity 15,\nan edge from node 15 to node 2 with capacity 8,\nan edge from node 15 to node 6 with capacity 5,\nan edge from node 16 to node 13 with capacity 20,\nan edge from node 16 to node 7 with capacity 13,\nan edge from node 16 to node 12 with capacity 4,\nan edge from node 16 to node 11 with capacity 2,\nan edge from node 17 to node 3 with capacity 17,\nan edge from node 17 to node 7 with capacity 11,\nan edge from node 17 to node 18 with capacity 16,\nan edge from node 17 to node 10 with capacity 6,\nan edge from node 17 to node 8 with capacity 13,\nan edge from node 17 to node 0 with capacity 11,\nan edge from node 18 to node 3 with capacity 13,\nan edge from node 18 to node 15 with capacity 18,\nan edge from node 18 to node 4 with capacity 13,\nan edge from node 18 to node 17 with capacity 8,\nan edge from node 18 to node 12 with capacity 9,\nan edge from node 18 to node 19 with capacity 15,\nan edge from node 18 to node 6 with capacity 14,\nan edge from node 19 to node 7 with capacity 4,\nan edge from node 19 to node 17 with capacity 3,\nan edge from node 19 to node 12 with capacity 15.\nQ: What is the maximum flow from node 8 to node 7?\nA:", "answer": "The maximum flow from node 8 to node 7 is 51.", "difficulty": "hard", "doc_id": "118"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 1,\nan edge from node 0 to node 2 with capacity 9,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 1 to node 4 with capacity 2,\nan edge from node 1 to node 0 with capacity 1,\nan edge from node 1 to node 2 with capacity 3,\nan edge from node 3 to node 0 with capacity 1,\nan edge from node 4 to node 0 with capacity 3.\nQ: What is the maximum flow from node 1 to node 0?\nA:", "answer": "The maximum flow from node 1 to node 0 is 4.", "difficulty": "easy", "doc_id": "119"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 1 with capacity 8,\nan edge from node 1 to node 2 with capacity 1,\nan edge from node 2 to node 5 with capacity 7,\nan edge from node 2 to node 3 with capacity 7,\nan edge from node 3 to node 5 with capacity 8,\nan edge from node 3 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 0 with capacity 3,\nan edge from node 5 to node 3 with capacity 8,\nan edge from node 5 to node 1 with capacity 3.\nQ: What is the maximum flow from node 2 to node 1?\nA:", "answer": "The maximum flow from node 2 to node 1 is 8.", "difficulty": "easy", "doc_id": "120"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 3 with capacity 10,\nan edge from node 0 to node 1 with capacity 7,\nan edge from node 1 to node 4 with capacity 1,\nan edge from node 1 to node 2 with capacity 1,\nan edge from node 2 to node 0 with capacity 6,\nan edge from node 3 to node 7 with capacity 3,\nan edge from node 4 to node 3 with capacity 7,\nan edge from node 5 to node 3 with capacity 4,\nan edge from node 5 to node 2 with capacity 3,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 7 to node 0 with capacity 4,\nan edge from node 7 to node 5 with capacity 6.\nQ: What is the maximum flow from node 1 to node 0?\nA:", "answer": "The maximum flow from node 1 to node 0 is 2.", "difficulty": "easy", "doc_id": "121"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 2 with capacity 11,\nan edge from node 0 to node 18 with capacity 6,\nan edge from node 0 to node 15 with capacity 9,\nan edge from node 0 to node 17 with capacity 3,\nan edge from node 1 to node 7 with capacity 14,\nan edge from node 1 to node 15 with capacity 3,\nan edge from node 1 to node 12 with capacity 2,\nan edge from node 1 to node 10 with capacity 12,\nan edge from node 2 to node 5 with capacity 7,\nan edge from node 2 to node 6 with capacity 17,\nan edge from node 2 to node 15 with capacity 4,\nan edge from node 2 to node 14 with capacity 5,\nan edge from node 2 to node 13 with capacity 12,\nan edge from node 3 to node 6 with capacity 18,\nan edge from node 3 to node 0 with capacity 14,\nan edge from node 3 to node 15 with capacity 10,\nan edge from node 3 to node 16 with capacity 14,\nan edge from node 3 to node 17 with capacity 7,\nan edge from node 4 to node 7 with capacity 3,\nan edge from node 4 to node 5 with capacity 11,\nan edge from node 4 to node 2 with capacity 11,\nan edge from node 4 to node 18 with capacity 4,\nan edge from node 4 to node 15 with capacity 4,\nan edge from node 4 to node 13 with capacity 3,\nan edge from node 4 to node 16 with capacity 15,\nan edge from node 4 to node 17 with capacity 20,\nan edge from node 5 to node 11 with capacity 15,\nan edge from node 6 to node 7 with capacity 20,\nan edge from node 6 to node 2 with capacity 3,\nan edge from node 6 to node 19 with capacity 19,\nan edge from node 6 to node 18 with capacity 18,\nan edge from node 6 to node 0 with capacity 19,\nan edge from node 6 to node 14 with capacity 5,\nan edge from node 6 to node 12 with capacity 16,\nan edge from node 6 to node 10 with capacity 8,\nan edge from node 6 to node 17 with capacity 2,\nan edge from node 7 to node 6 with capacity 2,\nan edge from node 7 to node 15 with capacity 2,\nan edge from node 7 to node 10 with capacity 2,\nan edge from node 7 to node 9 with capacity 6,\nan edge from node 8 to node 11 with capacity 18,\nan edge from node 8 to node 13 with capacity 5,\nan edge from node 8 to node 9 with capacity 7,\nan edge from node 9 to node 19 with capacity 18,\nan edge from node 9 to node 18 with capacity 2,\nan edge from node 9 to node 11 with capacity 9,\nan edge from node 9 to node 15 with capacity 14,\nan edge from node 10 to node 7 with capacity 6,\nan edge from node 10 to node 19 with capacity 2,\nan edge from node 10 to node 0 with capacity 16,\nan edge from node 10 to node 15 with capacity 7,\nan edge from node 10 to node 8 with capacity 4,\nan edge from node 10 to node 1 with capacity 10,\nan edge from node 11 to node 0 with capacity 15,\nan edge from node 11 to node 16 with capacity 5,\nan edge from node 11 to node 9 with capacity 19,\nan edge from node 12 to node 2 with capacity 13,\nan edge from node 12 to node 19 with capacity 17,\nan edge from node 12 to node 18 with capacity 2,\nan edge from node 13 to node 5 with capacity 19,\nan edge from node 13 to node 6 with capacity 15,\nan edge from node 13 to node 11 with capacity 7,\nan edge from node 13 to node 0 with capacity 20,\nan edge from node 13 to node 3 with capacity 18,\nan edge from node 13 to node 10 with capacity 16,\nan edge from node 14 to node 2 with capacity 8,\nan edge from node 14 to node 15 with capacity 1,\nan edge from node 14 to node 1 with capacity 16,\nan edge from node 15 to node 7 with capacity 15,\nan edge from node 15 to node 2 with capacity 1,\nan edge from node 15 to node 6 with capacity 16,\nan edge from node 15 to node 3 with capacity 18,\nan edge from node 15 to node 17 with capacity 9,\nan edge from node 16 to node 7 with capacity 12,\nan edge from node 16 to node 6 with capacity 19,\nan edge from node 16 to node 18 with capacity 16,\nan edge from node 16 to node 12 with capacity 7,\nan edge from node 17 to node 0 with capacity 10,\nan edge from node 17 to node 10 with capacity 3,\nan edge from node 18 to node 7 with capacity 11,\nan edge from node 18 to node 5 with capacity 16,\nan edge from node 18 to node 4 with capacity 15,\nan edge from node 18 to node 2 with capacity 18,\nan edge from node 18 to node 13 with capacity 18,\nan edge from node 18 to node 3 with capacity 19,\nan edge from node 19 to node 7 with capacity 16,\nan edge from node 19 to node 0 with capacity 20,\nan edge from node 19 to node 10 with capacity 11,\nan edge from node 19 to node 1 with capacity 11.\nQ: What is the maximum flow from node 4 to node 15?\nA:", "answer": "The maximum flow from node 4 to node 15 is 54.", "difficulty": "hard", "doc_id": "122"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 0 to node 6 with capacity 9,\nan edge from node 1 to node 4 with capacity 6,\nan edge from node 1 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 3 to node 5 with capacity 7,\nan edge from node 3 to node 1 with capacity 8,\nan edge from node 3 to node 6 with capacity 8,\nan edge from node 4 to node 1 with capacity 8,\nan edge from node 4 to node 2 with capacity 2,\nan edge from node 5 to node 6 with capacity 4,\nan edge from node 6 to node 3 with capacity 5,\nan edge from node 6 to node 1 with capacity 9,\nan edge from node 7 to node 4 with capacity 5.\nQ: What is the maximum flow from node 0 to node 2?\nA:", "answer": "The maximum flow from node 0 to node 2 is 2.", "difficulty": "easy", "doc_id": "123"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 1 to node 4 with capacity 4,\nan edge from node 2 to node 8 with capacity 3,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 3 to node 6 with capacity 10,\nan edge from node 3 to node 1 with capacity 6,\nan edge from node 4 to node 8 with capacity 2,\nan edge from node 5 to node 7 with capacity 4,\nan edge from node 5 to node 8 with capacity 2,\nan edge from node 6 to node 2 with capacity 10,\nan edge from node 7 to node 2 with capacity 9,\nan edge from node 8 to node 4 with capacity 1,\nan edge from node 8 to node 3 with capacity 1,\nan edge from node 8 to node 5 with capacity 3.\nQ: What is the maximum flow from node 4 to node 2?\nA:", "answer": "The maximum flow from node 4 to node 2 is 2.", "difficulty": "easy", "doc_id": "124"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 3 with capacity 19,\nan edge from node 0 to node 4 with capacity 19,\nan edge from node 0 to node 10 with capacity 18,\nan edge from node 1 to node 4 with capacity 6,\nan edge from node 1 to node 8 with capacity 2,\nan edge from node 1 to node 2 with capacity 4,\nan edge from node 1 to node 6 with capacity 15,\nan edge from node 1 to node 7 with capacity 16,\nan edge from node 2 to node 3 with capacity 14,\nan edge from node 2 to node 1 with capacity 3,\nan edge from node 2 to node 9 with capacity 14,\nan edge from node 3 to node 4 with capacity 17,\nan edge from node 3 to node 10 with capacity 10,\nan edge from node 3 to node 11 with capacity 20,\nan edge from node 4 to node 0 with capacity 13,\nan edge from node 4 to node 5 with capacity 5,\nan edge from node 4 to node 6 with capacity 8,\nan edge from node 4 to node 7 with capacity 3,\nan edge from node 5 to node 4 with capacity 7,\nan edge from node 5 to node 11 with capacity 19,\nan edge from node 6 to node 4 with capacity 13,\nan edge from node 6 to node 5 with capacity 17,\nan edge from node 6 to node 7 with capacity 14,\nan edge from node 7 to node 3 with capacity 6,\nan edge from node 7 to node 9 with capacity 10,\nan edge from node 7 to node 12 with capacity 3,\nan edge from node 7 to node 11 with capacity 14,\nan edge from node 8 to node 3 with capacity 4,\nan edge from node 8 to node 9 with capacity 20,\nan edge from node 8 to node 6 with capacity 3,\nan edge from node 8 to node 11 with capacity 19,\nan edge from node 9 to node 3 with capacity 2,\nan edge from node 9 to node 8 with capacity 16,\nan edge from node 9 to node 10 with capacity 18,\nan edge from node 10 to node 5 with capacity 10,\nan edge from node 10 to node 1 with capacity 9,\nan edge from node 11 to node 4 with capacity 1,\nan edge from node 11 to node 1 with capacity 13,\nan edge from node 11 to node 6 with capacity 20,\nan edge from node 11 to node 7 with capacity 14,\nan edge from node 12 to node 4 with capacity 18,\nan edge from node 12 to node 8 with capacity 11,\nan edge from node 12 to node 5 with capacity 10,\nan edge from node 12 to node 9 with capacity 3,\nan edge from node 12 to node 11 with capacity 6.\nQ: What is the maximum flow from node 8 to node 3?\nA:", "answer": "The maximum flow from node 8 to node 3 is 29.", "difficulty": "hard", "doc_id": "125"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 7 with capacity 18,\nan edge from node 0 to node 8 with capacity 20,\nan edge from node 1 to node 11 with capacity 2,\nan edge from node 1 to node 7 with capacity 11,\nan edge from node 1 to node 6 with capacity 5,\nan edge from node 1 to node 10 with capacity 2,\nan edge from node 1 to node 9 with capacity 3,\nan edge from node 1 to node 13 with capacity 8,\nan edge from node 2 to node 11 with capacity 14,\nan edge from node 2 to node 12 with capacity 12,\nan edge from node 2 to node 8 with capacity 17,\nan edge from node 2 to node 5 with capacity 20,\nan edge from node 3 to node 11 with capacity 8,\nan edge from node 3 to node 1 with capacity 2,\nan edge from node 3 to node 12 with capacity 3,\nan edge from node 3 to node 2 with capacity 11,\nan edge from node 3 to node 0 with capacity 6,\nan edge from node 3 to node 5 with capacity 10,\nan edge from node 3 to node 13 with capacity 8,\nan edge from node 4 to node 1 with capacity 18,\nan edge from node 4 to node 10 with capacity 4,\nan edge from node 4 to node 2 with capacity 19,\nan edge from node 4 to node 13 with capacity 5,\nan edge from node 5 to node 2 with capacity 1,\nan edge from node 5 to node 13 with capacity 19,\nan edge from node 6 to node 10 with capacity 4,\nan edge from node 6 to node 0 with capacity 13,\nan edge from node 6 to node 13 with capacity 20,\nan edge from node 7 to node 1 with capacity 15,\nan edge from node 8 to node 4 with capacity 15,\nan edge from node 8 to node 2 with capacity 18,\nan edge from node 9 to node 4 with capacity 6,\nan edge from node 9 to node 13 with capacity 16,\nan edge from node 10 to node 3 with capacity 17,\nan edge from node 10 to node 5 with capacity 7,\nan edge from node 11 to node 4 with capacity 20,\nan edge from node 12 to node 11 with capacity 2,\nan edge from node 12 to node 7 with capacity 12,\nan edge from node 12 to node 6 with capacity 11,\nan edge from node 12 to node 0 with capacity 7,\nan edge from node 12 to node 9 with capacity 4,\nan edge from node 12 to node 5 with capacity 8,\nan edge from node 13 to node 4 with capacity 4,\nan edge from node 13 to node 7 with capacity 15,\nan edge from node 13 to node 12 with capacity 11,\nan edge from node 13 to node 9 with capacity 8.\nQ: What is the maximum flow from node 4 to node 13?\nA:", "answer": "The maximum flow from node 4 to node 13 is 46.", "difficulty": "hard", "doc_id": "126"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 12 with capacity 18,\nan edge from node 0 to node 16 with capacity 13,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 15 with capacity 20,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 1 to node 5 with capacity 2,\nan edge from node 1 to node 12 with capacity 16,\nan edge from node 1 to node 11 with capacity 18,\nan edge from node 1 to node 0 with capacity 7,\nan edge from node 2 to node 13 with capacity 16,\nan edge from node 2 to node 10 with capacity 9,\nan edge from node 2 to node 6 with capacity 16,\nan edge from node 2 to node 16 with capacity 9,\nan edge from node 3 to node 0 with capacity 11,\nan edge from node 4 to node 7 with capacity 18,\nan edge from node 4 to node 12 with capacity 2,\nan edge from node 4 to node 1 with capacity 16,\nan edge from node 4 to node 0 with capacity 12,\nan edge from node 4 to node 2 with capacity 5,\nan edge from node 5 to node 15 with capacity 6,\nan edge from node 5 to node 3 with capacity 5,\nan edge from node 5 to node 2 with capacity 17,\nan edge from node 6 to node 15 with capacity 17,\nan edge from node 6 to node 7 with capacity 8,\nan edge from node 6 to node 10 with capacity 5,\nan edge from node 6 to node 5 with capacity 3,\nan edge from node 6 to node 12 with capacity 3,\nan edge from node 7 to node 15 with capacity 1,\nan edge from node 7 to node 3 with capacity 12,\nan edge from node 7 to node 4 with capacity 5,\nan edge from node 7 to node 8 with capacity 12,\nan edge from node 8 to node 3 with capacity 15,\nan edge from node 8 to node 11 with capacity 16,\nan edge from node 8 to node 1 with capacity 2,\nan edge from node 9 to node 13 with capacity 14,\nan edge from node 9 to node 3 with capacity 14,\nan edge from node 9 to node 1 with capacity 17,\nan edge from node 9 to node 14 with capacity 2,\nan edge from node 10 to node 13 with capacity 9,\nan edge from node 10 to node 5 with capacity 16,\nan edge from node 10 to node 16 with capacity 20,\nan edge from node 10 to node 1 with capacity 8,\nan edge from node 10 to node 0 with capacity 13,\nan edge from node 11 to node 6 with capacity 10,\nan edge from node 11 to node 5 with capacity 19,\nan edge from node 11 to node 1 with capacity 19,\nan edge from node 11 to node 14 with capacity 4,\nan edge from node 12 to node 15 with capacity 13,\nan edge from node 12 to node 7 with capacity 19,\nan edge from node 12 to node 3 with capacity 4,\nan edge from node 12 to node 9 with capacity 8,\nan edge from node 13 to node 10 with capacity 2,\nan edge from node 13 to node 4 with capacity 15,\nan edge from node 14 to node 7 with capacity 3,\nan edge from node 14 to node 4 with capacity 13,\nan edge from node 14 to node 12 with capacity 5,\nan edge from node 14 to node 9 with capacity 13,\nan edge from node 14 to node 8 with capacity 12,\nan edge from node 14 to node 11 with capacity 1,\nan edge from node 14 to node 2 with capacity 11,\nan edge from node 15 to node 10 with capacity 11,\nan edge from node 15 to node 5 with capacity 5,\nan edge from node 15 to node 1 with capacity 7,\nan edge from node 16 to node 1 with capacity 5,\nan edge from node 16 to node 0 with capacity 8,\nan edge from node 16 to node 2 with capacity 19.\nQ: What is the maximum flow from node 8 to node 11?\nA:", "answer": "The maximum flow from node 8 to node 11 is 29.", "difficulty": "hard", "doc_id": "127"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 9 with capacity 16,\nan edge from node 1 to node 8 with capacity 17,\nan edge from node 2 to node 8 with capacity 10,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 7 with capacity 9,\nan edge from node 3 to node 8 with capacity 3,\nan edge from node 3 to node 10 with capacity 6,\nan edge from node 4 to node 0 with capacity 17,\nan edge from node 4 to node 10 with capacity 3,\nan edge from node 5 to node 8 with capacity 14,\nan edge from node 5 to node 11 with capacity 18,\nan edge from node 6 to node 2 with capacity 17,\nan edge from node 6 to node 8 with capacity 7,\nan edge from node 7 to node 1 with capacity 18,\nan edge from node 8 to node 6 with capacity 17,\nan edge from node 8 to node 1 with capacity 4,\nan edge from node 8 to node 0 with capacity 5,\nan edge from node 9 to node 5 with capacity 11,\nan edge from node 9 to node 8 with capacity 9,\nan edge from node 9 to node 0 with capacity 9,\nan edge from node 9 to node 3 with capacity 17,\nan edge from node 10 to node 1 with capacity 17,\nan edge from node 11 to node 6 with capacity 17,\nan edge from node 11 to node 1 with capacity 13,\nan edge from node 11 to node 2 with capacity 8,\nan edge from node 11 to node 4 with capacity 11.\nQ: What is the maximum flow from node 7 to node 2?\nA:", "answer": "The maximum flow from node 7 to node 2 is 18.", "difficulty": "hard", "doc_id": "128"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 1 with capacity 13,\nan edge from node 0 to node 4 with capacity 19,\nan edge from node 0 to node 3 with capacity 12,\nan edge from node 0 to node 2 with capacity 7,\nan edge from node 1 to node 9 with capacity 20,\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 2 to node 6 with capacity 10,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 2 to node 10 with capacity 5,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 9 with capacity 9,\nan edge from node 4 to node 10 with capacity 15,\nan edge from node 4 to node 9 with capacity 20,\nan edge from node 4 to node 2 with capacity 4,\nan edge from node 5 to node 10 with capacity 20,\nan edge from node 5 to node 0 with capacity 20,\nan edge from node 6 to node 1 with capacity 19,\nan edge from node 6 to node 3 with capacity 3,\nan edge from node 6 to node 7 with capacity 15,\nan edge from node 6 to node 8 with capacity 11,\nan edge from node 7 to node 10 with capacity 6,\nan edge from node 8 to node 6 with capacity 5,\nan edge from node 8 to node 5 with capacity 11,\nan edge from node 8 to node 9 with capacity 19,\nan edge from node 8 to node 0 with capacity 19,\nan edge from node 9 to node 6 with capacity 13,\nan edge from node 9 to node 7 with capacity 14,\nan edge from node 9 to node 2 with capacity 14,\nan edge from node 9 to node 0 with capacity 2,\nan edge from node 10 to node 3 with capacity 12,\nan edge from node 10 to node 0 with capacity 14.\nQ: What is the maximum flow from node 7 to node 9?\nA:", "answer": "The maximum flow from node 7 to node 9 is 6.", "difficulty": "hard", "doc_id": "129"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 9 with capacity 4,\nan edge from node 0 to node 2 with capacity 5,\nan edge from node 1 to node 0 with capacity 18,\nan edge from node 1 to node 5 with capacity 1,\nan edge from node 2 to node 4 with capacity 12,\nan edge from node 2 to node 11 with capacity 7,\nan edge from node 3 to node 5 with capacity 10,\nan edge from node 3 to node 12 with capacity 19,\nan edge from node 4 to node 5 with capacity 17,\nan edge from node 4 to node 10 with capacity 16,\nan edge from node 4 to node 6 with capacity 17,\nan edge from node 5 to node 8 with capacity 8,\nan edge from node 5 to node 4 with capacity 18,\nan edge from node 5 to node 11 with capacity 9,\nan edge from node 6 to node 5 with capacity 14,\nan edge from node 6 to node 4 with capacity 8,\nan edge from node 6 to node 9 with capacity 19,\nan edge from node 6 to node 1 with capacity 4,\nan edge from node 6 to node 10 with capacity 4,\nan edge from node 7 to node 3 with capacity 6,\nan edge from node 7 to node 4 with capacity 5,\nan edge from node 7 to node 10 with capacity 5,\nan edge from node 8 to node 9 with capacity 13,\nan edge from node 9 to node 4 with capacity 17,\nan edge from node 10 to node 8 with capacity 5,\nan edge from node 10 to node 5 with capacity 9,\nan edge from node 10 to node 1 with capacity 20,\nan edge from node 10 to node 12 with capacity 6,\nan edge from node 11 to node 4 with capacity 10,\nan edge from node 11 to node 1 with capacity 8,\nan edge from node 11 to node 12 with capacity 19,\nan edge from node 12 to node 5 with capacity 16,\nan edge from node 12 to node 4 with capacity 1,\nan edge from node 12 to node 7 with capacity 1.\nQ: What is the maximum flow from node 10 to node 11?\nA:", "answer": "The maximum flow from node 10 to node 11 is 14.", "difficulty": "hard", "doc_id": "130"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 0 to node 1 with capacity 8,\nan edge from node 1 to node 4 with capacity 4,\nan edge from node 1 to node 3 with capacity 2,\nan edge from node 3 to node 2 with capacity 8,\nan edge from node 4 to node 7 with capacity 5,\nan edge from node 4 to node 6 with capacity 1,\nan edge from node 4 to node 2 with capacity 9,\nan edge from node 4 to node 1 with capacity 5,\nan edge from node 5 to node 0 with capacity 8,\nan edge from node 5 to node 4 with capacity 10,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 7 to node 5 with capacity 9,\nan edge from node 7 to node 3 with capacity 1.\nQ: What is the maximum flow from node 1 to node 2?\nA:", "answer": "The maximum flow from node 1 to node 2 is 6.", "difficulty": "easy", "doc_id": "131"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 1 to node 7 with capacity 5,\nan edge from node 2 to node 4 with capacity 1,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 2 with capacity 5,\nan edge from node 4 to node 6 with capacity 9,\nan edge from node 5 to node 2 with capacity 8,\nan edge from node 5 to node 3 with capacity 6,\nan edge from node 6 to node 7 with capacity 7,\nan edge from node 6 to node 1 with capacity 8,\nan edge from node 7 to node 3 with capacity 2.\nQ: What is the maximum flow from node 5 to node 3?\nA:", "answer": "The maximum flow from node 5 to node 3 is 8.", "difficulty": "easy", "doc_id": "132"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 1 to node 3 with capacity 8,\nan edge from node 1 to node 5 with capacity 2,\nan edge from node 1 to node 6 with capacity 3,\nan edge from node 1 to node 2 with capacity 2,\nan edge from node 3 to node 0 with capacity 3,\nan edge from node 4 to node 8 with capacity 3,\nan edge from node 4 to node 0 with capacity 9,\nan edge from node 4 to node 6 with capacity 8,\nan edge from node 8 to node 1 with capacity 8,\nan edge from node 8 to node 3 with capacity 2.\nQ: What is the maximum flow from node 4 to node 0?\nA:", "answer": "The maximum flow from node 4 to node 0 is 12.", "difficulty": "easy", "doc_id": "133"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 3,\nan edge from node 0 to node 2 with capacity 3,\nan edge from node 0 to node 4 with capacity 3,\nan edge from node 1 to node 3 with capacity 1,\nan edge from node 2 to node 0 with capacity 10,\nan edge from node 3 to node 2 with capacity 8,\nan edge from node 3 to node 4 with capacity 2,\nan edge from node 4 to node 2 with capacity 8,\nan edge from node 4 to node 0 with capacity 10.\nQ: What is the maximum flow from node 0 to node 2?\nA:", "answer": "The maximum flow from node 0 to node 2 is 7.", "difficulty": "easy", "doc_id": "134"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 1 to node 6 with capacity 4,\nan edge from node 1 to node 5 with capacity 5,\nan edge from node 2 to node 3 with capacity 6,\nan edge from node 3 to node 1 with capacity 9,\nan edge from node 3 to node 7 with capacity 5,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 4 to node 6 with capacity 7,\nan edge from node 4 to node 5 with capacity 7,\nan edge from node 5 to node 0 with capacity 1,\nan edge from node 5 to node 7 with capacity 3,\nan edge from node 6 to node 1 with capacity 5,\nan edge from node 6 to node 2 with capacity 2.\nQ: What is the maximum flow from node 6 to node 7?\nA:", "answer": "The maximum flow from node 6 to node 7 is 5.", "difficulty": "easy", "doc_id": "135"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 4 with capacity 10,\nan edge from node 2 to node 0 with capacity 3,\nan edge from node 2 to node 1 with capacity 4,\nan edge from node 3 to node 4 with capacity 2,\nan edge from node 3 to node 1 with capacity 6,\nan edge from node 3 to node 5 with capacity 6,\nan edge from node 4 to node 3 with capacity 2,\nan edge from node 5 to node 1 with capacity 5,\nan edge from node 5 to node 3 with capacity 2.\nQ: What is the maximum flow from node 2 to node 1?\nA:", "answer": "The maximum flow from node 2 to node 1 is 6.", "difficulty": "easy", "doc_id": "136"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 12 with capacity 1,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 0 to node 18 with capacity 18,\nan edge from node 0 to node 14 with capacity 17,\nan edge from node 0 to node 1 with capacity 2,\nan edge from node 0 to node 13 with capacity 6,\nan edge from node 1 to node 7 with capacity 19,\nan edge from node 1 to node 8 with capacity 8,\nan edge from node 2 to node 12 with capacity 9,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 2 to node 7 with capacity 18,\nan edge from node 2 to node 5 with capacity 16,\nan edge from node 2 to node 0 with capacity 8,\nan edge from node 2 to node 13 with capacity 4,\nan edge from node 2 to node 11 with capacity 4,\nan edge from node 3 to node 4 with capacity 6,\nan edge from node 3 to node 15 with capacity 17,\nan edge from node 3 to node 0 with capacity 3,\nan edge from node 3 to node 6 with capacity 20,\nan edge from node 4 to node 7 with capacity 10,\nan edge from node 4 to node 15 with capacity 12,\nan edge from node 4 to node 16 with capacity 12,\nan edge from node 4 to node 5 with capacity 20,\nan edge from node 4 to node 1 with capacity 12,\nan edge from node 4 to node 2 with capacity 17,\nan edge from node 4 to node 13 with capacity 1,\nan edge from node 5 to node 4 with capacity 4,\nan edge from node 5 to node 14 with capacity 11,\nan edge from node 5 to node 1 with capacity 20,\nan edge from node 5 to node 0 with capacity 6,\nan edge from node 5 to node 13 with capacity 2,\nan edge from node 5 to node 6 with capacity 5,\nan edge from node 6 to node 4 with capacity 14,\nan edge from node 6 to node 5 with capacity 20,\nan edge from node 6 to node 9 with capacity 5,\nan edge from node 6 to node 0 with capacity 20,\nan edge from node 6 to node 17 with capacity 8,\nan edge from node 7 to node 3 with capacity 14,\nan edge from node 7 to node 15 with capacity 19,\nan edge from node 7 to node 18 with capacity 4,\nan edge from node 7 to node 10 with capacity 5,\nan edge from node 7 to node 5 with capacity 20,\nan edge from node 7 to node 1 with capacity 20,\nan edge from node 7 to node 8 with capacity 12,\nan edge from node 8 to node 12 with capacity 19,\nan edge from node 8 to node 15 with capacity 14,\nan edge from node 8 to node 10 with capacity 16,\nan edge from node 8 to node 14 with capacity 20,\nan edge from node 8 to node 5 with capacity 15,\nan edge from node 8 to node 11 with capacity 3,\nan edge from node 9 to node 7 with capacity 3,\nan edge from node 9 to node 15 with capacity 15,\nan edge from node 9 to node 16 with capacity 4,\nan edge from node 9 to node 10 with capacity 15,\nan edge from node 9 to node 14 with capacity 8,\nan edge from node 9 to node 13 with capacity 15,\nan edge from node 9 to node 17 with capacity 2,\nan edge from node 9 to node 11 with capacity 12,\nan edge from node 10 to node 4 with capacity 11,\nan edge from node 10 to node 3 with capacity 20,\nan edge from node 10 to node 11 with capacity 14,\nan edge from node 11 to node 7 with capacity 11,\nan edge from node 11 to node 16 with capacity 17,\nan edge from node 11 to node 14 with capacity 11,\nan edge from node 11 to node 9 with capacity 20,\nan edge from node 11 to node 8 with capacity 20,\nan edge from node 12 to node 7 with capacity 6,\nan edge from node 12 to node 3 with capacity 5,\nan edge from node 12 to node 18 with capacity 18,\nan edge from node 13 to node 12 with capacity 11,\nan edge from node 13 to node 4 with capacity 16,\nan edge from node 13 to node 15 with capacity 6,\nan edge from node 13 to node 16 with capacity 18,\nan edge from node 13 to node 18 with capacity 17,\nan edge from node 13 to node 17 with capacity 2,\nan edge from node 13 to node 6 with capacity 19,\nan edge from node 14 to node 13 with capacity 13,\nan edge from node 15 to node 7 with capacity 2,\nan edge from node 15 to node 18 with capacity 20,\nan edge from node 15 to node 1 with capacity 15,\nan edge from node 15 to node 13 with capacity 6,\nan edge from node 15 to node 8 with capacity 5,\nan edge from node 16 to node 12 with capacity 16,\nan edge from node 16 to node 7 with capacity 16,\nan edge from node 16 to node 10 with capacity 1,\nan edge from node 16 to node 9 with capacity 9,\nan edge from node 16 to node 0 with capacity 5,\nan edge from node 17 to node 3 with capacity 4,\nan edge from node 17 to node 9 with capacity 19,\nan edge from node 17 to node 2 with capacity 7,\nan edge from node 17 to node 11 with capacity 19,\nan edge from node 18 to node 12 with capacity 8,\nan edge from node 18 to node 16 with capacity 16,\nan edge from node 18 to node 1 with capacity 7,\nan edge from node 18 to node 13 with capacity 16,\nan edge from node 18 to node 17 with capacity 8.\nQ: What is the maximum flow from node 18 to node 12?\nA:", "answer": "The maximum flow from node 18 to node 12 is 55.", "difficulty": "hard", "doc_id": "137"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 16 with capacity 11,\nan edge from node 0 to node 7 with capacity 20,\nan edge from node 0 to node 17 with capacity 7,\nan edge from node 0 to node 6 with capacity 10,\nan edge from node 1 to node 18 with capacity 15,\nan edge from node 1 to node 16 with capacity 19,\nan edge from node 1 to node 10 with capacity 16,\nan edge from node 1 to node 4 with capacity 13,\nan edge from node 1 to node 3 with capacity 19,\nan edge from node 1 to node 13 with capacity 20,\nan edge from node 2 to node 18 with capacity 14,\nan edge from node 2 to node 15 with capacity 1,\nan edge from node 2 to node 16 with capacity 2,\nan edge from node 2 to node 5 with capacity 8,\nan edge from node 3 to node 16 with capacity 19,\nan edge from node 3 to node 8 with capacity 17,\nan edge from node 3 to node 11 with capacity 5,\nan edge from node 3 to node 17 with capacity 7,\nan edge from node 3 to node 13 with capacity 1,\nan edge from node 3 to node 9 with capacity 5,\nan edge from node 4 to node 18 with capacity 6,\nan edge from node 4 to node 8 with capacity 7,\nan edge from node 4 to node 12 with capacity 11,\nan edge from node 4 to node 14 with capacity 3,\nan edge from node 4 to node 1 with capacity 5,\nan edge from node 5 to node 4 with capacity 8,\nan edge from node 5 to node 12 with capacity 16,\nan edge from node 5 to node 17 with capacity 4,\nan edge from node 5 to node 9 with capacity 17,\nan edge from node 6 to node 15 with capacity 16,\nan edge from node 6 to node 5 with capacity 2,\nan edge from node 6 to node 13 with capacity 20,\nan edge from node 7 to node 2 with capacity 16,\nan edge from node 7 to node 15 with capacity 14,\nan edge from node 7 to node 4 with capacity 13,\nan edge from node 7 to node 12 with capacity 11,\nan edge from node 7 to node 1 with capacity 16,\nan edge from node 8 to node 16 with capacity 6,\nan edge from node 8 to node 0 with capacity 13,\nan edge from node 8 to node 10 with capacity 4,\nan edge from node 8 to node 3 with capacity 16,\nan edge from node 8 to node 17 with capacity 6,\nan edge from node 8 to node 1 with capacity 8,\nan edge from node 8 to node 9 with capacity 6,\nan edge from node 9 to node 2 with capacity 6,\nan edge from node 9 to node 0 with capacity 18,\nan edge from node 10 to node 18 with capacity 2,\nan edge from node 10 to node 15 with capacity 4,\nan edge from node 10 to node 0 with capacity 11,\nan edge from node 10 to node 5 with capacity 14,\nan edge from node 10 to node 7 with capacity 13,\nan edge from node 10 to node 12 with capacity 19,\nan edge from node 10 to node 17 with capacity 3,\nan edge from node 11 to node 7 with capacity 12,\nan edge from node 11 to node 8 with capacity 12,\nan edge from node 11 to node 14 with capacity 20,\nan edge from node 11 to node 6 with capacity 5,\nan edge from node 12 to node 18 with capacity 18,\nan edge from node 12 to node 7 with capacity 4,\nan edge from node 12 to node 8 with capacity 9,\nan edge from node 12 to node 3 with capacity 20,\nan edge from node 12 to node 11 with capacity 10,\nan edge from node 12 to node 14 with capacity 12,\nan edge from node 13 to node 2 with capacity 7,\nan edge from node 13 to node 0 with capacity 20,\nan edge from node 13 to node 12 with capacity 7,\nan edge from node 13 to node 3 with capacity 15,\nan edge from node 13 to node 14 with capacity 4,\nan edge from node 13 to node 9 with capacity 1,\nan edge from node 14 to node 15 with capacity 12,\nan edge from node 14 to node 0 with capacity 15,\nan edge from node 14 to node 12 with capacity 20,\nan edge from node 14 to node 13 with capacity 6,\nan edge from node 15 to node 12 with capacity 20,\nan edge from node 15 to node 17 with capacity 2,\nan edge from node 15 to node 13 with capacity 1,\nan edge from node 16 to node 5 with capacity 5,\nan edge from node 16 to node 7 with capacity 4,\nan edge from node 16 to node 4 with capacity 4,\nan edge from node 16 to node 3 with capacity 10,\nan edge from node 16 to node 11 with capacity 14,\nan edge from node 16 to node 17 with capacity 16,\nan edge from node 16 to node 14 with capacity 7,\nan edge from node 17 to node 0 with capacity 5,\nan edge from node 17 to node 5 with capacity 17,\nan edge from node 17 to node 3 with capacity 4,\nan edge from node 17 to node 13 with capacity 1,\nan edge from node 17 to node 14 with capacity 6,\nan edge from node 18 to node 16 with capacity 5,\nan edge from node 18 to node 4 with capacity 16,\nan edge from node 18 to node 3 with capacity 13,\nan edge from node 18 to node 11 with capacity 19,\nan edge from node 18 to node 9 with capacity 18.\nQ: What is the maximum flow from node 5 to node 15?\nA:", "answer": "The maximum flow from node 5 to node 15 is 45.", "difficulty": "hard", "doc_id": "138"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 12 with capacity 6,\nan edge from node 0 to node 7 with capacity 3,\nan edge from node 0 to node 13 with capacity 4,\nan edge from node 1 to node 11 with capacity 14,\nan edge from node 1 to node 3 with capacity 20,\nan edge from node 1 to node 7 with capacity 11,\nan edge from node 1 to node 5 with capacity 11,\nan edge from node 2 to node 11 with capacity 13,\nan edge from node 2 to node 13 with capacity 12,\nan edge from node 2 to node 9 with capacity 4,\nan edge from node 2 to node 10 with capacity 11,\nan edge from node 3 to node 11 with capacity 15,\nan edge from node 3 to node 13 with capacity 1,\nan edge from node 4 to node 6 with capacity 12,\nan edge from node 5 to node 12 with capacity 5,\nan edge from node 5 to node 0 with capacity 20,\nan edge from node 5 to node 1 with capacity 4,\nan edge from node 5 to node 3 with capacity 15,\nan edge from node 5 to node 2 with capacity 16,\nan edge from node 5 to node 13 with capacity 13,\nan edge from node 6 to node 11 with capacity 7,\nan edge from node 6 to node 13 with capacity 8,\nan edge from node 6 to node 5 with capacity 4,\nan edge from node 7 to node 13 with capacity 14,\nan edge from node 7 to node 5 with capacity 3,\nan edge from node 7 to node 10 with capacity 5,\nan edge from node 8 to node 12 with capacity 1,\nan edge from node 8 to node 3 with capacity 1,\nan edge from node 8 to node 2 with capacity 12,\nan edge from node 9 to node 0 with capacity 14,\nan edge from node 9 to node 3 with capacity 5,\nan edge from node 9 to node 2 with capacity 13,\nan edge from node 9 to node 13 with capacity 20,\nan edge from node 9 to node 10 with capacity 12,\nan edge from node 10 to node 8 with capacity 1,\nan edge from node 10 to node 12 with capacity 20,\nan edge from node 10 to node 1 with capacity 8,\nan edge from node 10 to node 2 with capacity 14,\nan edge from node 10 to node 7 with capacity 13,\nan edge from node 11 to node 12 with capacity 4,\nan edge from node 11 to node 2 with capacity 12,\nan edge from node 11 to node 5 with capacity 20,\nan edge from node 11 to node 9 with capacity 20,\nan edge from node 12 to node 4 with capacity 8,\nan edge from node 12 to node 11 with capacity 12,\nan edge from node 12 to node 1 with capacity 18,\nan edge from node 12 to node 2 with capacity 13,\nan edge from node 12 to node 5 with capacity 1,\nan edge from node 12 to node 10 with capacity 6,\nan edge from node 13 to node 8 with capacity 16,\nan edge from node 13 to node 11 with capacity 3,\nan edge from node 13 to node 10 with capacity 19.\nQ: What is the maximum flow from node 0 to node 3?\nA:", "answer": "The maximum flow from node 0 to node 3 is 13.", "difficulty": "hard", "doc_id": "139"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 10 with capacity 3,\nan edge from node 0 to node 5 with capacity 4,\nan edge from node 0 to node 13 with capacity 14,\nan edge from node 0 to node 8 with capacity 11,\nan edge from node 0 to node 9 with capacity 4,\nan edge from node 1 to node 3 with capacity 5,\nan edge from node 1 to node 11 with capacity 4,\nan edge from node 1 to node 13 with capacity 10,\nan edge from node 2 to node 3 with capacity 7,\nan edge from node 2 to node 7 with capacity 1,\nan edge from node 3 to node 10 with capacity 12,\nan edge from node 3 to node 7 with capacity 15,\nan edge from node 4 to node 13 with capacity 17,\nan edge from node 5 to node 10 with capacity 5,\nan edge from node 5 to node 2 with capacity 17,\nan edge from node 6 to node 0 with capacity 18,\nan edge from node 6 to node 7 with capacity 18,\nan edge from node 6 to node 9 with capacity 6,\nan edge from node 7 to node 4 with capacity 6,\nan edge from node 7 to node 6 with capacity 10,\nan edge from node 7 to node 13 with capacity 19,\nan edge from node 7 to node 8 with capacity 9,\nan edge from node 7 to node 9 with capacity 20,\nan edge from node 8 to node 4 with capacity 3,\nan edge from node 8 to node 1 with capacity 15,\nan edge from node 8 to node 13 with capacity 4,\nan edge from node 8 to node 7 with capacity 13,\nan edge from node 9 to node 6 with capacity 18,\nan edge from node 9 to node 5 with capacity 10,\nan edge from node 9 to node 12 with capacity 6,\nan edge from node 10 to node 7 with capacity 10,\nan edge from node 10 to node 12 with capacity 6,\nan edge from node 11 to node 4 with capacity 5,\nan edge from node 11 to node 0 with capacity 9,\nan edge from node 11 to node 1 with capacity 20,\nan edge from node 11 to node 2 with capacity 4,\nan edge from node 11 to node 13 with capacity 11,\nan edge from node 11 to node 8 with capacity 7,\nan edge from node 12 to node 10 with capacity 10,\nan edge from node 12 to node 13 with capacity 18,\nan edge from node 12 to node 9 with capacity 15,\nan edge from node 13 to node 6 with capacity 5,\nan edge from node 13 to node 2 with capacity 4.\nQ: What is the maximum flow from node 11 to node 5?\nA:", "answer": "The maximum flow from node 11 to node 5 is 14.", "difficulty": "hard", "doc_id": "140"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 12 with capacity 1,\nan edge from node 1 to node 3 with capacity 2,\nan edge from node 1 to node 10 with capacity 10,\nan edge from node 1 to node 2 with capacity 17,\nan edge from node 2 to node 0 with capacity 10,\nan edge from node 2 to node 3 with capacity 15,\nan edge from node 2 to node 10 with capacity 2,\nan edge from node 2 to node 4 with capacity 15,\nan edge from node 3 to node 7 with capacity 15,\nan edge from node 4 to node 0 with capacity 6,\nan edge from node 4 to node 7 with capacity 12,\nan edge from node 4 to node 11 with capacity 6,\nan edge from node 4 to node 2 with capacity 13,\nan edge from node 5 to node 11 with capacity 8,\nan edge from node 6 to node 0 with capacity 8,\nan edge from node 7 to node 2 with capacity 2,\nan edge from node 7 to node 1 with capacity 4,\nan edge from node 8 to node 0 with capacity 8,\nan edge from node 8 to node 9 with capacity 12,\nan edge from node 8 to node 10 with capacity 11,\nan edge from node 8 to node 7 with capacity 8,\nan edge from node 8 to node 2 with capacity 17,\nan edge from node 8 to node 6 with capacity 3,\nan edge from node 8 to node 1 with capacity 8,\nan edge from node 9 to node 3 with capacity 4,\nan edge from node 9 to node 11 with capacity 13,\nan edge from node 9 to node 4 with capacity 13,\nan edge from node 9 to node 1 with capacity 11,\nan edge from node 9 to node 8 with capacity 13,\nan edge from node 10 to node 9 with capacity 4,\nan edge from node 10 to node 2 with capacity 13,\nan edge from node 10 to node 5 with capacity 19,\nan edge from node 11 to node 3 with capacity 11,\nan edge from node 11 to node 9 with capacity 14,\nan edge from node 11 to node 10 with capacity 9,\nan edge from node 11 to node 4 with capacity 18,\nan edge from node 11 to node 1 with capacity 6,\nan edge from node 12 to node 0 with capacity 6,\nan edge from node 12 to node 7 with capacity 1,\nan edge from node 12 to node 6 with capacity 13,\nan edge from node 12 to node 5 with capacity 13.\nQ: What is the maximum flow from node 1 to node 6?\nA:", "answer": "The maximum flow from node 1 to node 6 is 4.", "difficulty": "hard", "doc_id": "141"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 14 with capacity 13,\nan edge from node 1 to node 4 with capacity 15,\nan edge from node 1 to node 9 with capacity 9,\nan edge from node 1 to node 14 with capacity 7,\nan edge from node 2 to node 4 with capacity 13,\nan edge from node 2 to node 9 with capacity 2,\nan edge from node 2 to node 14 with capacity 13,\nan edge from node 3 to node 2 with capacity 10,\nan edge from node 3 to node 1 with capacity 19,\nan edge from node 3 to node 10 with capacity 3,\nan edge from node 3 to node 6 with capacity 16,\nan edge from node 3 to node 9 with capacity 3,\nan edge from node 3 to node 7 with capacity 18,\nan edge from node 3 to node 13 with capacity 2,\nan edge from node 4 to node 12 with capacity 9,\nan edge from node 4 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 14,\nan edge from node 4 to node 6 with capacity 7,\nan edge from node 4 to node 13 with capacity 12,\nan edge from node 5 to node 4 with capacity 6,\nan edge from node 5 to node 14 with capacity 7,\nan edge from node 6 to node 8 with capacity 14,\nan edge from node 6 to node 1 with capacity 15,\nan edge from node 6 to node 9 with capacity 16,\nan edge from node 6 to node 14 with capacity 14,\nan edge from node 6 to node 7 with capacity 13,\nan edge from node 6 to node 13 with capacity 6,\nan edge from node 6 to node 0 with capacity 19,\nan edge from node 7 to node 2 with capacity 3,\nan edge from node 7 to node 1 with capacity 17,\nan edge from node 7 to node 3 with capacity 11,\nan edge from node 7 to node 9 with capacity 5,\nan edge from node 7 to node 0 with capacity 11,\nan edge from node 8 to node 12 with capacity 18,\nan edge from node 8 to node 6 with capacity 16,\nan edge from node 8 to node 9 with capacity 18,\nan edge from node 8 to node 11 with capacity 14,\nan edge from node 9 to node 5 with capacity 15,\nan edge from node 9 to node 3 with capacity 8,\nan edge from node 9 to node 13 with capacity 7,\nan edge from node 10 to node 1 with capacity 6,\nan edge from node 10 to node 4 with capacity 7,\nan edge from node 10 to node 0 with capacity 14,\nan edge from node 11 to node 2 with capacity 1,\nan edge from node 11 to node 5 with capacity 4,\nan edge from node 11 to node 3 with capacity 5,\nan edge from node 11 to node 14 with capacity 19,\nan edge from node 11 to node 13 with capacity 7,\nan edge from node 12 to node 2 with capacity 10,\nan edge from node 12 to node 5 with capacity 1,\nan edge from node 12 to node 13 with capacity 14,\nan edge from node 13 to node 12 with capacity 5,\nan edge from node 13 to node 4 with capacity 20,\nan edge from node 13 to node 3 with capacity 15,\nan edge from node 13 to node 7 with capacity 8,\nan edge from node 14 to node 8 with capacity 8,\nan edge from node 14 to node 10 with capacity 7.\nQ: What is the maximum flow from node 0 to node 13?\nA:", "answer": "The maximum flow from node 0 to node 13 is 13.", "difficulty": "hard", "doc_id": "142"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 1 to node 0 with capacity 2,\nan edge from node 1 to node 12 with capacity 17,\nan edge from node 1 to node 7 with capacity 11,\nan edge from node 2 to node 3 with capacity 9,\nan edge from node 2 to node 9 with capacity 18,\nan edge from node 2 to node 4 with capacity 7,\nan edge from node 2 to node 7 with capacity 6,\nan edge from node 2 to node 5 with capacity 15,\nan edge from node 3 to node 8 with capacity 2,\nan edge from node 5 to node 8 with capacity 5,\nan edge from node 5 to node 1 with capacity 18,\nan edge from node 5 to node 2 with capacity 3,\nan edge from node 6 to node 3 with capacity 5,\nan edge from node 6 to node 1 with capacity 11,\nan edge from node 6 to node 9 with capacity 17,\nan edge from node 7 to node 0 with capacity 7,\nan edge from node 7 to node 10 with capacity 6,\nan edge from node 7 to node 11 with capacity 3,\nan edge from node 7 to node 6 with capacity 20,\nan edge from node 8 to node 1 with capacity 15,\nan edge from node 8 to node 12 with capacity 8,\nan edge from node 8 to node 4 with capacity 20,\nan edge from node 8 to node 11 with capacity 1,\nan edge from node 9 to node 12 with capacity 19,\nan edge from node 9 to node 10 with capacity 13,\nan edge from node 9 to node 5 with capacity 20,\nan edge from node 10 to node 8 with capacity 10,\nan edge from node 10 to node 12 with capacity 12,\nan edge from node 10 to node 2 with capacity 19,\nan edge from node 10 to node 11 with capacity 17,\nan edge from node 11 to node 0 with capacity 5,\nan edge from node 11 to node 9 with capacity 12,\nan edge from node 11 to node 12 with capacity 11,\nan edge from node 11 to node 2 with capacity 9,\nan edge from node 12 to node 1 with capacity 12,\nan edge from node 12 to node 4 with capacity 10.\nQ: What is the maximum flow from node 9 to node 4?\nA:", "answer": "The maximum flow from node 9 to node 4 is 34.", "difficulty": "hard", "doc_id": "143"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 1 to node 3 with capacity 5,\nan edge from node 1 to node 0 with capacity 7,\nan edge from node 1 to node 6 with capacity 3,\nan edge from node 1 to node 7 with capacity 1,\nan edge from node 1 to node 4 with capacity 1,\nan edge from node 3 to node 1 with capacity 8,\nan edge from node 4 to node 0 with capacity 2,\nan edge from node 4 to node 5 with capacity 5,\nan edge from node 4 to node 7 with capacity 8,\nan edge from node 5 to node 2 with capacity 9,\nan edge from node 5 to node 1 with capacity 1,\nan edge from node 6 to node 0 with capacity 4,\nan edge from node 6 to node 5 with capacity 10.\nQ: What is the maximum flow from node 3 to node 0?\nA:", "answer": "The maximum flow from node 3 to node 0 is 8.", "difficulty": "easy", "doc_id": "144"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 9 with capacity 4,\nan edge from node 0 to node 14 with capacity 7,\nan edge from node 0 to node 13 with capacity 2,\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 8 with capacity 3,\nan edge from node 0 to node 2 with capacity 13,\nan edge from node 1 to node 12 with capacity 3,\nan edge from node 1 to node 10 with capacity 11,\nan edge from node 1 to node 11 with capacity 8,\nan edge from node 1 to node 4 with capacity 2,\nan edge from node 1 to node 7 with capacity 7,\nan edge from node 2 to node 5 with capacity 4,\nan edge from node 2 to node 9 with capacity 11,\nan edge from node 2 to node 14 with capacity 13,\nan edge from node 2 to node 13 with capacity 11,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 3 to node 12 with capacity 8,\nan edge from node 3 to node 14 with capacity 12,\nan edge from node 3 to node 13 with capacity 11,\nan edge from node 3 to node 8 with capacity 3,\nan edge from node 3 to node 10 with capacity 14,\nan edge from node 4 to node 12 with capacity 2,\nan edge from node 4 to node 9 with capacity 12,\nan edge from node 4 to node 13 with capacity 16,\nan edge from node 4 to node 3 with capacity 13,\nan edge from node 5 to node 0 with capacity 1,\nan edge from node 5 to node 4 with capacity 12,\nan edge from node 5 to node 7 with capacity 4,\nan edge from node 5 to node 1 with capacity 16,\nan edge from node 6 to node 5 with capacity 12,\nan edge from node 6 to node 0 with capacity 16,\nan edge from node 6 to node 9 with capacity 17,\nan edge from node 6 to node 4 with capacity 14,\nan edge from node 7 to node 5 with capacity 20,\nan edge from node 7 to node 0 with capacity 20,\nan edge from node 7 to node 8 with capacity 8,\nan edge from node 7 to node 10 with capacity 3,\nan edge from node 7 to node 6 with capacity 16,\nan edge from node 8 to node 0 with capacity 11,\nan edge from node 8 to node 12 with capacity 15,\nan edge from node 8 to node 2 with capacity 20,\nan edge from node 9 to node 5 with capacity 10,\nan edge from node 9 to node 8 with capacity 8,\nan edge from node 9 to node 11 with capacity 3,\nan edge from node 11 to node 5 with capacity 15,\nan edge from node 11 to node 13 with capacity 5,\nan edge from node 11 to node 3 with capacity 2,\nan edge from node 11 to node 4 with capacity 18,\nan edge from node 12 to node 6 with capacity 19,\nan edge from node 13 to node 5 with capacity 2,\nan edge from node 13 to node 12 with capacity 17,\nan edge from node 13 to node 14 with capacity 17,\nan edge from node 13 to node 8 with capacity 1,\nan edge from node 14 to node 0 with capacity 11,\nan edge from node 14 to node 12 with capacity 8,\nan edge from node 14 to node 11 with capacity 8.\nQ: What is the maximum flow from node 11 to node 4?\nA:", "answer": "The maximum flow from node 11 to node 4 is 40.", "difficulty": "hard", "doc_id": "145"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 6 with capacity 6,\nan edge from node 1 to node 3 with capacity 2,\nan edge from node 1 to node 8 with capacity 8,\nan edge from node 2 to node 1 with capacity 1,\nan edge from node 2 to node 5 with capacity 7,\nan edge from node 2 to node 6 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 3 to node 6 with capacity 8,\nan edge from node 4 to node 5 with capacity 1,\nan edge from node 5 to node 2 with capacity 2,\nan edge from node 5 to node 4 with capacity 7,\nan edge from node 5 to node 3 with capacity 10,\nan edge from node 6 to node 2 with capacity 9,\nan edge from node 7 to node 5 with capacity 6,\nan edge from node 7 to node 0 with capacity 5,\nan edge from node 8 to node 0 with capacity 3.\nQ: What is the maximum flow from node 7 to node 1?\nA:", "answer": "The maximum flow from node 7 to node 1 is 1.", "difficulty": "easy", "doc_id": "146"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 1 with capacity 5,\nan edge from node 0 to node 12 with capacity 18,\nan edge from node 0 to node 13 with capacity 13,\nan edge from node 1 to node 4 with capacity 6,\nan edge from node 1 to node 9 with capacity 20,\nan edge from node 2 to node 10 with capacity 11,\nan edge from node 2 to node 8 with capacity 18,\nan edge from node 2 to node 9 with capacity 7,\nan edge from node 3 to node 6 with capacity 14,\nan edge from node 3 to node 10 with capacity 7,\nan edge from node 3 to node 8 with capacity 15,\nan edge from node 3 to node 5 with capacity 13,\nan edge from node 3 to node 12 with capacity 1,\nan edge from node 4 to node 3 with capacity 4,\nan edge from node 4 to node 2 with capacity 11,\nan edge from node 4 to node 13 with capacity 12,\nan edge from node 5 to node 4 with capacity 14,\nan edge from node 5 to node 2 with capacity 12,\nan edge from node 5 to node 13 with capacity 4,\nan edge from node 6 to node 10 with capacity 2,\nan edge from node 6 to node 12 with capacity 18,\nan edge from node 7 to node 4 with capacity 6,\nan edge from node 7 to node 6 with capacity 13,\nan edge from node 7 to node 9 with capacity 7,\nan edge from node 7 to node 2 with capacity 15,\nan edge from node 8 to node 0 with capacity 20,\nan edge from node 8 to node 4 with capacity 9,\nan edge from node 8 to node 6 with capacity 7,\nan edge from node 8 to node 5 with capacity 12,\nan edge from node 8 to node 2 with capacity 20,\nan edge from node 8 to node 12 with capacity 4,\nan edge from node 8 to node 13 with capacity 10,\nan edge from node 9 to node 10 with capacity 11,\nan edge from node 10 to node 3 with capacity 11,\nan edge from node 10 to node 13 with capacity 17,\nan edge from node 11 to node 7 with capacity 7,\nan edge from node 11 to node 6 with capacity 15,\nan edge from node 11 to node 10 with capacity 14,\nan edge from node 11 to node 8 with capacity 20,\nan edge from node 12 to node 11 with capacity 12,\nan edge from node 12 to node 5 with capacity 16,\nan edge from node 13 to node 10 with capacity 5,\nan edge from node 13 to node 5 with capacity 6,\nan edge from node 13 to node 2 with capacity 14.\nQ: What is the maximum flow from node 11 to node 1?\nA:", "answer": "The maximum flow from node 11 to node 1 is 5.", "difficulty": "hard", "doc_id": "147"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 6,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 2 with capacity 1,\nan edge from node 1 to node 3 with capacity 9,\nan edge from node 1 to node 0 with capacity 7,\nan edge from node 2 to node 4 with capacity 7,\nan edge from node 3 to node 0 with capacity 7,\nan edge from node 3 to node 4 with capacity 3,\nan edge from node 4 to node 2 with capacity 9.\nQ: What is the maximum flow from node 3 to node 2?\nA:", "answer": "The maximum flow from node 3 to node 2 is 10.", "difficulty": "easy", "doc_id": "148"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 6 with capacity 8,\nan edge from node 0 to node 5 with capacity 15,\nan edge from node 0 to node 8 with capacity 20,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 1 to node 6 with capacity 5,\nan edge from node 1 to node 13 with capacity 9,\nan edge from node 1 to node 11 with capacity 2,\nan edge from node 1 to node 4 with capacity 13,\nan edge from node 1 to node 15 with capacity 15,\nan edge from node 1 to node 7 with capacity 11,\nan edge from node 2 to node 6 with capacity 15,\nan edge from node 2 to node 3 with capacity 15,\nan edge from node 2 to node 5 with capacity 17,\nan edge from node 2 to node 4 with capacity 7,\nan edge from node 2 to node 15 with capacity 8,\nan edge from node 2 to node 1 with capacity 20,\nan edge from node 3 to node 0 with capacity 4,\nan edge from node 3 to node 16 with capacity 4,\nan edge from node 3 to node 2 with capacity 8,\nan edge from node 4 to node 10 with capacity 8,\nan edge from node 4 to node 16 with capacity 12,\nan edge from node 4 to node 8 with capacity 15,\nan edge from node 4 to node 2 with capacity 5,\nan edge from node 5 to node 6 with capacity 16,\nan edge from node 5 to node 3 with capacity 3,\nan edge from node 5 to node 11 with capacity 7,\nan edge from node 5 to node 1 with capacity 6,\nan edge from node 6 to node 10 with capacity 20,\nan edge from node 6 to node 14 with capacity 12,\nan edge from node 6 to node 8 with capacity 13,\nan edge from node 8 to node 6 with capacity 5,\nan edge from node 8 to node 10 with capacity 17,\nan edge from node 8 to node 4 with capacity 7,\nan edge from node 8 to node 15 with capacity 10,\nan edge from node 8 to node 12 with capacity 5,\nan edge from node 8 to node 1 with capacity 15,\nan edge from node 9 to node 1 with capacity 8,\nan edge from node 9 to node 8 with capacity 13,\nan edge from node 9 to node 2 with capacity 15,\nan edge from node 10 to node 9 with capacity 17,\nan edge from node 10 to node 12 with capacity 8,\nan edge from node 11 to node 0 with capacity 18,\nan edge from node 11 to node 5 with capacity 4,\nan edge from node 11 to node 16 with capacity 20,\nan edge from node 11 to node 4 with capacity 7,\nan edge from node 11 to node 9 with capacity 14,\nan edge from node 11 to node 2 with capacity 4,\nan edge from node 12 to node 13 with capacity 5,\nan edge from node 12 to node 5 with capacity 13,\nan edge from node 12 to node 15 with capacity 9,\nan edge from node 12 to node 1 with capacity 16,\nan edge from node 12 to node 2 with capacity 5,\nan edge from node 13 to node 0 with capacity 20,\nan edge from node 13 to node 4 with capacity 12,\nan edge from node 13 to node 9 with capacity 12,\nan edge from node 13 to node 8 with capacity 9,\nan edge from node 13 to node 2 with capacity 19,\nan edge from node 14 to node 16 with capacity 12,\nan edge from node 14 to node 7 with capacity 16,\nan edge from node 14 to node 1 with capacity 14,\nan edge from node 15 to node 6 with capacity 17,\nan edge from node 15 to node 13 with capacity 17,\nan edge from node 15 to node 14 with capacity 18,\nan edge from node 15 to node 4 with capacity 9,\nan edge from node 15 to node 9 with capacity 19,\nan edge from node 15 to node 7 with capacity 10,\nan edge from node 15 to node 1 with capacity 15,\nan edge from node 16 to node 0 with capacity 18,\nan edge from node 16 to node 3 with capacity 10,\nan edge from node 16 to node 9 with capacity 18,\nan edge from node 16 to node 7 with capacity 6,\nan edge from node 16 to node 12 with capacity 5.\nQ: What is the maximum flow from node 6 to node 1?\nA:", "answer": "The maximum flow from node 6 to node 1 is 45.", "difficulty": "hard", "doc_id": "149"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 6 with capacity 10,\nan edge from node 0 to node 1 with capacity 2,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 2 with capacity 1,\nan edge from node 1 to node 0 with capacity 5,\nan edge from node 1 to node 7 with capacity 3,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 3 to node 2 with capacity 10,\nan edge from node 3 to node 7 with capacity 4,\nan edge from node 4 to node 2 with capacity 6,\nan edge from node 4 to node 5 with capacity 1,\nan edge from node 6 to node 3 with capacity 9,\nan edge from node 6 to node 0 with capacity 4,\nan edge from node 6 to node 5 with capacity 9,\nan edge from node 7 to node 3 with capacity 8,\nan edge from node 7 to node 1 with capacity 7,\nan edge from node 7 to node 2 with capacity 7,\nan edge from node 8 to node 0 with capacity 6,\nan edge from node 8 to node 5 with capacity 3.\nQ: What is the maximum flow from node 0 to node 7?\nA:", "answer": "The maximum flow from node 0 to node 7 is 6.", "difficulty": "easy", "doc_id": "150"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 3 with capacity 5,\nan edge from node 0 to node 4 with capacity 4,\nan edge from node 1 to node 5 with capacity 9,\nan edge from node 1 to node 3 with capacity 2,\nan edge from node 1 to node 4 with capacity 10,\nan edge from node 1 to node 9 with capacity 3,\nan edge from node 2 to node 8 with capacity 18,\nan edge from node 2 to node 3 with capacity 1,\nan edge from node 3 to node 5 with capacity 1,\nan edge from node 3 to node 2 with capacity 9,\nan edge from node 4 to node 2 with capacity 8,\nan edge from node 4 to node 3 with capacity 12,\nan edge from node 5 to node 8 with capacity 2,\nan edge from node 5 to node 1 with capacity 17,\nan edge from node 5 to node 0 with capacity 16,\nan edge from node 6 to node 3 with capacity 8,\nan edge from node 6 to node 1 with capacity 18,\nan edge from node 6 to node 0 with capacity 9,\nan edge from node 7 to node 5 with capacity 19,\nan edge from node 7 to node 6 with capacity 20,\nan edge from node 7 to node 10 with capacity 9,\nan edge from node 7 to node 4 with capacity 9,\nan edge from node 7 to node 9 with capacity 12,\nan edge from node 9 to node 6 with capacity 3,\nan edge from node 9 to node 4 with capacity 12,\nan edge from node 9 to node 0 with capacity 9,\nan edge from node 10 to node 5 with capacity 11,\nan edge from node 10 to node 1 with capacity 16.\nQ: What is the maximum flow from node 5 to node 3?\nA:", "answer": "The maximum flow from node 5 to node 3 is 23.", "difficulty": "hard", "doc_id": "151"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 0 to node 3 with capacity 9,\nan edge from node 0 to node 1 with capacity 4,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 2 to node 0 with capacity 7,\nan edge from node 3 to node 4 with capacity 2,\nan edge from node 3 to node 0 with capacity 9,\nan edge from node 4 to node 2 with capacity 10.\nQ: What is the maximum flow from node 0 to node 2?\nA:", "answer": "The maximum flow from node 0 to node 2 is 8.", "difficulty": "easy", "doc_id": "152"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 15 with capacity 10,\nan edge from node 0 to node 4 with capacity 9,\nan edge from node 0 to node 11 with capacity 16,\nan edge from node 0 to node 8 with capacity 15,\nan edge from node 0 to node 10 with capacity 11,\nan edge from node 1 to node 13 with capacity 11,\nan edge from node 1 to node 14 with capacity 10,\nan edge from node 2 to node 7 with capacity 16,\nan edge from node 2 to node 8 with capacity 17,\nan edge from node 2 to node 10 with capacity 6,\nan edge from node 3 to node 0 with capacity 7,\nan edge from node 3 to node 8 with capacity 2,\nan edge from node 3 to node 1 with capacity 17,\nan edge from node 4 to node 0 with capacity 19,\nan edge from node 4 to node 11 with capacity 20,\nan edge from node 5 to node 0 with capacity 10,\nan edge from node 5 to node 11 with capacity 1,\nan edge from node 5 to node 8 with capacity 1,\nan edge from node 5 to node 3 with capacity 2,\nan edge from node 5 to node 9 with capacity 17,\nan edge from node 6 to node 14 with capacity 9,\nan edge from node 7 to node 0 with capacity 1,\nan edge from node 7 to node 15 with capacity 2,\nan edge from node 7 to node 4 with capacity 11,\nan edge from node 7 to node 13 with capacity 8,\nan edge from node 7 to node 6 with capacity 9,\nan edge from node 8 to node 13 with capacity 10,\nan edge from node 8 to node 6 with capacity 5,\nan edge from node 8 to node 2 with capacity 18,\nan edge from node 8 to node 3 with capacity 9,\nan edge from node 9 to node 13 with capacity 3,\nan edge from node 9 to node 12 with capacity 16,\nan edge from node 10 to node 5 with capacity 3,\nan edge from node 10 to node 15 with capacity 17,\nan edge from node 10 to node 7 with capacity 11,\nan edge from node 10 to node 14 with capacity 15,\nan edge from node 10 to node 11 with capacity 1,\nan edge from node 10 to node 1 with capacity 18,\nan edge from node 10 to node 3 with capacity 5,\nan edge from node 10 to node 9 with capacity 2,\nan edge from node 11 to node 5 with capacity 16,\nan edge from node 11 to node 2 with capacity 11,\nan edge from node 12 to node 5 with capacity 19,\nan edge from node 12 to node 4 with capacity 18,\nan edge from node 12 to node 13 with capacity 10,\nan edge from node 12 to node 14 with capacity 4,\nan edge from node 12 to node 11 with capacity 1,\nan edge from node 12 to node 8 with capacity 1,\nan edge from node 13 to node 9 with capacity 14,\nan edge from node 14 to node 11 with capacity 17,\nan edge from node 14 to node 12 with capacity 5,\nan edge from node 14 to node 2 with capacity 4,\nan edge from node 15 to node 5 with capacity 14,\nan edge from node 15 to node 7 with capacity 13,\nan edge from node 15 to node 6 with capacity 16,\nan edge from node 15 to node 1 with capacity 7,\nan edge from node 15 to node 3 with capacity 9.\nQ: What is the maximum flow from node 0 to node 11?\nA:", "answer": "The maximum flow from node 0 to node 11 is 56.", "difficulty": "hard", "doc_id": "153"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 6 with capacity 2,\nan edge from node 1 to node 3 with capacity 2,\nan edge from node 1 to node 6 with capacity 6,\nan edge from node 1 to node 11 with capacity 17,\nan edge from node 1 to node 7 with capacity 16,\nan edge from node 1 to node 8 with capacity 13,\nan edge from node 1 to node 5 with capacity 12,\nan edge from node 2 to node 4 with capacity 1,\nan edge from node 2 to node 0 with capacity 16,\nan edge from node 2 to node 9 with capacity 7,\nan edge from node 2 to node 6 with capacity 10,\nan edge from node 2 to node 7 with capacity 13,\nan edge from node 2 to node 13 with capacity 15,\nan edge from node 2 to node 10 with capacity 11,\nan edge from node 2 to node 8 with capacity 17,\nan edge from node 2 to node 5 with capacity 7,\nan edge from node 3 to node 4 with capacity 16,\nan edge from node 3 to node 0 with capacity 19,\nan edge from node 3 to node 7 with capacity 2,\nan edge from node 3 to node 1 with capacity 15,\nan edge from node 4 to node 3 with capacity 3,\nan edge from node 4 to node 9 with capacity 12,\nan edge from node 4 to node 13 with capacity 8,\nan edge from node 5 to node 12 with capacity 9,\nan edge from node 5 to node 3 with capacity 6,\nan edge from node 5 to node 2 with capacity 12,\nan edge from node 5 to node 4 with capacity 11,\nan edge from node 5 to node 8 with capacity 2,\nan edge from node 7 to node 3 with capacity 16,\nan edge from node 7 to node 0 with capacity 10,\nan edge from node 7 to node 13 with capacity 10,\nan edge from node 7 to node 8 with capacity 3,\nan edge from node 7 to node 1 with capacity 17,\nan edge from node 8 to node 2 with capacity 20,\nan edge from node 8 to node 0 with capacity 20,\nan edge from node 8 to node 6 with capacity 5,\nan edge from node 8 to node 11 with capacity 8,\nan edge from node 9 to node 11 with capacity 2,\nan edge from node 9 to node 8 with capacity 13,\nan edge from node 10 to node 6 with capacity 7,\nan edge from node 11 to node 9 with capacity 11,\nan edge from node 11 to node 7 with capacity 20,\nan edge from node 11 to node 1 with capacity 10,\nan edge from node 12 to node 4 with capacity 16,\nan edge from node 12 to node 0 with capacity 9,\nan edge from node 12 to node 9 with capacity 16,\nan edge from node 12 to node 6 with capacity 9,\nan edge from node 12 to node 7 with capacity 1,\nan edge from node 12 to node 10 with capacity 19,\nan edge from node 13 to node 9 with capacity 4,\nan edge from node 13 to node 11 with capacity 8,\nan edge from node 13 to node 8 with capacity 19,\nan edge from node 13 to node 5 with capacity 12,\nan edge from node 13 to node 1 with capacity 2.\nQ: What is the maximum flow from node 1 to node 6?\nA:", "answer": "The maximum flow from node 1 to node 6 is 39.", "difficulty": "hard", "doc_id": "154"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 0 to node 4 with capacity 6,\nan edge from node 0 to node 6 with capacity 11,\nan edge from node 0 to node 3 with capacity 12,\nan edge from node 0 to node 2 with capacity 13,\nan edge from node 0 to node 8 with capacity 1,\nan edge from node 1 to node 10 with capacity 14,\nan edge from node 1 to node 5 with capacity 15,\nan edge from node 2 to node 7 with capacity 5,\nan edge from node 2 to node 5 with capacity 11,\nan edge from node 3 to node 7 with capacity 17,\nan edge from node 3 to node 10 with capacity 12,\nan edge from node 3 to node 9 with capacity 9,\nan edge from node 3 to node 0 with capacity 12,\nan edge from node 4 to node 6 with capacity 11,\nan edge from node 4 to node 2 with capacity 13,\nan edge from node 5 to node 1 with capacity 1,\nan edge from node 5 to node 8 with capacity 17,\nan edge from node 6 to node 3 with capacity 5,\nan edge from node 6 to node 5 with capacity 5,\nan edge from node 6 to node 2 with capacity 15,\nan edge from node 7 to node 0 with capacity 12,\nan edge from node 7 to node 8 with capacity 5,\nan edge from node 8 to node 7 with capacity 15,\nan edge from node 8 to node 3 with capacity 19,\nan edge from node 9 to node 3 with capacity 13,\nan edge from node 9 to node 8 with capacity 16,\nan edge from node 10 to node 4 with capacity 11,\nan edge from node 10 to node 5 with capacity 3,\nan edge from node 10 to node 9 with capacity 16.\nQ: What is the maximum flow from node 2 to node 6?\nA:", "answer": "The maximum flow from node 2 to node 6 is 16.", "difficulty": "hard", "doc_id": "155"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 3 with capacity 6,\nan edge from node 0 to node 8 with capacity 4,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 1 to node 3 with capacity 4,\nan edge from node 2 to node 5 with capacity 7,\nan edge from node 2 to node 8 with capacity 4,\nan edge from node 4 to node 3 with capacity 6,\nan edge from node 4 to node 0 with capacity 1,\nan edge from node 5 to node 7 with capacity 1,\nan edge from node 6 to node 2 with capacity 10,\nan edge from node 7 to node 2 with capacity 8,\nan edge from node 7 to node 1 with capacity 4,\nan edge from node 7 to node 8 with capacity 6,\nan edge from node 8 to node 1 with capacity 1.\nQ: What is the maximum flow from node 6 to node 1?\nA:", "answer": "The maximum flow from node 6 to node 1 is 2.", "difficulty": "easy", "doc_id": "156"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 9 with capacity 3,\nan edge from node 1 to node 7 with capacity 4,\nan edge from node 1 to node 8 with capacity 8,\nan edge from node 2 to node 1 with capacity 3,\nan edge from node 2 to node 8 with capacity 7,\nan edge from node 3 to node 7 with capacity 1,\nan edge from node 4 to node 0 with capacity 8,\nan edge from node 4 to node 1 with capacity 3,\nan edge from node 4 to node 7 with capacity 9,\nan edge from node 4 to node 2 with capacity 3,\nan edge from node 5 to node 0 with capacity 10,\nan edge from node 5 to node 7 with capacity 6,\nan edge from node 6 to node 5 with capacity 3,\nan edge from node 6 to node 1 with capacity 2,\nan edge from node 6 to node 7 with capacity 7,\nan edge from node 6 to node 2 with capacity 6,\nan edge from node 8 to node 7 with capacity 2,\nan edge from node 8 to node 4 with capacity 7,\nan edge from node 9 to node 0 with capacity 1,\nan edge from node 9 to node 8 with capacity 4.\nQ: What is the maximum flow from node 8 to node 7?\nA:", "answer": "The maximum flow from node 8 to node 7 is 9.", "difficulty": "easy", "doc_id": "157"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 1 to node 2 with capacity 3,\nan edge from node 1 to node 6 with capacity 5,\nan edge from node 2 to node 5 with capacity 1,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 3 to node 2 with capacity 1,\nan edge from node 3 to node 6 with capacity 5,\nan edge from node 3 to node 5 with capacity 1,\nan edge from node 3 to node 0 with capacity 8,\nan edge from node 5 to node 2 with capacity 6,\nan edge from node 6 to node 2 with capacity 6,\nan edge from node 6 to node 1 with capacity 3,\nan edge from node 6 to node 0 with capacity 5.\nQ: What is the maximum flow from node 3 to node 2?\nA:", "answer": "The maximum flow from node 3 to node 2 is 7.", "difficulty": "easy", "doc_id": "158"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 7 with capacity 4,\nan edge from node 1 to node 3 with capacity 10,\nan edge from node 2 to node 0 with capacity 5,\nan edge from node 3 to node 6 with capacity 7,\nan edge from node 3 to node 4 with capacity 2,\nan edge from node 3 to node 7 with capacity 7,\nan edge from node 3 to node 2 with capacity 5,\nan edge from node 4 to node 7 with capacity 6,\nan edge from node 5 to node 7 with capacity 1,\nan edge from node 6 to node 4 with capacity 5,\nan edge from node 6 to node 0 with capacity 8,\nan edge from node 6 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 7?\nA:", "answer": "The maximum flow from node 1 to node 7 is 10.", "difficulty": "easy", "doc_id": "159"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 18 with capacity 11,\nan edge from node 0 to node 16 with capacity 15,\nan edge from node 0 to node 6 with capacity 2,\nan edge from node 0 to node 15 with capacity 14,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 16 with capacity 11,\nan edge from node 1 to node 15 with capacity 13,\nan edge from node 2 to node 7 with capacity 3,\nan edge from node 2 to node 0 with capacity 10,\nan edge from node 2 to node 12 with capacity 17,\nan edge from node 2 to node 6 with capacity 4,\nan edge from node 2 to node 17 with capacity 14,\nan edge from node 3 to node 4 with capacity 18,\nan edge from node 4 to node 5 with capacity 16,\nan edge from node 4 to node 3 with capacity 2,\nan edge from node 4 to node 13 with capacity 9,\nan edge from node 4 to node 17 with capacity 4,\nan edge from node 4 to node 10 with capacity 6,\nan edge from node 5 to node 11 with capacity 18,\nan edge from node 5 to node 4 with capacity 15,\nan edge from node 5 to node 2 with capacity 16,\nan edge from node 5 to node 0 with capacity 11,\nan edge from node 5 to node 17 with capacity 1,\nan edge from node 6 to node 14 with capacity 17,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 6 to node 3 with capacity 13,\nan edge from node 6 to node 18 with capacity 12,\nan edge from node 6 to node 0 with capacity 11,\nan edge from node 6 to node 15 with capacity 11,\nan edge from node 7 to node 4 with capacity 13,\nan edge from node 7 to node 5 with capacity 2,\nan edge from node 7 to node 3 with capacity 20,\nan edge from node 7 to node 13 with capacity 4,\nan edge from node 7 to node 0 with capacity 14,\nan edge from node 7 to node 16 with capacity 2,\nan edge from node 7 to node 17 with capacity 14,\nan edge from node 7 to node 15 with capacity 13,\nan edge from node 8 to node 3 with capacity 1,\nan edge from node 8 to node 2 with capacity 4,\nan edge from node 8 to node 1 with capacity 11,\nan edge from node 8 to node 12 with capacity 3,\nan edge from node 8 to node 6 with capacity 19,\nan edge from node 8 to node 9 with capacity 15,\nan edge from node 9 to node 5 with capacity 2,\nan edge from node 9 to node 7 with capacity 16,\nan edge from node 9 to node 8 with capacity 13,\nan edge from node 10 to node 11 with capacity 12,\nan edge from node 10 to node 2 with capacity 14,\nan edge from node 10 to node 6 with capacity 17,\nan edge from node 10 to node 17 with capacity 12,\nan edge from node 10 to node 9 with capacity 2,\nan edge from node 11 to node 14 with capacity 13,\nan edge from node 11 to node 5 with capacity 2,\nan edge from node 11 to node 18 with capacity 1,\nan edge from node 11 to node 15 with capacity 7,\nan edge from node 12 to node 11 with capacity 19,\nan edge from node 12 to node 5 with capacity 9,\nan edge from node 12 to node 8 with capacity 3,\nan edge from node 12 to node 15 with capacity 5,\nan edge from node 13 to node 3 with capacity 8,\nan edge from node 13 to node 2 with capacity 14,\nan edge from node 13 to node 18 with capacity 8,\nan edge from node 13 to node 16 with capacity 14,\nan edge from node 13 to node 6 with capacity 20,\nan edge from node 14 to node 3 with capacity 19,\nan edge from node 14 to node 16 with capacity 5,\nan edge from node 14 to node 17 with capacity 9,\nan edge from node 14 to node 9 with capacity 15,\nan edge from node 14 to node 15 with capacity 19,\nan edge from node 15 to node 3 with capacity 11,\nan edge from node 15 to node 18 with capacity 20,\nan edge from node 16 to node 4 with capacity 18,\nan edge from node 16 to node 1 with capacity 1,\nan edge from node 16 to node 18 with capacity 15,\nan edge from node 16 to node 0 with capacity 7,\nan edge from node 16 to node 6 with capacity 20,\nan edge from node 16 to node 8 with capacity 7,\nan edge from node 17 to node 4 with capacity 15,\nan edge from node 17 to node 3 with capacity 3,\nan edge from node 17 to node 18 with capacity 5,\nan edge from node 17 to node 0 with capacity 7,\nan edge from node 17 to node 6 with capacity 9,\nan edge from node 17 to node 9 with capacity 11,\nan edge from node 18 to node 1 with capacity 9.\nQ: What is the maximum flow from node 10 to node 6?\nA:", "answer": "The maximum flow from node 10 to node 6 is 57.", "difficulty": "hard", "doc_id": "160"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 9 with capacity 14,\nan edge from node 0 to node 1 with capacity 15,\nan edge from node 0 to node 5 with capacity 13,\nan edge from node 0 to node 10 with capacity 4,\nan edge from node 1 to node 4 with capacity 12,\nan edge from node 1 to node 0 with capacity 8,\nan edge from node 1 to node 3 with capacity 17,\nan edge from node 2 to node 9 with capacity 17,\nan edge from node 2 to node 1 with capacity 18,\nan edge from node 2 to node 12 with capacity 14,\nan edge from node 2 to node 14 with capacity 19,\nan edge from node 2 to node 5 with capacity 13,\nan edge from node 2 to node 10 with capacity 15,\nan edge from node 3 to node 9 with capacity 3,\nan edge from node 3 to node 2 with capacity 18,\nan edge from node 3 to node 13 with capacity 1,\nan edge from node 4 to node 9 with capacity 15,\nan edge from node 4 to node 12 with capacity 15,\nan edge from node 4 to node 14 with capacity 1,\nan edge from node 4 to node 8 with capacity 14,\nan edge from node 5 to node 9 with capacity 17,\nan edge from node 5 to node 2 with capacity 9,\nan edge from node 5 to node 7 with capacity 9,\nan edge from node 6 to node 12 with capacity 14,\nan edge from node 6 to node 11 with capacity 3,\nan edge from node 6 to node 5 with capacity 10,\nan edge from node 6 to node 0 with capacity 14,\nan edge from node 6 to node 3 with capacity 12,\nan edge from node 6 to node 8 with capacity 14,\nan edge from node 7 to node 12 with capacity 2,\nan edge from node 7 to node 0 with capacity 1,\nan edge from node 7 to node 6 with capacity 11,\nan edge from node 8 to node 1 with capacity 7,\nan edge from node 8 to node 13 with capacity 11,\nan edge from node 8 to node 15 with capacity 16,\nan edge from node 9 to node 4 with capacity 16,\nan edge from node 9 to node 5 with capacity 17,\nan edge from node 9 to node 3 with capacity 16,\nan edge from node 9 to node 8 with capacity 6,\nan edge from node 10 to node 9 with capacity 10,\nan edge from node 10 to node 12 with capacity 2,\nan edge from node 10 to node 7 with capacity 1,\nan edge from node 10 to node 11 with capacity 5,\nan edge from node 10 to node 3 with capacity 17,\nan edge from node 10 to node 13 with capacity 14,\nan edge from node 10 to node 8 with capacity 8,\nan edge from node 11 to node 7 with capacity 4,\nan edge from node 11 to node 13 with capacity 1,\nan edge from node 11 to node 15 with capacity 17,\nan edge from node 12 to node 9 with capacity 11,\nan edge from node 12 to node 1 with capacity 13,\nan edge from node 12 to node 7 with capacity 16,\nan edge from node 13 to node 1 with capacity 1,\nan edge from node 13 to node 15 with capacity 1,\nan edge from node 14 to node 9 with capacity 19,\nan edge from node 14 to node 2 with capacity 13,\nan edge from node 15 to node 0 with capacity 5,\nan edge from node 15 to node 3 with capacity 14.\nQ: What is the maximum flow from node 4 to node 10?\nA:", "answer": "The maximum flow from node 4 to node 10 is 19.", "difficulty": "hard", "doc_id": "161"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 7 with capacity 15,\nan edge from node 0 to node 4 with capacity 2,\nan edge from node 0 to node 5 with capacity 4,\nan edge from node 0 to node 8 with capacity 16,\nan edge from node 1 to node 6 with capacity 6,\nan edge from node 1 to node 11 with capacity 7,\nan edge from node 1 to node 10 with capacity 6,\nan edge from node 2 to node 6 with capacity 2,\nan edge from node 2 to node 4 with capacity 11,\nan edge from node 2 to node 3 with capacity 14,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 2 to node 0 with capacity 3,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 0 with capacity 15,\nan edge from node 4 to node 3 with capacity 9,\nan edge from node 4 to node 0 with capacity 12,\nan edge from node 5 to node 4 with capacity 20,\nan edge from node 5 to node 2 with capacity 14,\nan edge from node 6 to node 10 with capacity 7,\nan edge from node 6 to node 3 with capacity 8,\nan edge from node 6 to node 2 with capacity 18,\nan edge from node 7 to node 10 with capacity 10,\nan edge from node 7 to node 1 with capacity 1,\nan edge from node 7 to node 4 with capacity 14,\nan edge from node 7 to node 0 with capacity 5,\nan edge from node 8 to node 11 with capacity 8,\nan edge from node 8 to node 7 with capacity 4,\nan edge from node 9 to node 11 with capacity 3,\nan edge from node 9 to node 0 with capacity 14,\nan edge from node 10 to node 6 with capacity 1,\nan edge from node 10 to node 7 with capacity 3,\nan edge from node 10 to node 0 with capacity 11,\nan edge from node 11 to node 9 with capacity 9,\nan edge from node 11 to node 3 with capacity 16,\nan edge from node 11 to node 2 with capacity 14.\nQ: What is the maximum flow from node 7 to node 6?\nA:", "answer": "The maximum flow from node 7 to node 6 is 4.", "difficulty": "hard", "doc_id": "162"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 4 with capacity 6,\nan edge from node 0 to node 3 with capacity 13,\nan edge from node 1 to node 8 with capacity 11,\nan edge from node 1 to node 0 with capacity 15,\nan edge from node 1 to node 3 with capacity 10,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 2 to node 13 with capacity 5,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 3 with capacity 5,\nan edge from node 3 to node 5 with capacity 4,\nan edge from node 3 to node 8 with capacity 7,\nan edge from node 3 to node 4 with capacity 20,\nan edge from node 3 to node 7 with capacity 9,\nan edge from node 3 to node 11 with capacity 8,\nan edge from node 3 to node 6 with capacity 4,\nan edge from node 4 to node 5 with capacity 9,\nan edge from node 4 to node 1 with capacity 10,\nan edge from node 4 to node 7 with capacity 6,\nan edge from node 4 to node 2 with capacity 11,\nan edge from node 5 to node 2 with capacity 18,\nan edge from node 6 to node 8 with capacity 20,\nan edge from node 6 to node 4 with capacity 2,\nan edge from node 6 to node 1 with capacity 17,\nan edge from node 6 to node 3 with capacity 8,\nan edge from node 7 to node 12 with capacity 4,\nan edge from node 7 to node 5 with capacity 4,\nan edge from node 7 to node 11 with capacity 1,\nan edge from node 7 to node 6 with capacity 10,\nan edge from node 8 to node 12 with capacity 2,\nan edge from node 8 to node 5 with capacity 6,\nan edge from node 8 to node 9 with capacity 12,\nan edge from node 8 to node 13 with capacity 4,\nan edge from node 8 to node 0 with capacity 10,\nan edge from node 8 to node 3 with capacity 20,\nan edge from node 8 to node 2 with capacity 9,\nan edge from node 9 to node 2 with capacity 5,\nan edge from node 9 to node 10 with capacity 10,\nan edge from node 10 to node 8 with capacity 13,\nan edge from node 10 to node 7 with capacity 5,\nan edge from node 11 to node 4 with capacity 9,\nan edge from node 11 to node 2 with capacity 6,\nan edge from node 11 to node 10 with capacity 9,\nan edge from node 12 to node 5 with capacity 18,\nan edge from node 12 to node 11 with capacity 2,\nan edge from node 12 to node 3 with capacity 4,\nan edge from node 13 to node 12 with capacity 16,\nan edge from node 13 to node 3 with capacity 3,\nan edge from node 13 to node 10 with capacity 17.\nQ: What is the maximum flow from node 2 to node 4?\nA:", "answer": "The maximum flow from node 2 to node 4 is 22.", "difficulty": "hard", "doc_id": "163"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 1 to node 7 with capacity 2,\nan edge from node 1 to node 3 with capacity 18,\nan edge from node 1 to node 9 with capacity 16,\nan edge from node 1 to node 6 with capacity 5,\nan edge from node 1 to node 8 with capacity 16,\nan edge from node 2 to node 5 with capacity 11,\nan edge from node 2 to node 13 with capacity 8,\nan edge from node 2 to node 1 with capacity 14,\nan edge from node 2 to node 10 with capacity 7,\nan edge from node 3 to node 6 with capacity 17,\nan edge from node 6 to node 12 with capacity 3,\nan edge from node 6 to node 13 with capacity 9,\nan edge from node 7 to node 15 with capacity 8,\nan edge from node 7 to node 2 with capacity 8,\nan edge from node 7 to node 9 with capacity 3,\nan edge from node 7 to node 4 with capacity 13,\nan edge from node 7 to node 1 with capacity 6,\nan edge from node 7 to node 10 with capacity 11,\nan edge from node 8 to node 13 with capacity 2,\nan edge from node 8 to node 3 with capacity 17,\nan edge from node 8 to node 1 with capacity 13,\nan edge from node 8 to node 10 with capacity 11,\nan edge from node 9 to node 12 with capacity 9,\nan edge from node 9 to node 13 with capacity 18,\nan edge from node 9 to node 10 with capacity 3,\nan edge from node 9 to node 0 with capacity 7,\nan edge from node 9 to node 14 with capacity 1,\nan edge from node 10 to node 15 with capacity 5,\nan edge from node 10 to node 13 with capacity 15,\nan edge from node 10 to node 3 with capacity 13,\nan edge from node 10 to node 6 with capacity 4,\nan edge from node 10 to node 0 with capacity 4,\nan edge from node 11 to node 2 with capacity 12,\nan edge from node 11 to node 4 with capacity 12,\nan edge from node 11 to node 6 with capacity 20,\nan edge from node 11 to node 10 with capacity 15,\nan edge from node 12 to node 15 with capacity 7,\nan edge from node 12 to node 11 with capacity 2,\nan edge from node 12 to node 2 with capacity 11,\nan edge from node 12 to node 3 with capacity 6,\nan edge from node 12 to node 4 with capacity 13,\nan edge from node 12 to node 0 with capacity 17,\nan edge from node 13 to node 0 with capacity 8,\nan edge from node 14 to node 12 with capacity 18,\nan edge from node 14 to node 15 with capacity 1,\nan edge from node 14 to node 5 with capacity 6,\nan edge from node 14 to node 13 with capacity 7,\nan edge from node 14 to node 7 with capacity 3,\nan edge from node 14 to node 9 with capacity 4,\nan edge from node 14 to node 0 with capacity 19,\nan edge from node 15 to node 5 with capacity 15,\nan edge from node 15 to node 10 with capacity 15,\nan edge from node 15 to node 0 with capacity 11.\nQ: What is the maximum flow from node 15 to node 14?\nA:", "answer": "The maximum flow from node 15 to node 14 is 1.", "difficulty": "hard", "doc_id": "164"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 4 with capacity 14,\nan edge from node 0 to node 2 with capacity 11,\nan edge from node 1 to node 0 with capacity 9,\nan edge from node 1 to node 5 with capacity 13,\nan edge from node 1 to node 7 with capacity 17,\nan edge from node 2 to node 6 with capacity 19,\nan edge from node 2 to node 0 with capacity 3,\nan edge from node 2 to node 9 with capacity 14,\nan edge from node 3 to node 9 with capacity 4,\nan edge from node 3 to node 2 with capacity 9,\nan edge from node 4 to node 6 with capacity 15,\nan edge from node 4 to node 1 with capacity 7,\nan edge from node 5 to node 6 with capacity 13,\nan edge from node 5 to node 0 with capacity 6,\nan edge from node 5 to node 10 with capacity 20,\nan edge from node 6 to node 4 with capacity 14,\nan edge from node 6 to node 9 with capacity 4,\nan edge from node 6 to node 10 with capacity 9,\nan edge from node 7 to node 10 with capacity 15,\nan edge from node 9 to node 6 with capacity 18,\nan edge from node 9 to node 7 with capacity 17,\nan edge from node 9 to node 3 with capacity 14,\nan edge from node 10 to node 2 with capacity 8.\nQ: What is the maximum flow from node 9 to node 10?\nA:", "answer": "The maximum flow from node 9 to node 10 is 31.", "difficulty": "hard", "doc_id": "165"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 9,\nan edge from node 1 to node 2 with capacity 2,\nan edge from node 3 to node 0 with capacity 4,\nan edge from node 4 to node 0 with capacity 3,\nan edge from node 4 to node 3 with capacity 5,\nan edge from node 4 to node 2 with capacity 9,\nan edge from node 4 to node 1 with capacity 2.\nQ: What is the maximum flow from node 4 to node 2?\nA:", "answer": "The maximum flow from node 4 to node 2 is 11.", "difficulty": "easy", "doc_id": "166"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 6 with capacity 9,\nan edge from node 0 to node 1 with capacity 3,\nan edge from node 0 to node 2 with capacity 9,\nan edge from node 1 to node 4 with capacity 1,\nan edge from node 1 to node 8 with capacity 8,\nan edge from node 2 to node 0 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 4 to node 7 with capacity 3,\nan edge from node 4 to node 2 with capacity 2,\nan edge from node 5 to node 1 with capacity 4,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 6 to node 3 with capacity 7,\nan edge from node 6 to node 1 with capacity 7,\nan edge from node 6 to node 2 with capacity 6,\nan edge from node 7 to node 4 with capacity 8,\nan edge from node 7 to node 8 with capacity 7,\nan edge from node 8 to node 6 with capacity 4.\nQ: What is the maximum flow from node 3 to node 1?\nA:", "answer": "The maximum flow from node 3 to node 1 is 4.", "difficulty": "easy", "doc_id": "167"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 4 with capacity 9,\nan edge from node 0 to node 10 with capacity 18,\nan edge from node 1 to node 6 with capacity 9,\nan edge from node 1 to node 3 with capacity 19,\nan edge from node 1 to node 10 with capacity 1,\nan edge from node 1 to node 5 with capacity 4,\nan edge from node 2 to node 8 with capacity 18,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 0 with capacity 11,\nan edge from node 2 to node 9 with capacity 1,\nan edge from node 3 to node 1 with capacity 13,\nan edge from node 3 to node 10 with capacity 19,\nan edge from node 4 to node 10 with capacity 19,\nan edge from node 4 to node 5 with capacity 20,\nan edge from node 5 to node 1 with capacity 10,\nan edge from node 5 to node 9 with capacity 8,\nan edge from node 6 to node 1 with capacity 4,\nan edge from node 6 to node 7 with capacity 9,\nan edge from node 6 to node 0 with capacity 17,\nan edge from node 6 to node 2 with capacity 16,\nan edge from node 6 to node 5 with capacity 5,\nan edge from node 7 to node 10 with capacity 9,\nan edge from node 7 to node 5 with capacity 16,\nan edge from node 8 to node 4 with capacity 8,\nan edge from node 8 to node 7 with capacity 7,\nan edge from node 8 to node 5 with capacity 12,\nan edge from node 8 to node 9 with capacity 16,\nan edge from node 9 to node 8 with capacity 12,\nan edge from node 9 to node 10 with capacity 1,\nan edge from node 10 to node 4 with capacity 15.\nQ: What is the maximum flow from node 1 to node 4?\nA:", "answer": "The maximum flow from node 1 to node 4 is 28.", "difficulty": "hard", "doc_id": "168"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 13 with capacity 17,\nan edge from node 1 to node 7 with capacity 2,\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 1 to node 6 with capacity 16,\nan edge from node 1 to node 5 with capacity 19,\nan edge from node 2 to node 12 with capacity 12,\nan edge from node 2 to node 8 with capacity 5,\nan edge from node 2 to node 9 with capacity 4,\nan edge from node 2 to node 6 with capacity 8,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 2 to node 5 with capacity 7,\nan edge from node 3 to node 1 with capacity 17,\nan edge from node 3 to node 10 with capacity 15,\nan edge from node 3 to node 0 with capacity 1,\nan edge from node 3 to node 9 with capacity 19,\nan edge from node 4 to node 2 with capacity 18,\nan edge from node 4 to node 12 with capacity 2,\nan edge from node 4 to node 7 with capacity 3,\nan edge from node 5 to node 3 with capacity 10,\nan edge from node 5 to node 0 with capacity 12,\nan edge from node 5 to node 4 with capacity 3,\nan edge from node 6 to node 10 with capacity 6,\nan edge from node 8 to node 3 with capacity 8,\nan edge from node 8 to node 7 with capacity 2,\nan edge from node 8 to node 11 with capacity 16,\nan edge from node 9 to node 3 with capacity 10,\nan edge from node 10 to node 12 with capacity 14,\nan edge from node 10 to node 0 with capacity 10,\nan edge from node 10 to node 8 with capacity 6,\nan edge from node 10 to node 9 with capacity 7,\nan edge from node 10 to node 4 with capacity 12,\nan edge from node 11 to node 1 with capacity 14,\nan edge from node 11 to node 7 with capacity 8,\nan edge from node 11 to node 0 with capacity 13,\nan edge from node 11 to node 9 with capacity 16,\nan edge from node 12 to node 10 with capacity 1,\nan edge from node 12 to node 13 with capacity 3,\nan edge from node 12 to node 7 with capacity 4,\nan edge from node 12 to node 0 with capacity 20,\nan edge from node 12 to node 9 with capacity 10,\nan edge from node 13 to node 9 with capacity 19.\nQ: What is the maximum flow from node 9 to node 11?\nA:", "answer": "The maximum flow from node 9 to node 11 is 10.", "difficulty": "hard", "doc_id": "169"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 6 with capacity 8,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 2 with capacity 3,\nan edge from node 2 to node 6 with capacity 8,\nan edge from node 2 to node 1 with capacity 3,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 6 with capacity 8,\nan edge from node 4 to node 3 with capacity 1,\nan edge from node 4 to node 2 with capacity 4,\nan edge from node 5 to node 0 with capacity 5,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 6 to node 2 with capacity 9,\nan edge from node 7 to node 0 with capacity 9.\nQ: What is the maximum flow from node 3 to node 5?\nA:", "answer": "The maximum flow from node 3 to node 5 is 1.", "difficulty": "easy", "doc_id": "170"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 4 with capacity 3,\nan edge from node 0 to node 2 with capacity 13,\nan edge from node 0 to node 13 with capacity 20,\nan edge from node 0 to node 6 with capacity 6,\nan edge from node 1 to node 0 with capacity 18,\nan edge from node 1 to node 5 with capacity 2,\nan edge from node 1 to node 15 with capacity 13,\nan edge from node 1 to node 8 with capacity 7,\nan edge from node 1 to node 9 with capacity 20,\nan edge from node 1 to node 13 with capacity 18,\nan edge from node 1 to node 11 with capacity 2,\nan edge from node 2 to node 0 with capacity 16,\nan edge from node 2 to node 4 with capacity 15,\nan edge from node 2 to node 8 with capacity 9,\nan edge from node 2 to node 11 with capacity 5,\nan edge from node 2 to node 6 with capacity 13,\nan edge from node 3 to node 10 with capacity 7,\nan edge from node 3 to node 15 with capacity 10,\nan edge from node 3 to node 14 with capacity 15,\nan edge from node 4 to node 10 with capacity 8,\nan edge from node 4 to node 2 with capacity 3,\nan edge from node 4 to node 3 with capacity 16,\nan edge from node 4 to node 9 with capacity 6,\nan edge from node 4 to node 13 with capacity 9,\nan edge from node 4 to node 11 with capacity 20,\nan edge from node 4 to node 6 with capacity 2,\nan edge from node 5 to node 6 with capacity 6,\nan edge from node 6 to node 0 with capacity 8,\nan edge from node 6 to node 7 with capacity 15,\nan edge from node 6 to node 15 with capacity 19,\nan edge from node 6 to node 14 with capacity 2,\nan edge from node 6 to node 8 with capacity 1,\nan edge from node 6 to node 9 with capacity 17,\nan edge from node 7 to node 0 with capacity 17,\nan edge from node 7 to node 8 with capacity 5,\nan edge from node 8 to node 9 with capacity 5,\nan edge from node 8 to node 11 with capacity 17,\nan edge from node 8 to node 6 with capacity 12,\nan edge from node 9 to node 0 with capacity 13,\nan edge from node 9 to node 5 with capacity 20,\nan edge from node 9 to node 7 with capacity 3,\nan edge from node 9 to node 2 with capacity 20,\nan edge from node 9 to node 12 with capacity 14,\nan edge from node 9 to node 3 with capacity 11,\nan edge from node 10 to node 5 with capacity 19,\nan edge from node 10 to node 14 with capacity 7,\nan edge from node 11 to node 0 with capacity 3,\nan edge from node 11 to node 15 with capacity 6,\nan edge from node 11 to node 12 with capacity 15,\nan edge from node 11 to node 8 with capacity 2,\nan edge from node 11 to node 3 with capacity 13,\nan edge from node 12 to node 0 with capacity 13,\nan edge from node 12 to node 4 with capacity 11,\nan edge from node 12 to node 2 with capacity 2,\nan edge from node 12 to node 15 with capacity 14,\nan edge from node 12 to node 6 with capacity 5,\nan edge from node 13 to node 4 with capacity 14,\nan edge from node 13 to node 2 with capacity 10,\nan edge from node 13 to node 15 with capacity 20,\nan edge from node 13 to node 8 with capacity 14,\nan edge from node 13 to node 3 with capacity 11,\nan edge from node 13 to node 1 with capacity 12,\nan edge from node 13 to node 11 with capacity 5,\nan edge from node 13 to node 6 with capacity 8,\nan edge from node 14 to node 10 with capacity 14,\nan edge from node 14 to node 4 with capacity 15,\nan edge from node 14 to node 12 with capacity 9,\nan edge from node 14 to node 3 with capacity 9,\nan edge from node 14 to node 13 with capacity 8,\nan edge from node 14 to node 6 with capacity 8,\nan edge from node 15 to node 10 with capacity 1,\nan edge from node 15 to node 4 with capacity 3.\nQ: What is the maximum flow from node 15 to node 13?\nA:", "answer": "The maximum flow from node 15 to node 13 is 4.", "difficulty": "hard", "doc_id": "171"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 8 with capacity 10,\nan edge from node 0 to node 2 with capacity 5,\nan edge from node 0 to node 4 with capacity 2,\nan edge from node 1 to node 2 with capacity 2,\nan edge from node 1 to node 9 with capacity 4,\nan edge from node 1 to node 6 with capacity 7,\nan edge from node 1 to node 4 with capacity 3,\nan edge from node 2 to node 5 with capacity 8,\nan edge from node 2 to node 3 with capacity 8,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 9 with capacity 7,\nan edge from node 5 to node 6 with capacity 5,\nan edge from node 5 to node 0 with capacity 10,\nan edge from node 5 to node 1 with capacity 10,\nan edge from node 6 to node 3 with capacity 4,\nan edge from node 7 to node 3 with capacity 8,\nan edge from node 7 to node 4 with capacity 7,\nan edge from node 8 to node 7 with capacity 7,\nan edge from node 8 to node 5 with capacity 9,\nan edge from node 9 to node 5 with capacity 5,\nan edge from node 9 to node 4 with capacity 2.\nQ: What is the maximum flow from node 4 to node 3?\nA:", "answer": "The maximum flow from node 4 to node 3 is 5.", "difficulty": "easy", "doc_id": "172"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 3 with capacity 5,\nan edge from node 0 to node 18 with capacity 9,\nan edge from node 0 to node 6 with capacity 1,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 11 with capacity 13,\nan edge from node 1 to node 6 with capacity 7,\nan edge from node 2 to node 10 with capacity 5,\nan edge from node 2 to node 18 with capacity 1,\nan edge from node 3 to node 16 with capacity 11,\nan edge from node 3 to node 17 with capacity 4,\nan edge from node 3 to node 14 with capacity 4,\nan edge from node 3 to node 10 with capacity 12,\nan edge from node 3 to node 11 with capacity 7,\nan edge from node 3 to node 7 with capacity 4,\nan edge from node 3 to node 15 with capacity 12,\nan edge from node 4 to node 16 with capacity 8,\nan edge from node 4 to node 2 with capacity 10,\nan edge from node 4 to node 3 with capacity 5,\nan edge from node 4 to node 9 with capacity 2,\nan edge from node 5 to node 13 with capacity 10,\nan edge from node 5 to node 4 with capacity 20,\nan edge from node 6 to node 17 with capacity 15,\nan edge from node 6 to node 14 with capacity 14,\nan edge from node 6 to node 10 with capacity 8,\nan edge from node 6 to node 3 with capacity 14,\nan edge from node 6 to node 9 with capacity 4,\nan edge from node 6 to node 11 with capacity 5,\nan edge from node 6 to node 7 with capacity 9,\nan edge from node 6 to node 15 with capacity 8,\nan edge from node 7 to node 14 with capacity 1,\nan edge from node 7 to node 10 with capacity 14,\nan edge from node 7 to node 4 with capacity 7,\nan edge from node 7 to node 12 with capacity 17,\nan edge from node 8 to node 14 with capacity 10,\nan edge from node 8 to node 18 with capacity 7,\nan edge from node 8 to node 11 with capacity 5,\nan edge from node 8 to node 7 with capacity 3,\nan edge from node 8 to node 6 with capacity 8,\nan edge from node 9 to node 5 with capacity 16,\nan edge from node 9 to node 14 with capacity 14,\nan edge from node 9 to node 1 with capacity 14,\nan edge from node 10 to node 3 with capacity 2,\nan edge from node 10 to node 4 with capacity 14,\nan edge from node 10 to node 1 with capacity 15,\nan edge from node 10 to node 6 with capacity 18,\nan edge from node 11 to node 16 with capacity 10,\nan edge from node 11 to node 10 with capacity 9,\nan edge from node 11 to node 1 with capacity 17,\nan edge from node 11 to node 7 with capacity 8,\nan edge from node 11 to node 6 with capacity 4,\nan edge from node 12 to node 0 with capacity 14,\nan edge from node 12 to node 13 with capacity 17,\nan edge from node 12 to node 5 with capacity 4,\nan edge from node 12 to node 14 with capacity 16,\nan edge from node 12 to node 10 with capacity 10,\nan edge from node 12 to node 3 with capacity 17,\nan edge from node 12 to node 15 with capacity 8,\nan edge from node 13 to node 0 with capacity 19,\nan edge from node 13 to node 3 with capacity 5,\nan edge from node 13 to node 18 with capacity 4,\nan edge from node 13 to node 12 with capacity 1,\nan edge from node 14 to node 16 with capacity 11,\nan edge from node 14 to node 5 with capacity 11,\nan edge from node 14 to node 10 with capacity 18,\nan edge from node 14 to node 11 with capacity 17,\nan edge from node 14 to node 15 with capacity 9,\nan edge from node 15 to node 5 with capacity 7,\nan edge from node 15 to node 3 with capacity 10,\nan edge from node 15 to node 9 with capacity 3,\nan edge from node 16 to node 10 with capacity 3,\nan edge from node 16 to node 9 with capacity 17,\nan edge from node 16 to node 8 with capacity 7,\nan edge from node 16 to node 6 with capacity 2,\nan edge from node 16 to node 15 with capacity 3,\nan edge from node 17 to node 5 with capacity 9,\nan edge from node 17 to node 14 with capacity 5,\nan edge from node 17 to node 12 with capacity 18,\nan edge from node 18 to node 16 with capacity 3,\nan edge from node 18 to node 17 with capacity 3,\nan edge from node 18 to node 6 with capacity 6.\nQ: What is the maximum flow from node 14 to node 15?\nA:", "answer": "The maximum flow from node 14 to node 15 is 40.", "difficulty": "hard", "doc_id": "173"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 8 with capacity 10,\nan edge from node 0 to node 1 with capacity 8,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 0 to node 9 with capacity 17,\nan edge from node 1 to node 5 with capacity 20,\nan edge from node 1 to node 0 with capacity 18,\nan edge from node 1 to node 2 with capacity 11,\nan edge from node 2 to node 8 with capacity 5,\nan edge from node 2 to node 5 with capacity 5,\nan edge from node 2 to node 0 with capacity 14,\nan edge from node 2 to node 3 with capacity 16,\nan edge from node 2 to node 12 with capacity 2,\nan edge from node 3 to node 5 with capacity 17,\nan edge from node 3 to node 7 with capacity 19,\nan edge from node 3 to node 11 with capacity 15,\nan edge from node 3 to node 12 with capacity 11,\nan edge from node 4 to node 5 with capacity 7,\nan edge from node 4 to node 3 with capacity 6,\nan edge from node 4 to node 9 with capacity 4,\nan edge from node 4 to node 2 with capacity 10,\nan edge from node 5 to node 6 with capacity 7,\nan edge from node 5 to node 1 with capacity 16,\nan edge from node 6 to node 9 with capacity 7,\nan edge from node 6 to node 2 with capacity 5,\nan edge from node 7 to node 10 with capacity 17,\nan edge from node 7 to node 9 with capacity 9,\nan edge from node 8 to node 9 with capacity 3,\nan edge from node 9 to node 8 with capacity 9,\nan edge from node 9 to node 7 with capacity 1,\nan edge from node 9 to node 1 with capacity 19,\nan edge from node 9 to node 2 with capacity 4,\nan edge from node 10 to node 8 with capacity 10,\nan edge from node 10 to node 6 with capacity 17,\nan edge from node 11 to node 1 with capacity 5,\nan edge from node 11 to node 12 with capacity 17.\nQ: What is the maximum flow from node 5 to node 0?\nA:", "answer": "The maximum flow from node 5 to node 0 is 23.", "difficulty": "hard", "doc_id": "174"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 17 with capacity 17,\nan edge from node 0 to node 5 with capacity 17,\nan edge from node 0 to node 11 with capacity 16,\nan edge from node 0 to node 16 with capacity 10,\nan edge from node 1 to node 15 with capacity 20,\nan edge from node 1 to node 5 with capacity 13,\nan edge from node 1 to node 0 with capacity 18,\nan edge from node 1 to node 16 with capacity 6,\nan edge from node 2 to node 8 with capacity 12,\nan edge from node 2 to node 1 with capacity 7,\nan edge from node 2 to node 6 with capacity 6,\nan edge from node 3 to node 15 with capacity 14,\nan edge from node 3 to node 13 with capacity 13,\nan edge from node 3 to node 9 with capacity 20,\nan edge from node 4 to node 12 with capacity 18,\nan edge from node 4 to node 1 with capacity 11,\nan edge from node 4 to node 14 with capacity 9,\nan edge from node 4 to node 11 with capacity 19,\nan edge from node 4 to node 9 with capacity 16,\nan edge from node 4 to node 16 with capacity 18,\nan edge from node 5 to node 17 with capacity 5,\nan edge from node 5 to node 8 with capacity 14,\nan edge from node 5 to node 3 with capacity 12,\nan edge from node 5 to node 12 with capacity 15,\nan edge from node 5 to node 10 with capacity 7,\nan edge from node 5 to node 6 with capacity 8,\nan edge from node 6 to node 2 with capacity 20,\nan edge from node 6 to node 4 with capacity 2,\nan edge from node 6 to node 12 with capacity 7,\nan edge from node 6 to node 10 with capacity 1,\nan edge from node 6 to node 11 with capacity 5,\nan edge from node 7 to node 15 with capacity 15,\nan edge from node 7 to node 17 with capacity 14,\nan edge from node 7 to node 2 with capacity 11,\nan edge from node 7 to node 4 with capacity 8,\nan edge from node 8 to node 13 with capacity 1,\nan edge from node 8 to node 12 with capacity 9,\nan edge from node 9 to node 17 with capacity 9,\nan edge from node 9 to node 0 with capacity 9,\nan edge from node 10 to node 7 with capacity 13,\nan edge from node 11 to node 17 with capacity 12,\nan edge from node 11 to node 1 with capacity 20,\nan edge from node 11 to node 6 with capacity 20,\nan edge from node 12 to node 5 with capacity 12,\nan edge from node 12 to node 14 with capacity 1,\nan edge from node 12 to node 10 with capacity 15,\nan edge from node 13 to node 7 with capacity 17,\nan edge from node 13 to node 4 with capacity 12,\nan edge from node 13 to node 14 with capacity 4,\nan edge from node 14 to node 13 with capacity 4,\nan edge from node 14 to node 3 with capacity 2,\nan edge from node 14 to node 2 with capacity 3,\nan edge from node 14 to node 12 with capacity 18,\nan edge from node 15 to node 5 with capacity 11,\nan edge from node 15 to node 2 with capacity 9,\nan edge from node 15 to node 1 with capacity 16,\nan edge from node 16 to node 13 with capacity 14,\nan edge from node 16 to node 7 with capacity 3,\nan edge from node 16 to node 4 with capacity 14,\nan edge from node 16 to node 6 with capacity 15,\nan edge from node 17 to node 15 with capacity 9,\nan edge from node 17 to node 3 with capacity 7,\nan edge from node 17 to node 11 with capacity 5.\nQ: What is the maximum flow from node 12 to node 14?\nA:", "answer": "The maximum flow from node 12 to node 14 is 14.", "difficulty": "hard", "doc_id": "175"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 14 with capacity 10,\nan edge from node 0 to node 13 with capacity 7,\nan edge from node 0 to node 1 with capacity 6,\nan edge from node 0 to node 10 with capacity 11,\nan edge from node 0 to node 2 with capacity 4,\nan edge from node 0 to node 11 with capacity 17,\nan edge from node 0 to node 4 with capacity 17,\nan edge from node 1 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 20,\nan edge from node 3 to node 16 with capacity 12,\nan edge from node 3 to node 1 with capacity 1,\nan edge from node 3 to node 10 with capacity 13,\nan edge from node 3 to node 12 with capacity 13,\nan edge from node 3 to node 5 with capacity 5,\nan edge from node 3 to node 11 with capacity 4,\nan edge from node 4 to node 10 with capacity 8,\nan edge from node 4 to node 6 with capacity 2,\nan edge from node 4 to node 5 with capacity 10,\nan edge from node 4 to node 0 with capacity 11,\nan edge from node 5 to node 13 with capacity 20,\nan edge from node 5 to node 9 with capacity 7,\nan edge from node 5 to node 10 with capacity 9,\nan edge from node 5 to node 11 with capacity 14,\nan edge from node 5 to node 8 with capacity 2,\nan edge from node 6 to node 16 with capacity 8,\nan edge from node 6 to node 2 with capacity 9,\nan edge from node 6 to node 12 with capacity 1,\nan edge from node 6 to node 15 with capacity 8,\nan edge from node 8 to node 16 with capacity 9,\nan edge from node 8 to node 2 with capacity 12,\nan edge from node 8 to node 7 with capacity 4,\nan edge from node 8 to node 5 with capacity 16,\nan edge from node 8 to node 15 with capacity 1,\nan edge from node 8 to node 0 with capacity 19,\nan edge from node 9 to node 0 with capacity 12,\nan edge from node 10 to node 14 with capacity 1,\nan edge from node 10 to node 7 with capacity 13,\nan edge from node 10 to node 0 with capacity 20,\nan edge from node 10 to node 8 with capacity 12,\nan edge from node 11 to node 13 with capacity 7,\nan edge from node 11 to node 1 with capacity 3,\nan edge from node 11 to node 10 with capacity 13,\nan edge from node 11 to node 3 with capacity 1,\nan edge from node 11 to node 5 with capacity 19,\nan edge from node 12 to node 16 with capacity 5,\nan edge from node 12 to node 10 with capacity 15,\nan edge from node 12 to node 0 with capacity 3,\nan edge from node 12 to node 4 with capacity 17,\nan edge from node 13 to node 16 with capacity 2,\nan edge from node 13 to node 9 with capacity 2,\nan edge from node 13 to node 1 with capacity 18,\nan edge from node 13 to node 10 with capacity 4,\nan edge from node 13 to node 2 with capacity 9,\nan edge from node 13 to node 7 with capacity 1,\nan edge from node 14 to node 1 with capacity 16,\nan edge from node 14 to node 2 with capacity 14,\nan edge from node 14 to node 5 with capacity 11,\nan edge from node 14 to node 8 with capacity 1,\nan edge from node 15 to node 14 with capacity 19,\nan edge from node 15 to node 9 with capacity 12,\nan edge from node 15 to node 2 with capacity 14,\nan edge from node 16 to node 6 with capacity 12,\nan edge from node 16 to node 8 with capacity 16.\nQ: What is the maximum flow from node 13 to node 4?\nA:", "answer": "The maximum flow from node 13 to node 4 is 19.", "difficulty": "hard", "doc_id": "176"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 12 with capacity 1,\nan edge from node 0 to node 18 with capacity 14,\nan edge from node 0 to node 7 with capacity 18,\nan edge from node 0 to node 19 with capacity 14,\nan edge from node 1 to node 14 with capacity 11,\nan edge from node 1 to node 18 with capacity 14,\nan edge from node 1 to node 6 with capacity 20,\nan edge from node 1 to node 15 with capacity 20,\nan edge from node 1 to node 9 with capacity 12,\nan edge from node 1 to node 4 with capacity 20,\nan edge from node 1 to node 2 with capacity 15,\nan edge from node 2 to node 1 with capacity 9,\nan edge from node 2 to node 12 with capacity 12,\nan edge from node 2 to node 18 with capacity 8,\nan edge from node 2 to node 6 with capacity 4,\nan edge from node 2 to node 7 with capacity 16,\nan edge from node 2 to node 19 with capacity 1,\nan edge from node 2 to node 9 with capacity 5,\nan edge from node 2 to node 0 with capacity 12,\nan edge from node 3 to node 1 with capacity 9,\nan edge from node 3 to node 12 with capacity 12,\nan edge from node 3 to node 15 with capacity 3,\nan edge from node 3 to node 19 with capacity 10,\nan edge from node 4 to node 6 with capacity 16,\nan edge from node 4 to node 7 with capacity 11,\nan edge from node 4 to node 9 with capacity 4,\nan edge from node 4 to node 0 with capacity 13,\nan edge from node 4 to node 17 with capacity 6,\nan edge from node 5 to node 1 with capacity 6,\nan edge from node 5 to node 10 with capacity 12,\nan edge from node 5 to node 4 with capacity 11,\nan edge from node 6 to node 10 with capacity 8,\nan edge from node 6 to node 9 with capacity 16,\nan edge from node 6 to node 4 with capacity 16,\nan edge from node 6 to node 13 with capacity 13,\nan edge from node 7 to node 18 with capacity 20,\nan edge from node 7 to node 3 with capacity 9,\nan edge from node 7 to node 10 with capacity 12,\nan edge from node 7 to node 9 with capacity 11,\nan edge from node 7 to node 11 with capacity 3,\nan edge from node 8 to node 14 with capacity 12,\nan edge from node 8 to node 18 with capacity 12,\nan edge from node 8 to node 7 with capacity 14,\nan edge from node 8 to node 19 with capacity 6,\nan edge from node 8 to node 4 with capacity 4,\nan edge from node 9 to node 14 with capacity 6,\nan edge from node 9 to node 12 with capacity 8,\nan edge from node 9 to node 2 with capacity 9,\nan edge from node 9 to node 0 with capacity 11,\nan edge from node 10 to node 14 with capacity 16,\nan edge from node 10 to node 18 with capacity 9,\nan edge from node 10 to node 9 with capacity 14,\nan edge from node 10 to node 4 with capacity 14,\nan edge from node 10 to node 17 with capacity 2,\nan edge from node 10 to node 11 with capacity 8,\nan edge from node 11 to node 1 with capacity 1,\nan edge from node 11 to node 16 with capacity 13,\nan edge from node 11 to node 12 with capacity 6,\nan edge from node 11 to node 18 with capacity 1,\nan edge from node 11 to node 3 with capacity 10,\nan edge from node 11 to node 15 with capacity 6,\nan edge from node 11 to node 17 with capacity 15,\nan edge from node 12 to node 3 with capacity 5,\nan edge from node 12 to node 0 with capacity 17,\nan edge from node 14 to node 5 with capacity 5,\nan edge from node 14 to node 3 with capacity 10,\nan edge from node 14 to node 15 with capacity 9,\nan edge from node 14 to node 19 with capacity 12,\nan edge from node 14 to node 4 with capacity 6,\nan edge from node 14 to node 0 with capacity 4,\nan edge from node 15 to node 1 with capacity 20,\nan edge from node 15 to node 16 with capacity 1,\nan edge from node 15 to node 10 with capacity 8,\nan edge from node 15 to node 4 with capacity 16,\nan edge from node 15 to node 0 with capacity 14,\nan edge from node 16 to node 3 with capacity 16,\nan edge from node 16 to node 7 with capacity 17,\nan edge from node 16 to node 10 with capacity 15,\nan edge from node 16 to node 9 with capacity 14,\nan edge from node 16 to node 2 with capacity 2,\nan edge from node 16 to node 0 with capacity 4,\nan edge from node 16 to node 11 with capacity 2,\nan edge from node 17 to node 1 with capacity 14,\nan edge from node 17 to node 18 with capacity 12,\nan edge from node 17 to node 8 with capacity 18,\nan edge from node 17 to node 6 with capacity 8,\nan edge from node 17 to node 10 with capacity 18,\nan edge from node 17 to node 9 with capacity 9,\nan edge from node 17 to node 2 with capacity 14,\nan edge from node 18 to node 1 with capacity 20,\nan edge from node 18 to node 12 with capacity 19,\nan edge from node 18 to node 19 with capacity 16,\nan edge from node 18 to node 2 with capacity 16,\nan edge from node 19 to node 1 with capacity 8,\nan edge from node 19 to node 14 with capacity 10,\nan edge from node 19 to node 5 with capacity 7,\nan edge from node 19 to node 3 with capacity 7,\nan edge from node 19 to node 8 with capacity 7,\nan edge from node 19 to node 15 with capacity 19,\nan edge from node 19 to node 9 with capacity 7,\nan edge from node 19 to node 0 with capacity 3.\nQ: What is the maximum flow from node 6 to node 13?\nA:", "answer": "The maximum flow from node 6 to node 13 is 13.", "difficulty": "hard", "doc_id": "177"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 18 with capacity 6,\nan edge from node 0 to node 5 with capacity 18,\nan edge from node 0 to node 7 with capacity 11,\nan edge from node 1 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 13,\nan edge from node 1 to node 0 with capacity 1,\nan edge from node 1 to node 10 with capacity 10,\nan edge from node 1 to node 15 with capacity 19,\nan edge from node 2 to node 4 with capacity 14,\nan edge from node 2 to node 6 with capacity 20,\nan edge from node 2 to node 1 with capacity 9,\nan edge from node 2 to node 13 with capacity 9,\nan edge from node 2 to node 16 with capacity 7,\nan edge from node 2 to node 18 with capacity 20,\nan edge from node 2 to node 17 with capacity 17,\nan edge from node 2 to node 5 with capacity 7,\nan edge from node 3 to node 9 with capacity 12,\nan edge from node 3 to node 0 with capacity 7,\nan edge from node 3 to node 1 with capacity 17,\nan edge from node 3 to node 14 with capacity 10,\nan edge from node 3 to node 15 with capacity 2,\nan edge from node 3 to node 17 with capacity 4,\nan edge from node 3 to node 7 with capacity 16,\nan edge from node 3 to node 12 with capacity 9,\nan edge from node 4 to node 1 with capacity 11,\nan edge from node 4 to node 8 with capacity 19,\nan edge from node 5 to node 10 with capacity 7,\nan edge from node 6 to node 3 with capacity 2,\nan edge from node 6 to node 4 with capacity 9,\nan edge from node 6 to node 13 with capacity 18,\nan edge from node 6 to node 10 with capacity 19,\nan edge from node 6 to node 19 with capacity 2,\nan edge from node 6 to node 5 with capacity 9,\nan edge from node 6 to node 12 with capacity 12,\nan edge from node 7 to node 3 with capacity 20,\nan edge from node 7 to node 6 with capacity 1,\nan edge from node 7 to node 10 with capacity 5,\nan edge from node 7 to node 15 with capacity 3,\nan edge from node 7 to node 17 with capacity 2,\nan edge from node 7 to node 12 with capacity 5,\nan edge from node 8 to node 9 with capacity 20,\nan edge from node 8 to node 4 with capacity 13,\nan edge from node 8 to node 16 with capacity 16,\nan edge from node 8 to node 5 with capacity 1,\nan edge from node 8 to node 7 with capacity 16,\nan edge from node 8 to node 12 with capacity 16,\nan edge from node 9 to node 4 with capacity 12,\nan edge from node 9 to node 6 with capacity 3,\nan edge from node 9 to node 0 with capacity 10,\nan edge from node 9 to node 10 with capacity 5,\nan edge from node 9 to node 19 with capacity 16,\nan edge from node 9 to node 12 with capacity 10,\nan edge from node 10 to node 9 with capacity 8,\nan edge from node 10 to node 2 with capacity 18,\nan edge from node 10 to node 0 with capacity 6,\nan edge from node 10 to node 14 with capacity 9,\nan edge from node 10 to node 19 with capacity 2,\nan edge from node 10 to node 5 with capacity 10,\nan edge from node 10 to node 7 with capacity 8,\nan edge from node 11 to node 9 with capacity 8,\nan edge from node 11 to node 0 with capacity 7,\nan edge from node 11 to node 16 with capacity 7,\nan edge from node 11 to node 19 with capacity 2,\nan edge from node 11 to node 12 with capacity 17,\nan edge from node 12 to node 3 with capacity 19,\nan edge from node 12 to node 4 with capacity 15,\nan edge from node 12 to node 11 with capacity 18,\nan edge from node 12 to node 17 with capacity 6,\nan edge from node 12 to node 5 with capacity 1,\nan edge from node 12 to node 7 with capacity 15,\nan edge from node 13 to node 6 with capacity 17,\nan edge from node 13 to node 0 with capacity 16,\nan edge from node 13 to node 16 with capacity 7,\nan edge from node 13 to node 7 with capacity 13,\nan edge from node 14 to node 9 with capacity 7,\nan edge from node 14 to node 4 with capacity 15,\nan edge from node 14 to node 16 with capacity 8,\nan edge from node 14 to node 11 with capacity 1,\nan edge from node 14 to node 15 with capacity 8,\nan edge from node 14 to node 7 with capacity 6,\nan edge from node 15 to node 3 with capacity 9,\nan edge from node 15 to node 2 with capacity 10,\nan edge from node 15 to node 1 with capacity 12,\nan edge from node 15 to node 16 with capacity 19,\nan edge from node 15 to node 10 with capacity 10,\nan edge from node 16 to node 3 with capacity 14,\nan edge from node 16 to node 0 with capacity 12,\nan edge from node 16 to node 14 with capacity 19,\nan edge from node 16 to node 17 with capacity 9,\nan edge from node 16 to node 19 with capacity 3,\nan edge from node 17 to node 3 with capacity 13,\nan edge from node 17 to node 4 with capacity 18,\nan edge from node 17 to node 1 with capacity 15,\nan edge from node 17 to node 16 with capacity 14,\nan edge from node 17 to node 19 with capacity 8,\nan edge from node 17 to node 8 with capacity 12,\nan edge from node 18 to node 6 with capacity 2,\nan edge from node 18 to node 0 with capacity 7,\nan edge from node 18 to node 13 with capacity 5,\nan edge from node 18 to node 10 with capacity 3,\nan edge from node 18 to node 11 with capacity 17,\nan edge from node 18 to node 14 with capacity 20,\nan edge from node 18 to node 15 with capacity 17,\nan edge from node 18 to node 5 with capacity 8,\nan edge from node 19 to node 6 with capacity 8,\nan edge from node 19 to node 1 with capacity 8,\nan edge from node 19 to node 16 with capacity 19,\nan edge from node 19 to node 18 with capacity 11,\nan edge from node 19 to node 12 with capacity 17.\nQ: What is the maximum flow from node 16 to node 10?\nA:", "answer": "The maximum flow from node 16 to node 10 is 57.", "difficulty": "hard", "doc_id": "178"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 14 with capacity 19,\nan edge from node 0 to node 1 with capacity 19,\nan edge from node 1 to node 14 with capacity 18,\nan edge from node 1 to node 5 with capacity 2,\nan edge from node 1 to node 13 with capacity 7,\nan edge from node 2 to node 0 with capacity 3,\nan edge from node 2 to node 3 with capacity 10,\nan edge from node 2 to node 12 with capacity 8,\nan edge from node 2 to node 1 with capacity 20,\nan edge from node 3 to node 6 with capacity 12,\nan edge from node 3 to node 9 with capacity 10,\nan edge from node 3 to node 8 with capacity 18,\nan edge from node 3 to node 4 with capacity 7,\nan edge from node 4 to node 6 with capacity 2,\nan edge from node 4 to node 0 with capacity 3,\nan edge from node 4 to node 9 with capacity 9,\nan edge from node 4 to node 3 with capacity 13,\nan edge from node 4 to node 12 with capacity 10,\nan edge from node 4 to node 13 with capacity 18,\nan edge from node 5 to node 6 with capacity 8,\nan edge from node 5 to node 9 with capacity 3,\nan edge from node 5 to node 13 with capacity 14,\nan edge from node 6 to node 10 with capacity 8,\nan edge from node 6 to node 5 with capacity 11,\nan edge from node 6 to node 2 with capacity 1,\nan edge from node 6 to node 13 with capacity 5,\nan edge from node 8 to node 0 with capacity 7,\nan edge from node 8 to node 14 with capacity 13,\nan edge from node 8 to node 13 with capacity 2,\nan edge from node 9 to node 0 with capacity 5,\nan edge from node 9 to node 13 with capacity 12,\nan edge from node 9 to node 4 with capacity 15,\nan edge from node 10 to node 5 with capacity 16,\nan edge from node 11 to node 0 with capacity 7,\nan edge from node 11 to node 12 with capacity 9,\nan edge from node 11 to node 13 with capacity 19,\nan edge from node 11 to node 4 with capacity 14,\nan edge from node 12 to node 3 with capacity 19,\nan edge from node 12 to node 7 with capacity 4,\nan edge from node 13 to node 10 with capacity 17,\nan edge from node 13 to node 14 with capacity 6,\nan edge from node 13 to node 4 with capacity 1.\nQ: What is the maximum flow from node 11 to node 2?\nA:", "answer": "The maximum flow from node 11 to node 2 is 1.", "difficulty": "hard", "doc_id": "179"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 19 with capacity 19,\nan edge from node 0 to node 18 with capacity 2,\nan edge from node 0 to node 7 with capacity 18,\nan edge from node 1 to node 17 with capacity 16,\nan edge from node 1 to node 19 with capacity 14,\nan edge from node 2 to node 13 with capacity 1,\nan edge from node 2 to node 18 with capacity 3,\nan edge from node 2 to node 7 with capacity 7,\nan edge from node 2 to node 11 with capacity 14,\nan edge from node 3 to node 17 with capacity 7,\nan edge from node 3 to node 13 with capacity 11,\nan edge from node 3 to node 15 with capacity 9,\nan edge from node 3 to node 12 with capacity 20,\nan edge from node 3 to node 0 with capacity 16,\nan edge from node 4 to node 9 with capacity 1,\nan edge from node 4 to node 14 with capacity 1,\nan edge from node 4 to node 12 with capacity 13,\nan edge from node 4 to node 18 with capacity 1,\nan edge from node 5 to node 3 with capacity 17,\nan edge from node 5 to node 6 with capacity 5,\nan edge from node 5 to node 14 with capacity 16,\nan edge from node 5 to node 8 with capacity 14,\nan edge from node 5 to node 2 with capacity 15,\nan edge from node 5 to node 11 with capacity 2,\nan edge from node 6 to node 3 with capacity 4,\nan edge from node 6 to node 19 with capacity 17,\nan edge from node 6 to node 12 with capacity 15,\nan edge from node 7 to node 17 with capacity 9,\nan edge from node 7 to node 6 with capacity 14,\nan edge from node 7 to node 14 with capacity 15,\nan edge from node 7 to node 16 with capacity 3,\nan edge from node 7 to node 15 with capacity 13,\nan edge from node 7 to node 18 with capacity 14,\nan edge from node 7 to node 11 with capacity 11,\nan edge from node 8 to node 0 with capacity 18,\nan edge from node 9 to node 3 with capacity 8,\nan edge from node 9 to node 10 with capacity 17,\nan edge from node 9 to node 4 with capacity 12,\nan edge from node 10 to node 13 with capacity 19,\nan edge from node 10 to node 2 with capacity 13,\nan edge from node 10 to node 11 with capacity 15,\nan edge from node 11 to node 19 with capacity 8,\nan edge from node 11 to node 13 with capacity 6,\nan edge from node 11 to node 12 with capacity 19,\nan edge from node 11 to node 0 with capacity 18,\nan edge from node 12 to node 13 with capacity 18,\nan edge from node 12 to node 14 with capacity 5,\nan edge from node 12 to node 8 with capacity 16,\nan edge from node 12 to node 1 with capacity 3,\nan edge from node 12 to node 15 with capacity 10,\nan edge from node 13 to node 17 with capacity 2,\nan edge from node 13 to node 8 with capacity 19,\nan edge from node 13 to node 16 with capacity 16,\nan edge from node 13 to node 1 with capacity 3,\nan edge from node 13 to node 12 with capacity 8,\nan edge from node 13 to node 18 with capacity 14,\nan edge from node 13 to node 0 with capacity 1,\nan edge from node 14 to node 5 with capacity 18,\nan edge from node 14 to node 9 with capacity 6,\nan edge from node 14 to node 10 with capacity 14,\nan edge from node 14 to node 15 with capacity 16,\nan edge from node 14 to node 11 with capacity 12,\nan edge from node 15 to node 14 with capacity 7,\nan edge from node 15 to node 8 with capacity 17,\nan edge from node 15 to node 7 with capacity 12,\nan edge from node 16 to node 17 with capacity 19,\nan edge from node 16 to node 19 with capacity 10,\nan edge from node 16 to node 8 with capacity 15,\nan edge from node 16 to node 10 with capacity 1,\nan edge from node 16 to node 12 with capacity 6,\nan edge from node 17 to node 13 with capacity 20,\nan edge from node 17 to node 2 with capacity 4,\nan edge from node 18 to node 5 with capacity 2,\nan edge from node 18 to node 6 with capacity 9,\nan edge from node 18 to node 13 with capacity 1,\nan edge from node 18 to node 8 with capacity 19,\nan edge from node 18 to node 10 with capacity 1,\nan edge from node 18 to node 15 with capacity 12,\nan edge from node 18 to node 4 with capacity 1,\nan edge from node 18 to node 7 with capacity 9,\nan edge from node 19 to node 17 with capacity 11,\nan edge from node 19 to node 13 with capacity 11,\nan edge from node 19 to node 9 with capacity 15,\nan edge from node 19 to node 8 with capacity 20,\nan edge from node 19 to node 10 with capacity 10.\nQ: What is the maximum flow from node 7 to node 6?\nA:", "answer": "The maximum flow from node 7 to node 6 is 28.", "difficulty": "hard", "doc_id": "180"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 18 with capacity 9,\nan edge from node 0 to node 1 with capacity 17,\nan edge from node 0 to node 4 with capacity 12,\nan edge from node 0 to node 2 with capacity 1,\nan edge from node 0 to node 5 with capacity 15,\nan edge from node 0 to node 9 with capacity 10,\nan edge from node 0 to node 15 with capacity 1,\nan edge from node 0 to node 19 with capacity 3,\nan edge from node 0 to node 13 with capacity 5,\nan edge from node 0 to node 14 with capacity 9,\nan edge from node 1 to node 3 with capacity 14,\nan edge from node 1 to node 12 with capacity 19,\nan edge from node 1 to node 5 with capacity 5,\nan edge from node 1 to node 9 with capacity 18,\nan edge from node 1 to node 7 with capacity 14,\nan edge from node 2 to node 18 with capacity 8,\nan edge from node 2 to node 4 with capacity 2,\nan edge from node 2 to node 0 with capacity 19,\nan edge from node 2 to node 8 with capacity 12,\nan edge from node 3 to node 16 with capacity 8,\nan edge from node 3 to node 2 with capacity 20,\nan edge from node 3 to node 12 with capacity 11,\nan edge from node 3 to node 15 with capacity 10,\nan edge from node 3 to node 19 with capacity 2,\nan edge from node 3 to node 17 with capacity 10,\nan edge from node 4 to node 18 with capacity 10,\nan edge from node 4 to node 1 with capacity 17,\nan edge from node 4 to node 15 with capacity 3,\nan edge from node 4 to node 7 with capacity 4,\nan edge from node 4 to node 14 with capacity 1,\nan edge from node 5 to node 3 with capacity 5,\nan edge from node 5 to node 1 with capacity 17,\nan edge from node 5 to node 4 with capacity 17,\nan edge from node 5 to node 2 with capacity 18,\nan edge from node 5 to node 12 with capacity 4,\nan edge from node 5 to node 9 with capacity 9,\nan edge from node 5 to node 0 with capacity 9,\nan edge from node 5 to node 15 with capacity 2,\nan edge from node 5 to node 17 with capacity 6,\nan edge from node 5 to node 13 with capacity 7,\nan edge from node 5 to node 10 with capacity 5,\nan edge from node 6 to node 9 with capacity 1,\nan edge from node 6 to node 0 with capacity 13,\nan edge from node 6 to node 15 with capacity 9,\nan edge from node 7 to node 18 with capacity 16,\nan edge from node 7 to node 16 with capacity 20,\nan edge from node 7 to node 1 with capacity 17,\nan edge from node 7 to node 12 with capacity 10,\nan edge from node 7 to node 9 with capacity 13,\nan edge from node 8 to node 18 with capacity 17,\nan edge from node 8 to node 16 with capacity 11,\nan edge from node 8 to node 3 with capacity 12,\nan edge from node 8 to node 2 with capacity 8,\nan edge from node 8 to node 6 with capacity 14,\nan edge from node 8 to node 15 with capacity 11,\nan edge from node 8 to node 19 with capacity 6,\nan edge from node 8 to node 10 with capacity 19,\nan edge from node 9 to node 16 with capacity 9,\nan edge from node 9 to node 12 with capacity 5,\nan edge from node 9 to node 11 with capacity 1,\nan edge from node 9 to node 13 with capacity 20,\nan edge from node 10 to node 1 with capacity 8,\nan edge from node 10 to node 2 with capacity 18,\nan edge from node 10 to node 9 with capacity 1,\nan edge from node 10 to node 11 with capacity 14,\nan edge from node 10 to node 8 with capacity 10,\nan edge from node 10 to node 19 with capacity 17,\nan edge from node 10 to node 17 with capacity 11,\nan edge from node 10 to node 13 with capacity 9,\nan edge from node 10 to node 14 with capacity 17,\nan edge from node 11 to node 18 with capacity 14,\nan edge from node 11 to node 16 with capacity 13,\nan edge from node 11 to node 12 with capacity 6,\nan edge from node 11 to node 19 with capacity 13,\nan edge from node 11 to node 10 with capacity 12,\nan edge from node 11 to node 14 with capacity 15,\nan edge from node 12 to node 11 with capacity 10,\nan edge from node 12 to node 15 with capacity 20,\nan edge from node 12 to node 10 with capacity 3,\nan edge from node 13 to node 3 with capacity 10,\nan edge from node 13 to node 6 with capacity 9,\nan edge from node 13 to node 11 with capacity 11,\nan edge from node 13 to node 15 with capacity 15,\nan edge from node 13 to node 7 with capacity 11,\nan edge from node 14 to node 1 with capacity 5,\nan edge from node 14 to node 12 with capacity 14,\nan edge from node 15 to node 16 with capacity 6,\nan edge from node 15 to node 6 with capacity 15,\nan edge from node 15 to node 5 with capacity 20,\nan edge from node 15 to node 11 with capacity 18,\nan edge from node 15 to node 8 with capacity 11,\nan edge from node 16 to node 11 with capacity 15,\nan edge from node 16 to node 19 with capacity 16,\nan edge from node 16 to node 17 with capacity 11,\nan edge from node 17 to node 1 with capacity 20,\nan edge from node 17 to node 2 with capacity 15,\nan edge from node 17 to node 12 with capacity 3,\nan edge from node 17 to node 8 with capacity 6,\nan edge from node 18 to node 9 with capacity 7,\nan edge from node 18 to node 7 with capacity 16,\nan edge from node 18 to node 13 with capacity 2,\nan edge from node 19 to node 9 with capacity 18,\nan edge from node 19 to node 0 with capacity 3,\nan edge from node 19 to node 14 with capacity 8.\nQ: What is the maximum flow from node 5 to node 0?\nA:", "answer": "The maximum flow from node 5 to node 0 is 44.", "difficulty": "hard", "doc_id": "181"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 13 with capacity 10,\nan edge from node 0 to node 12 with capacity 10,\nan edge from node 0 to node 6 with capacity 4,\nan edge from node 1 to node 11 with capacity 19,\nan edge from node 1 to node 2 with capacity 4,\nan edge from node 1 to node 14 with capacity 17,\nan edge from node 2 to node 13 with capacity 10,\nan edge from node 2 to node 3 with capacity 9,\nan edge from node 2 to node 11 with capacity 12,\nan edge from node 2 to node 14 with capacity 14,\nan edge from node 2 to node 4 with capacity 17,\nan edge from node 3 to node 11 with capacity 8,\nan edge from node 3 to node 6 with capacity 5,\nan edge from node 3 to node 15 with capacity 5,\nan edge from node 3 to node 4 with capacity 3,\nan edge from node 4 to node 13 with capacity 7,\nan edge from node 4 to node 10 with capacity 8,\nan edge from node 4 to node 9 with capacity 14,\nan edge from node 4 to node 11 with capacity 12,\nan edge from node 4 to node 2 with capacity 2,\nan edge from node 4 to node 15 with capacity 9,\nan edge from node 5 to node 12 with capacity 12,\nan edge from node 5 to node 8 with capacity 11,\nan edge from node 5 to node 9 with capacity 7,\nan edge from node 6 to node 9 with capacity 1,\nan edge from node 6 to node 11 with capacity 7,\nan edge from node 6 to node 15 with capacity 18,\nan edge from node 7 to node 9 with capacity 12,\nan edge from node 7 to node 15 with capacity 4,\nan edge from node 7 to node 5 with capacity 11,\nan edge from node 8 to node 10 with capacity 9,\nan edge from node 8 to node 2 with capacity 9,\nan edge from node 8 to node 0 with capacity 19,\nan edge from node 8 to node 15 with capacity 9,\nan edge from node 9 to node 13 with capacity 5,\nan edge from node 9 to node 7 with capacity 15,\nan edge from node 9 to node 3 with capacity 15,\nan edge from node 9 to node 14 with capacity 5,\nan edge from node 10 to node 3 with capacity 9,\nan edge from node 10 to node 9 with capacity 10,\nan edge from node 10 to node 14 with capacity 4,\nan edge from node 11 to node 15 with capacity 2,\nan edge from node 12 to node 1 with capacity 6,\nan edge from node 12 to node 8 with capacity 8,\nan edge from node 12 to node 3 with capacity 15,\nan edge from node 12 to node 9 with capacity 10,\nan edge from node 12 to node 2 with capacity 11,\nan edge from node 12 to node 0 with capacity 13,\nan edge from node 12 to node 5 with capacity 15,\nan edge from node 13 to node 10 with capacity 16,\nan edge from node 13 to node 9 with capacity 18,\nan edge from node 13 to node 15 with capacity 10,\nan edge from node 13 to node 4 with capacity 12,\nan edge from node 14 to node 12 with capacity 16,\nan edge from node 14 to node 7 with capacity 1,\nan edge from node 14 to node 9 with capacity 5,\nan edge from node 14 to node 11 with capacity 6,\nan edge from node 15 to node 1 with capacity 16,\nan edge from node 15 to node 12 with capacity 2,\nan edge from node 15 to node 9 with capacity 7,\nan edge from node 15 to node 5 with capacity 14.\nQ: What is the maximum flow from node 10 to node 8?\nA:", "answer": "The maximum flow from node 10 to node 8 is 19.", "difficulty": "hard", "doc_id": "182"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 2,\nan edge from node 1 to node 5 with capacity 6,\nan edge from node 1 to node 0 with capacity 9,\nan edge from node 2 to node 0 with capacity 4,\nan edge from node 3 to node 4 with capacity 8,\nan edge from node 3 to node 0 with capacity 8,\nan edge from node 4 to node 5 with capacity 10,\nan edge from node 5 to node 4 with capacity 1,\nan edge from node 5 to node 0 with capacity 4,\nan edge from node 5 to node 3 with capacity 8.\nQ: What is the maximum flow from node 5 to node 4?\nA:", "answer": "The maximum flow from node 5 to node 4 is 11.", "difficulty": "easy", "doc_id": "183"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 1 to node 5 with capacity 8,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 1 to node 0 with capacity 1,\nan edge from node 2 to node 0 with capacity 5,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 5 to node 2 with capacity 4.\nQ: What is the maximum flow from node 1 to node 0?\nA:", "answer": "The maximum flow from node 1 to node 0 is 10.", "difficulty": "easy", "doc_id": "184"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 9,\nan edge from node 0 to node 6 with capacity 5,\nan edge from node 1 to node 5 with capacity 10,\nan edge from node 1 to node 6 with capacity 6,\nan edge from node 1 to node 2 with capacity 4,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 2 to node 6 with capacity 8,\nan edge from node 2 to node 0 with capacity 8,\nan edge from node 3 to node 0 with capacity 7,\nan edge from node 4 to node 3 with capacity 2,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 6 with capacity 3,\nan edge from node 5 to node 1 with capacity 5,\nan edge from node 5 to node 4 with capacity 2,\nan edge from node 6 to node 3 with capacity 1.\nQ: What is the maximum flow from node 4 to node 0?\nA:", "answer": "The maximum flow from node 4 to node 0 is 5.", "difficulty": "easy", "doc_id": "185"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 1 with capacity 13,\nan edge from node 0 to node 12 with capacity 12,\nan edge from node 0 to node 2 with capacity 15,\nan edge from node 0 to node 7 with capacity 9,\nan edge from node 1 to node 6 with capacity 19,\nan edge from node 1 to node 2 with capacity 18,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 2 to node 5 with capacity 9,\nan edge from node 2 to node 12 with capacity 4,\nan edge from node 2 to node 8 with capacity 12,\nan edge from node 2 to node 0 with capacity 6,\nan edge from node 2 to node 7 with capacity 8,\nan edge from node 3 to node 4 with capacity 9,\nan edge from node 3 to node 5 with capacity 20,\nan edge from node 3 to node 12 with capacity 15,\nan edge from node 3 to node 9 with capacity 9,\nan edge from node 3 to node 8 with capacity 18,\nan edge from node 3 to node 10 with capacity 5,\nan edge from node 3 to node 0 with capacity 20,\nan edge from node 3 to node 7 with capacity 12,\nan edge from node 4 to node 6 with capacity 15,\nan edge from node 4 to node 9 with capacity 19,\nan edge from node 4 to node 8 with capacity 17,\nan edge from node 4 to node 3 with capacity 7,\nan edge from node 5 to node 13 with capacity 15,\nan edge from node 5 to node 6 with capacity 2,\nan edge from node 5 to node 0 with capacity 15,\nan edge from node 6 to node 11 with capacity 19,\nan edge from node 6 to node 1 with capacity 9,\nan edge from node 6 to node 5 with capacity 7,\nan edge from node 6 to node 3 with capacity 4,\nan edge from node 7 to node 4 with capacity 14,\nan edge from node 7 to node 1 with capacity 20,\nan edge from node 7 to node 5 with capacity 9,\nan edge from node 7 to node 13 with capacity 20,\nan edge from node 7 to node 10 with capacity 14,\nan edge from node 8 to node 5 with capacity 9,\nan edge from node 9 to node 1 with capacity 20,\nan edge from node 9 to node 5 with capacity 19,\nan edge from node 9 to node 2 with capacity 1,\nan edge from node 9 to node 8 with capacity 18,\nan edge from node 10 to node 1 with capacity 14,\nan edge from node 10 to node 0 with capacity 18,\nan edge from node 10 to node 3 with capacity 16,\nan edge from node 11 to node 5 with capacity 15,\nan edge from node 11 to node 13 with capacity 14,\nan edge from node 11 to node 0 with capacity 15,\nan edge from node 12 to node 13 with capacity 13,\nan edge from node 12 to node 9 with capacity 2,\nan edge from node 13 to node 6 with capacity 17,\nan edge from node 13 to node 9 with capacity 9,\nan edge from node 13 to node 7 with capacity 4.\nQ: What is the maximum flow from node 13 to node 3?\nA:", "answer": "The maximum flow from node 13 to node 3 is 28.", "difficulty": "hard", "doc_id": "186"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 1 with capacity 5,\nan edge from node 0 to node 9 with capacity 1,\nan edge from node 1 to node 12 with capacity 7,\nan edge from node 1 to node 6 with capacity 9,\nan edge from node 1 to node 3 with capacity 8,\nan edge from node 1 to node 10 with capacity 17,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 9 with capacity 11,\nan edge from node 2 to node 12 with capacity 19,\nan edge from node 2 to node 5 with capacity 9,\nan edge from node 2 to node 0 with capacity 13,\nan edge from node 2 to node 7 with capacity 6,\nan edge from node 2 to node 13 with capacity 11,\nan edge from node 2 to node 11 with capacity 2,\nan edge from node 3 to node 10 with capacity 5,\nan edge from node 3 to node 4 with capacity 18,\nan edge from node 3 to node 7 with capacity 19,\nan edge from node 4 to node 10 with capacity 8,\nan edge from node 4 to node 5 with capacity 7,\nan edge from node 4 to node 8 with capacity 14,\nan edge from node 4 to node 0 with capacity 18,\nan edge from node 5 to node 4 with capacity 16,\nan edge from node 6 to node 12 with capacity 20,\nan edge from node 6 to node 3 with capacity 15,\nan edge from node 6 to node 10 with capacity 8,\nan edge from node 6 to node 8 with capacity 16,\nan edge from node 6 to node 2 with capacity 4,\nan edge from node 6 to node 0 with capacity 7,\nan edge from node 6 to node 4 with capacity 2,\nan edge from node 6 to node 11 with capacity 1,\nan edge from node 7 to node 6 with capacity 2,\nan edge from node 7 to node 3 with capacity 5,\nan edge from node 7 to node 13 with capacity 15,\nan edge from node 8 to node 12 with capacity 2,\nan edge from node 8 to node 6 with capacity 13,\nan edge from node 8 to node 1 with capacity 8,\nan edge from node 8 to node 3 with capacity 16,\nan edge from node 8 to node 9 with capacity 20,\nan edge from node 9 to node 3 with capacity 15,\nan edge from node 9 to node 10 with capacity 9,\nan edge from node 9 to node 5 with capacity 2,\nan edge from node 9 to node 2 with capacity 9,\nan edge from node 10 to node 12 with capacity 17,\nan edge from node 10 to node 1 with capacity 17,\nan edge from node 10 to node 5 with capacity 1,\nan edge from node 10 to node 7 with capacity 18,\nan edge from node 11 to node 8 with capacity 18,\nan edge from node 11 to node 7 with capacity 10,\nan edge from node 12 to node 8 with capacity 4,\nan edge from node 12 to node 2 with capacity 15,\nan edge from node 12 to node 7 with capacity 15,\nan edge from node 13 to node 8 with capacity 16,\nan edge from node 13 to node 7 with capacity 12,\nan edge from node 13 to node 11 with capacity 18.\nQ: What is the maximum flow from node 5 to node 9?\nA:", "answer": "The maximum flow from node 5 to node 9 is 16.", "difficulty": "hard", "doc_id": "187"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 2,\nan edge from node 0 to node 6 with capacity 10,\nan edge from node 1 to node 2 with capacity 2,\nan edge from node 1 to node 5 with capacity 9,\nan edge from node 1 to node 4 with capacity 8,\nan edge from node 2 to node 5 with capacity 4,\nan edge from node 2 to node 0 with capacity 3,\nan edge from node 2 to node 6 with capacity 3,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 3 with capacity 6,\nan edge from node 5 to node 6 with capacity 6,\nan edge from node 6 to node 2 with capacity 3,\nan edge from node 6 to node 1 with capacity 5,\nan edge from node 6 to node 0 with capacity 2,\nan edge from node 6 to node 3 with capacity 10,\nan edge from node 6 to node 4 with capacity 1.\nQ: What is the maximum flow from node 6 to node 3?\nA:", "answer": "The maximum flow from node 6 to node 3 is 18.", "difficulty": "easy", "doc_id": "188"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 9 with capacity 15,\nan edge from node 0 to node 6 with capacity 1,\nan edge from node 0 to node 3 with capacity 6,\nan edge from node 0 to node 4 with capacity 3,\nan edge from node 1 to node 3 with capacity 9,\nan edge from node 1 to node 5 with capacity 13,\nan edge from node 2 to node 8 with capacity 17,\nan edge from node 2 to node 5 with capacity 3,\nan edge from node 2 to node 1 with capacity 19,\nan edge from node 4 to node 2 with capacity 8,\nan edge from node 4 to node 8 with capacity 4,\nan edge from node 5 to node 10 with capacity 6,\nan edge from node 5 to node 7 with capacity 16,\nan edge from node 6 to node 10 with capacity 19,\nan edge from node 6 to node 7 with capacity 14,\nan edge from node 6 to node 0 with capacity 7,\nan edge from node 7 to node 10 with capacity 1,\nan edge from node 8 to node 3 with capacity 15,\nan edge from node 8 to node 1 with capacity 18,\nan edge from node 9 to node 7 with capacity 3,\nan edge from node 9 to node 6 with capacity 17,\nan edge from node 9 to node 0 with capacity 20,\nan edge from node 9 to node 2 with capacity 9,\nan edge from node 9 to node 4 with capacity 2,\nan edge from node 10 to node 2 with capacity 18.\nQ: What is the maximum flow from node 4 to node 3?\nA:", "answer": "The maximum flow from node 4 to node 3 is 12.", "difficulty": "hard", "doc_id": "189"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 18 with capacity 17,\nan edge from node 0 to node 7 with capacity 19,\nan edge from node 0 to node 4 with capacity 16,\nan edge from node 0 to node 15 with capacity 7,\nan edge from node 1 to node 7 with capacity 4,\nan edge from node 1 to node 6 with capacity 10,\nan edge from node 1 to node 10 with capacity 7,\nan edge from node 1 to node 15 with capacity 2,\nan edge from node 1 to node 14 with capacity 2,\nan edge from node 1 to node 17 with capacity 6,\nan edge from node 2 to node 7 with capacity 1,\nan edge from node 2 to node 11 with capacity 6,\nan edge from node 2 to node 13 with capacity 18,\nan edge from node 2 to node 15 with capacity 8,\nan edge from node 3 to node 5 with capacity 18,\nan edge from node 3 to node 6 with capacity 15,\nan edge from node 3 to node 9 with capacity 10,\nan edge from node 3 to node 15 with capacity 19,\nan edge from node 4 to node 0 with capacity 14,\nan edge from node 4 to node 10 with capacity 6,\nan edge from node 4 to node 3 with capacity 14,\nan edge from node 4 to node 12 with capacity 13,\nan edge from node 5 to node 18 with capacity 5,\nan edge from node 5 to node 7 with capacity 14,\nan edge from node 5 to node 2 with capacity 6,\nan edge from node 5 to node 4 with capacity 16,\nan edge from node 5 to node 17 with capacity 18,\nan edge from node 5 to node 3 with capacity 1,\nan edge from node 5 to node 1 with capacity 13,\nan edge from node 6 to node 0 with capacity 10,\nan edge from node 6 to node 7 with capacity 17,\nan edge from node 6 to node 11 with capacity 12,\nan edge from node 6 to node 13 with capacity 20,\nan edge from node 6 to node 2 with capacity 18,\nan edge from node 7 to node 5 with capacity 3,\nan edge from node 7 to node 11 with capacity 17,\nan edge from node 7 to node 13 with capacity 19,\nan edge from node 7 to node 10 with capacity 13,\nan edge from node 8 to node 18 with capacity 7,\nan edge from node 8 to node 0 with capacity 14,\nan edge from node 8 to node 9 with capacity 14,\nan edge from node 8 to node 2 with capacity 13,\nan edge from node 8 to node 17 with capacity 14,\nan edge from node 8 to node 12 with capacity 17,\nan edge from node 8 to node 1 with capacity 5,\nan edge from node 9 to node 0 with capacity 7,\nan edge from node 9 to node 2 with capacity 3,\nan edge from node 9 to node 8 with capacity 19,\nan edge from node 9 to node 15 with capacity 6,\nan edge from node 9 to node 3 with capacity 16,\nan edge from node 9 to node 1 with capacity 19,\nan edge from node 10 to node 5 with capacity 2,\nan edge from node 10 to node 18 with capacity 19,\nan edge from node 10 to node 6 with capacity 5,\nan edge from node 10 to node 13 with capacity 7,\nan edge from node 10 to node 15 with capacity 6,\nan edge from node 10 to node 3 with capacity 20,\nan edge from node 11 to node 18 with capacity 1,\nan edge from node 11 to node 7 with capacity 5,\nan edge from node 11 to node 9 with capacity 9,\nan edge from node 11 to node 13 with capacity 10,\nan edge from node 11 to node 4 with capacity 12,\nan edge from node 11 to node 14 with capacity 3,\nan edge from node 12 to node 18 with capacity 9,\nan edge from node 12 to node 9 with capacity 15,\nan edge from node 12 to node 2 with capacity 12,\nan edge from node 12 to node 4 with capacity 15,\nan edge from node 12 to node 15 with capacity 17,\nan edge from node 12 to node 3 with capacity 6,\nan edge from node 12 to node 1 with capacity 6,\nan edge from node 13 to node 11 with capacity 19,\nan edge from node 13 to node 2 with capacity 13,\nan edge from node 13 to node 14 with capacity 19,\nan edge from node 13 to node 17 with capacity 1,\nan edge from node 13 to node 12 with capacity 15,\nan edge from node 14 to node 18 with capacity 8,\nan edge from node 14 to node 7 with capacity 1,\nan edge from node 14 to node 11 with capacity 16,\nan edge from node 14 to node 6 with capacity 1,\nan edge from node 14 to node 2 with capacity 8,\nan edge from node 14 to node 16 with capacity 14,\nan edge from node 14 to node 4 with capacity 8,\nan edge from node 14 to node 15 with capacity 9,\nan edge from node 14 to node 17 with capacity 8,\nan edge from node 15 to node 10 with capacity 7,\nan edge from node 15 to node 2 with capacity 4,\nan edge from node 15 to node 14 with capacity 9,\nan edge from node 15 to node 12 with capacity 7,\nan edge from node 16 to node 13 with capacity 20,\nan edge from node 16 to node 8 with capacity 7,\nan edge from node 16 to node 15 with capacity 13,\nan edge from node 17 to node 0 with capacity 12,\nan edge from node 17 to node 6 with capacity 2,\nan edge from node 17 to node 10 with capacity 12,\nan edge from node 17 to node 8 with capacity 18,\nan edge from node 17 to node 12 with capacity 2,\nan edge from node 18 to node 7 with capacity 8,\nan edge from node 18 to node 2 with capacity 13,\nan edge from node 18 to node 4 with capacity 8,\nan edge from node 18 to node 8 with capacity 19,\nan edge from node 18 to node 15 with capacity 3,\nan edge from node 18 to node 12 with capacity 14.\nQ: What is the maximum flow from node 2 to node 15?\nA:", "answer": "The maximum flow from node 2 to node 15 is 33.", "difficulty": "hard", "doc_id": "190"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 2 with capacity 11,\nan edge from node 0 to node 1 with capacity 4,\nan edge from node 1 to node 13 with capacity 11,\nan edge from node 1 to node 8 with capacity 18,\nan edge from node 1 to node 11 with capacity 10,\nan edge from node 1 to node 6 with capacity 8,\nan edge from node 2 to node 7 with capacity 7,\nan edge from node 3 to node 13 with capacity 2,\nan edge from node 3 to node 2 with capacity 11,\nan edge from node 3 to node 5 with capacity 5,\nan edge from node 3 to node 6 with capacity 1,\nan edge from node 3 to node 1 with capacity 18,\nan edge from node 4 to node 6 with capacity 16,\nan edge from node 5 to node 7 with capacity 1,\nan edge from node 5 to node 4 with capacity 2,\nan edge from node 5 to node 9 with capacity 17,\nan edge from node 6 to node 2 with capacity 3,\nan edge from node 6 to node 8 with capacity 9,\nan edge from node 6 to node 11 with capacity 14,\nan edge from node 6 to node 12 with capacity 5,\nan edge from node 6 to node 5 with capacity 5,\nan edge from node 6 to node 0 with capacity 16,\nan edge from node 7 to node 13 with capacity 11,\nan edge from node 7 to node 9 with capacity 11,\nan edge from node 7 to node 2 with capacity 9,\nan edge from node 7 to node 11 with capacity 6,\nan edge from node 7 to node 1 with capacity 3,\nan edge from node 8 to node 13 with capacity 10,\nan edge from node 8 to node 9 with capacity 17,\nan edge from node 8 to node 5 with capacity 1,\nan edge from node 8 to node 6 with capacity 19,\nan edge from node 8 to node 1 with capacity 3,\nan edge from node 9 to node 3 with capacity 9,\nan edge from node 9 to node 6 with capacity 16,\nan edge from node 9 to node 0 with capacity 1,\nan edge from node 9 to node 1 with capacity 8,\nan edge from node 10 to node 3 with capacity 20,\nan edge from node 10 to node 2 with capacity 19,\nan edge from node 11 to node 10 with capacity 9,\nan edge from node 11 to node 5 with capacity 19,\nan edge from node 11 to node 0 with capacity 7,\nan edge from node 12 to node 10 with capacity 5,\nan edge from node 12 to node 4 with capacity 15,\nan edge from node 12 to node 3 with capacity 14,\nan edge from node 12 to node 8 with capacity 20,\nan edge from node 12 to node 14 with capacity 5,\nan edge from node 13 to node 10 with capacity 19,\nan edge from node 13 to node 3 with capacity 15,\nan edge from node 13 to node 0 with capacity 3,\nan edge from node 14 to node 7 with capacity 3,\nan edge from node 14 to node 9 with capacity 7,\nan edge from node 14 to node 1 with capacity 3.\nQ: What is the maximum flow from node 14 to node 2?\nA:", "answer": "The maximum flow from node 14 to node 2 is 13.", "difficulty": "hard", "doc_id": "191"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 5 with capacity 6,\nan edge from node 0 to node 3 with capacity 4,\nan edge from node 1 to node 6 with capacity 4,\nan edge from node 1 to node 0 with capacity 4,\nan edge from node 1 to node 3 with capacity 5,\nan edge from node 4 to node 6 with capacity 5,\nan edge from node 4 to node 3 with capacity 8,\nan edge from node 5 to node 2 with capacity 3,\nan edge from node 6 to node 0 with capacity 8.\nQ: What is the maximum flow from node 1 to node 3?\nA:", "answer": "The maximum flow from node 1 to node 3 is 9.", "difficulty": "easy", "doc_id": "192"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 1 with capacity 8,\nan edge from node 1 to node 6 with capacity 1,\nan edge from node 2 to node 6 with capacity 5,\nan edge from node 5 to node 1 with capacity 10,\nan edge from node 6 to node 7 with capacity 9,\nan edge from node 7 to node 3 with capacity 7,\nan edge from node 7 to node 5 with capacity 10,\nan edge from node 7 to node 6 with capacity 9,\nan edge from node 7 to node 0 with capacity 1.\nQ: What is the maximum flow from node 7 to node 6?\nA:", "answer": "The maximum flow from node 7 to node 6 is 10.", "difficulty": "easy", "doc_id": "193"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 4 with capacity 6,\nan edge from node 0 to node 5 with capacity 19,\nan edge from node 0 to node 10 with capacity 8,\nan edge from node 0 to node 13 with capacity 13,\nan edge from node 0 to node 19 with capacity 8,\nan edge from node 1 to node 3 with capacity 7,\nan edge from node 1 to node 17 with capacity 7,\nan edge from node 1 to node 15 with capacity 2,\nan edge from node 1 to node 16 with capacity 7,\nan edge from node 1 to node 10 with capacity 15,\nan edge from node 1 to node 13 with capacity 11,\nan edge from node 2 to node 12 with capacity 5,\nan edge from node 2 to node 19 with capacity 19,\nan edge from node 3 to node 18 with capacity 12,\nan edge from node 3 to node 17 with capacity 11,\nan edge from node 3 to node 4 with capacity 9,\nan edge from node 3 to node 5 with capacity 4,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 4 to node 2 with capacity 7,\nan edge from node 4 to node 1 with capacity 1,\nan edge from node 4 to node 16 with capacity 7,\nan edge from node 4 to node 13 with capacity 5,\nan edge from node 4 to node 19 with capacity 20,\nan edge from node 5 to node 18 with capacity 13,\nan edge from node 5 to node 7 with capacity 6,\nan edge from node 5 to node 12 with capacity 5,\nan edge from node 5 to node 6 with capacity 16,\nan edge from node 5 to node 16 with capacity 13,\nan edge from node 6 to node 17 with capacity 7,\nan edge from node 6 to node 7 with capacity 7,\nan edge from node 6 to node 2 with capacity 7,\nan edge from node 6 to node 12 with capacity 5,\nan edge from node 6 to node 16 with capacity 15,\nan edge from node 6 to node 19 with capacity 7,\nan edge from node 7 to node 15 with capacity 20,\nan edge from node 7 to node 4 with capacity 1,\nan edge from node 7 to node 1 with capacity 12,\nan edge from node 7 to node 16 with capacity 19,\nan edge from node 7 to node 11 with capacity 13,\nan edge from node 7 to node 13 with capacity 11,\nan edge from node 8 to node 15 with capacity 20,\nan edge from node 8 to node 0 with capacity 11,\nan edge from node 8 to node 14 with capacity 8,\nan edge from node 8 to node 6 with capacity 1,\nan edge from node 8 to node 11 with capacity 5,\nan edge from node 9 to node 5 with capacity 1,\nan edge from node 9 to node 8 with capacity 19,\nan edge from node 9 to node 10 with capacity 5,\nan edge from node 9 to node 11 with capacity 7,\nan edge from node 9 to node 13 with capacity 6,\nan edge from node 10 to node 4 with capacity 12,\nan edge from node 10 to node 2 with capacity 13,\nan edge from node 10 to node 1 with capacity 6,\nan edge from node 10 to node 11 with capacity 7,\nan edge from node 11 to node 3 with capacity 6,\nan edge from node 11 to node 17 with capacity 16,\nan edge from node 11 to node 8 with capacity 2,\nan edge from node 11 to node 16 with capacity 3,\nan edge from node 11 to node 9 with capacity 1,\nan edge from node 12 to node 3 with capacity 3,\nan edge from node 12 to node 15 with capacity 16,\nan edge from node 12 to node 0 with capacity 17,\nan edge from node 12 to node 1 with capacity 8,\nan edge from node 12 to node 5 with capacity 12,\nan edge from node 12 to node 16 with capacity 14,\nan edge from node 12 to node 10 with capacity 2,\nan edge from node 13 to node 3 with capacity 2,\nan edge from node 13 to node 2 with capacity 19,\nan edge from node 13 to node 16 with capacity 16,\nan edge from node 13 to node 11 with capacity 5,\nan edge from node 13 to node 19 with capacity 8,\nan edge from node 14 to node 5 with capacity 13,\nan edge from node 14 to node 9 with capacity 16,\nan edge from node 15 to node 18 with capacity 13,\nan edge from node 15 to node 17 with capacity 20,\nan edge from node 15 to node 4 with capacity 11,\nan edge from node 15 to node 12 with capacity 6,\nan edge from node 15 to node 10 with capacity 7,\nan edge from node 15 to node 11 with capacity 11,\nan edge from node 16 to node 18 with capacity 3,\nan edge from node 16 to node 15 with capacity 3,\nan edge from node 16 to node 12 with capacity 14,\nan edge from node 16 to node 1 with capacity 4,\nan edge from node 16 to node 5 with capacity 15,\nan edge from node 16 to node 6 with capacity 17,\nan edge from node 16 to node 9 with capacity 19,\nan edge from node 17 to node 15 with capacity 15,\nan edge from node 17 to node 0 with capacity 16,\nan edge from node 17 to node 12 with capacity 6,\nan edge from node 17 to node 1 with capacity 7,\nan edge from node 17 to node 5 with capacity 7,\nan edge from node 17 to node 11 with capacity 18,\nan edge from node 18 to node 7 with capacity 17,\nan edge from node 18 to node 1 with capacity 12,\nan edge from node 18 to node 5 with capacity 17,\nan edge from node 18 to node 8 with capacity 9,\nan edge from node 18 to node 13 with capacity 8,\nan edge from node 19 to node 18 with capacity 3,\nan edge from node 19 to node 17 with capacity 2,\nan edge from node 19 to node 0 with capacity 15,\nan edge from node 19 to node 4 with capacity 12,\nan edge from node 19 to node 2 with capacity 10,\nan edge from node 19 to node 5 with capacity 13.\nQ: What is the maximum flow from node 7 to node 15?\nA:", "answer": "The maximum flow from node 7 to node 15 is 76.", "difficulty": "hard", "doc_id": "194"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 2,\nan edge from node 2 to node 4 with capacity 9,\nan edge from node 2 to node 1 with capacity 8,\nan edge from node 3 to node 0 with capacity 1,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 0 with capacity 5.\nQ: What is the maximum flow from node 2 to node 1?\nA:", "answer": "The maximum flow from node 2 to node 1 is 10.", "difficulty": "easy", "doc_id": "195"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 4 with capacity 14,\nan edge from node 0 to node 5 with capacity 18,\nan edge from node 0 to node 6 with capacity 15,\nan edge from node 0 to node 12 with capacity 14,\nan edge from node 0 to node 18 with capacity 4,\nan edge from node 0 to node 3 with capacity 9,\nan edge from node 0 to node 8 with capacity 7,\nan edge from node 1 to node 5 with capacity 1,\nan edge from node 1 to node 13 with capacity 15,\nan edge from node 1 to node 2 with capacity 7,\nan edge from node 1 to node 7 with capacity 3,\nan edge from node 2 to node 0 with capacity 8,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 2 to node 13 with capacity 2,\nan edge from node 2 to node 6 with capacity 10,\nan edge from node 2 to node 16 with capacity 13,\nan edge from node 2 to node 9 with capacity 1,\nan edge from node 2 to node 7 with capacity 1,\nan edge from node 3 to node 11 with capacity 2,\nan edge from node 3 to node 12 with capacity 16,\nan edge from node 3 to node 14 with capacity 15,\nan edge from node 4 to node 11 with capacity 15,\nan edge from node 4 to node 19 with capacity 13,\nan edge from node 4 to node 6 with capacity 1,\nan edge from node 4 to node 17 with capacity 8,\nan edge from node 4 to node 9 with capacity 3,\nan edge from node 4 to node 14 with capacity 18,\nan edge from node 5 to node 0 with capacity 19,\nan edge from node 5 to node 13 with capacity 1,\nan edge from node 5 to node 18 with capacity 10,\nan edge from node 5 to node 2 with capacity 18,\nan edge from node 6 to node 0 with capacity 5,\nan edge from node 6 to node 19 with capacity 15,\nan edge from node 6 to node 1 with capacity 15,\nan edge from node 6 to node 12 with capacity 20,\nan edge from node 6 to node 14 with capacity 1,\nan edge from node 6 to node 2 with capacity 9,\nan edge from node 7 to node 13 with capacity 8,\nan edge from node 7 to node 1 with capacity 18,\nan edge from node 7 to node 6 with capacity 10,\nan edge from node 7 to node 10 with capacity 8,\nan edge from node 7 to node 17 with capacity 1,\nan edge from node 7 to node 14 with capacity 6,\nan edge from node 7 to node 2 with capacity 14,\nan edge from node 7 to node 3 with capacity 14,\nan edge from node 8 to node 19 with capacity 14,\nan edge from node 8 to node 12 with capacity 11,\nan edge from node 8 to node 16 with capacity 11,\nan edge from node 9 to node 0 with capacity 15,\nan edge from node 9 to node 5 with capacity 7,\nan edge from node 9 to node 11 with capacity 20,\nan edge from node 9 to node 6 with capacity 9,\nan edge from node 9 to node 17 with capacity 4,\nan edge from node 9 to node 14 with capacity 10,\nan edge from node 9 to node 2 with capacity 17,\nan edge from node 9 to node 7 with capacity 1,\nan edge from node 10 to node 0 with capacity 3,\nan edge from node 10 to node 17 with capacity 10,\nan edge from node 10 to node 3 with capacity 13,\nan edge from node 10 to node 8 with capacity 2,\nan edge from node 11 to node 0 with capacity 13,\nan edge from node 11 to node 4 with capacity 12,\nan edge from node 11 to node 6 with capacity 9,\nan edge from node 11 to node 17 with capacity 5,\nan edge from node 12 to node 5 with capacity 11,\nan edge from node 12 to node 11 with capacity 9,\nan edge from node 12 to node 6 with capacity 11,\nan edge from node 12 to node 10 with capacity 9,\nan edge from node 12 to node 14 with capacity 19,\nan edge from node 12 to node 2 with capacity 15,\nan edge from node 13 to node 11 with capacity 3,\nan edge from node 13 to node 2 with capacity 13,\nan edge from node 13 to node 8 with capacity 13,\nan edge from node 14 to node 11 with capacity 12,\nan edge from node 14 to node 10 with capacity 6,\nan edge from node 14 to node 15 with capacity 1,\nan edge from node 14 to node 16 with capacity 5,\nan edge from node 15 to node 0 with capacity 13,\nan edge from node 15 to node 5 with capacity 5,\nan edge from node 15 to node 1 with capacity 3,\nan edge from node 15 to node 12 with capacity 16,\nan edge from node 15 to node 18 with capacity 8,\nan edge from node 15 to node 14 with capacity 14,\nan edge from node 16 to node 0 with capacity 10,\nan edge from node 16 to node 4 with capacity 5,\nan edge from node 16 to node 5 with capacity 8,\nan edge from node 16 to node 19 with capacity 16,\nan edge from node 16 to node 12 with capacity 19,\nan edge from node 16 to node 10 with capacity 20,\nan edge from node 16 to node 18 with capacity 18,\nan edge from node 16 to node 3 with capacity 7,\nan edge from node 17 to node 0 with capacity 17,\nan edge from node 17 to node 4 with capacity 6,\nan edge from node 17 to node 12 with capacity 9,\nan edge from node 17 to node 14 with capacity 6,\nan edge from node 18 to node 1 with capacity 8,\nan edge from node 18 to node 15 with capacity 13,\nan edge from node 18 to node 8 with capacity 5,\nan edge from node 19 to node 2 with capacity 1,\nan edge from node 19 to node 8 with capacity 12.\nQ: What is the maximum flow from node 18 to node 19?\nA:", "answer": "The maximum flow from node 18 to node 19 is 26.", "difficulty": "hard", "doc_id": "196"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 1 to node 12 with capacity 6,\nan edge from node 1 to node 7 with capacity 20,\nan edge from node 2 to node 12 with capacity 1,\nan edge from node 2 to node 6 with capacity 7,\nan edge from node 2 to node 11 with capacity 11,\nan edge from node 2 to node 10 with capacity 14,\nan edge from node 2 to node 8 with capacity 17,\nan edge from node 2 to node 13 with capacity 10,\nan edge from node 3 to node 2 with capacity 19,\nan edge from node 3 to node 9 with capacity 7,\nan edge from node 3 to node 4 with capacity 16,\nan edge from node 4 to node 11 with capacity 18,\nan edge from node 4 to node 1 with capacity 20,\nan edge from node 4 to node 7 with capacity 6,\nan edge from node 4 to node 14 with capacity 13,\nan edge from node 5 to node 2 with capacity 4,\nan edge from node 5 to node 9 with capacity 2,\nan edge from node 6 to node 0 with capacity 20,\nan edge from node 6 to node 11 with capacity 8,\nan edge from node 6 to node 7 with capacity 7,\nan edge from node 6 to node 8 with capacity 12,\nan edge from node 7 to node 1 with capacity 8,\nan edge from node 7 to node 9 with capacity 6,\nan edge from node 8 to node 10 with capacity 8,\nan edge from node 8 to node 3 with capacity 10,\nan edge from node 8 to node 4 with capacity 7,\nan edge from node 9 to node 0 with capacity 18,\nan edge from node 9 to node 6 with capacity 14,\nan edge from node 9 to node 7 with capacity 10,\nan edge from node 9 to node 13 with capacity 5,\nan edge from node 9 to node 4 with capacity 7,\nan edge from node 10 to node 7 with capacity 2,\nan edge from node 11 to node 6 with capacity 15,\nan edge from node 11 to node 14 with capacity 19,\nan edge from node 11 to node 13 with capacity 15,\nan edge from node 12 to node 10 with capacity 6,\nan edge from node 12 to node 1 with capacity 14,\nan edge from node 12 to node 5 with capacity 19,\nan edge from node 13 to node 10 with capacity 18,\nan edge from node 13 to node 7 with capacity 8,\nan edge from node 14 to node 10 with capacity 12,\nan edge from node 14 to node 7 with capacity 12,\nan edge from node 14 to node 8 with capacity 12,\nan edge from node 14 to node 9 with capacity 20,\nan edge from node 14 to node 13 with capacity 3.\nQ: What is the maximum flow from node 1 to node 0?\nA:", "answer": "The maximum flow from node 1 to node 0 is 12.", "difficulty": "hard", "doc_id": "197"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 8 with capacity 7,\nan edge from node 0 to node 10 with capacity 17,\nan edge from node 0 to node 12 with capacity 17,\nan edge from node 1 to node 0 with capacity 20,\nan edge from node 2 to node 8 with capacity 5,\nan edge from node 2 to node 11 with capacity 15,\nan edge from node 3 to node 8 with capacity 8,\nan edge from node 3 to node 14 with capacity 3,\nan edge from node 3 to node 0 with capacity 11,\nan edge from node 3 to node 1 with capacity 1,\nan edge from node 3 to node 2 with capacity 15,\nan edge from node 3 to node 12 with capacity 6,\nan edge from node 3 to node 9 with capacity 16,\nan edge from node 4 to node 5 with capacity 12,\nan edge from node 4 to node 10 with capacity 12,\nan edge from node 4 to node 14 with capacity 13,\nan edge from node 4 to node 2 with capacity 14,\nan edge from node 4 to node 12 with capacity 14,\nan edge from node 4 to node 9 with capacity 3,\nan edge from node 5 to node 10 with capacity 17,\nan edge from node 5 to node 13 with capacity 20,\nan edge from node 5 to node 11 with capacity 4,\nan edge from node 5 to node 9 with capacity 3,\nan edge from node 6 to node 8 with capacity 17,\nan edge from node 6 to node 5 with capacity 15,\nan edge from node 6 to node 13 with capacity 13,\nan edge from node 6 to node 7 with capacity 15,\nan edge from node 6 to node 11 with capacity 10,\nan edge from node 6 to node 9 with capacity 20,\nan edge from node 7 to node 8 with capacity 19,\nan edge from node 7 to node 10 with capacity 8,\nan edge from node 7 to node 13 with capacity 18,\nan edge from node 7 to node 9 with capacity 6,\nan edge from node 8 to node 0 with capacity 9,\nan edge from node 8 to node 12 with capacity 1,\nan edge from node 8 to node 4 with capacity 14,\nan edge from node 8 to node 9 with capacity 20,\nan edge from node 9 to node 10 with capacity 4,\nan edge from node 9 to node 1 with capacity 20,\nan edge from node 9 to node 12 with capacity 16,\nan edge from node 9 to node 4 with capacity 10,\nan edge from node 10 to node 0 with capacity 16,\nan edge from node 10 to node 3 with capacity 2,\nan edge from node 10 to node 2 with capacity 10,\nan edge from node 11 to node 8 with capacity 15,\nan edge from node 11 to node 5 with capacity 9,\nan edge from node 11 to node 2 with capacity 18,\nan edge from node 12 to node 13 with capacity 7,\nan edge from node 12 to node 2 with capacity 16,\nan edge from node 13 to node 14 with capacity 20,\nan edge from node 13 to node 1 with capacity 11,\nan edge from node 13 to node 9 with capacity 20,\nan edge from node 14 to node 5 with capacity 11,\nan edge from node 14 to node 10 with capacity 9,\nan edge from node 14 to node 7 with capacity 15,\nan edge from node 14 to node 11 with capacity 8.\nQ: What is the maximum flow from node 13 to node 2?\nA:", "answer": "The maximum flow from node 13 to node 2 is 51.", "difficulty": "hard", "doc_id": "198"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 15 with capacity 12,\nan edge from node 0 to node 3 with capacity 10,\nan edge from node 0 to node 6 with capacity 9,\nan edge from node 0 to node 10 with capacity 1,\nan edge from node 0 to node 7 with capacity 6,\nan edge from node 1 to node 11 with capacity 17,\nan edge from node 1 to node 8 with capacity 11,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 7 with capacity 6,\nan edge from node 1 to node 0 with capacity 20,\nan edge from node 2 to node 15 with capacity 13,\nan edge from node 2 to node 11 with capacity 19,\nan edge from node 2 to node 13 with capacity 18,\nan edge from node 4 to node 10 with capacity 10,\nan edge from node 4 to node 0 with capacity 13,\nan edge from node 5 to node 3 with capacity 10,\nan edge from node 5 to node 8 with capacity 3,\nan edge from node 5 to node 12 with capacity 11,\nan edge from node 5 to node 1 with capacity 8,\nan edge from node 6 to node 15 with capacity 17,\nan edge from node 6 to node 0 with capacity 10,\nan edge from node 6 to node 13 with capacity 9,\nan edge from node 7 to node 15 with capacity 15,\nan edge from node 7 to node 9 with capacity 16,\nan edge from node 7 to node 4 with capacity 8,\nan edge from node 7 to node 14 with capacity 1,\nan edge from node 8 to node 15 with capacity 16,\nan edge from node 8 to node 5 with capacity 7,\nan edge from node 8 to node 10 with capacity 4,\nan edge from node 8 to node 12 with capacity 3,\nan edge from node 8 to node 16 with capacity 20,\nan edge from node 8 to node 14 with capacity 11,\nan edge from node 8 to node 0 with capacity 9,\nan edge from node 9 to node 3 with capacity 18,\nan edge from node 9 to node 6 with capacity 13,\nan edge from node 9 to node 8 with capacity 18,\nan edge from node 9 to node 2 with capacity 14,\nan edge from node 9 to node 14 with capacity 19,\nan edge from node 9 to node 13 with capacity 20,\nan edge from node 10 to node 11 with capacity 20,\nan edge from node 10 to node 9 with capacity 7,\nan edge from node 11 to node 5 with capacity 5,\nan edge from node 11 to node 7 with capacity 6,\nan edge from node 11 to node 16 with capacity 18,\nan edge from node 12 to node 15 with capacity 5,\nan edge from node 12 to node 6 with capacity 2,\nan edge from node 12 to node 2 with capacity 11,\nan edge from node 12 to node 16 with capacity 5,\nan edge from node 12 to node 9 with capacity 19,\nan edge from node 12 to node 4 with capacity 11,\nan edge from node 13 to node 11 with capacity 7,\nan edge from node 13 to node 3 with capacity 5,\nan edge from node 13 to node 6 with capacity 17,\nan edge from node 13 to node 5 with capacity 6,\nan edge from node 13 to node 7 with capacity 10,\nan edge from node 13 to node 0 with capacity 7,\nan edge from node 14 to node 15 with capacity 15,\nan edge from node 14 to node 4 with capacity 11,\nan edge from node 15 to node 11 with capacity 3,\nan edge from node 15 to node 1 with capacity 17,\nan edge from node 15 to node 16 with capacity 11,\nan edge from node 15 to node 9 with capacity 13,\nan edge from node 15 to node 14 with capacity 15,\nan edge from node 16 to node 15 with capacity 17,\nan edge from node 16 to node 6 with capacity 19,\nan edge from node 16 to node 8 with capacity 16.\nQ: What is the maximum flow from node 16 to node 7?\nA:", "answer": "The maximum flow from node 16 to node 7 is 28.", "difficulty": "hard", "doc_id": "199"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 4 with capacity 3,\nan edge from node 0 to node 16 with capacity 17,\nan edge from node 0 to node 13 with capacity 1,\nan edge from node 0 to node 8 with capacity 4,\nan edge from node 0 to node 1 with capacity 11,\nan edge from node 1 to node 15 with capacity 9,\nan edge from node 1 to node 6 with capacity 1,\nan edge from node 1 to node 12 with capacity 15,\nan edge from node 2 to node 15 with capacity 16,\nan edge from node 2 to node 4 with capacity 9,\nan edge from node 2 to node 9 with capacity 6,\nan edge from node 2 to node 3 with capacity 20,\nan edge from node 2 to node 6 with capacity 11,\nan edge from node 2 to node 5 with capacity 17,\nan edge from node 3 to node 15 with capacity 7,\nan edge from node 3 to node 11 with capacity 16,\nan edge from node 4 to node 15 with capacity 7,\nan edge from node 4 to node 16 with capacity 17,\nan edge from node 4 to node 8 with capacity 9,\nan edge from node 5 to node 4 with capacity 16,\nan edge from node 5 to node 0 with capacity 1,\nan edge from node 5 to node 1 with capacity 6,\nan edge from node 5 to node 12 with capacity 2,\nan edge from node 6 to node 4 with capacity 9,\nan edge from node 6 to node 3 with capacity 4,\nan edge from node 6 to node 5 with capacity 16,\nan edge from node 6 to node 14 with capacity 10,\nan edge from node 6 to node 13 with capacity 16,\nan edge from node 6 to node 8 with capacity 19,\nan edge from node 6 to node 11 with capacity 12,\nan edge from node 7 to node 0 with capacity 14,\nan edge from node 7 to node 6 with capacity 19,\nan edge from node 7 to node 1 with capacity 7,\nan edge from node 8 to node 15 with capacity 7,\nan edge from node 8 to node 16 with capacity 15,\nan edge from node 8 to node 2 with capacity 4,\nan edge from node 8 to node 5 with capacity 13,\nan edge from node 8 to node 1 with capacity 15,\nan edge from node 8 to node 12 with capacity 12,\nan edge from node 9 to node 4 with capacity 15,\nan edge from node 9 to node 16 with capacity 4,\nan edge from node 9 to node 5 with capacity 6,\nan edge from node 9 to node 13 with capacity 20,\nan edge from node 9 to node 10 with capacity 1,\nan edge from node 9 to node 8 with capacity 7,\nan edge from node 10 to node 4 with capacity 1,\nan edge from node 10 to node 0 with capacity 1,\nan edge from node 10 to node 6 with capacity 13,\nan edge from node 10 to node 12 with capacity 13,\nan edge from node 10 to node 17 with capacity 17,\nan edge from node 11 to node 4 with capacity 6,\nan edge from node 11 to node 7 with capacity 4,\nan edge from node 11 to node 5 with capacity 18,\nan edge from node 11 to node 14 with capacity 8,\nan edge from node 11 to node 13 with capacity 5,\nan edge from node 11 to node 12 with capacity 13,\nan edge from node 12 to node 10 with capacity 11,\nan edge from node 12 to node 1 with capacity 3,\nan edge from node 12 to node 11 with capacity 16,\nan edge from node 12 to node 17 with capacity 7,\nan edge from node 13 to node 15 with capacity 9,\nan edge from node 13 to node 9 with capacity 13,\nan edge from node 14 to node 15 with capacity 16,\nan edge from node 14 to node 2 with capacity 10,\nan edge from node 14 to node 13 with capacity 1,\nan edge from node 14 to node 12 with capacity 5,\nan edge from node 15 to node 7 with capacity 10,\nan edge from node 15 to node 5 with capacity 6,\nan edge from node 15 to node 10 with capacity 7,\nan edge from node 16 to node 15 with capacity 18,\nan edge from node 16 to node 0 with capacity 13,\nan edge from node 16 to node 3 with capacity 17,\nan edge from node 16 to node 6 with capacity 10,\nan edge from node 16 to node 10 with capacity 1,\nan edge from node 17 to node 0 with capacity 1,\nan edge from node 17 to node 7 with capacity 4,\nan edge from node 17 to node 13 with capacity 10.\nQ: What is the maximum flow from node 6 to node 5?\nA:", "answer": "The maximum flow from node 6 to node 5 is 73.", "difficulty": "hard", "doc_id": "200"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 10 with capacity 18,\nan edge from node 0 to node 7 with capacity 10,\nan edge from node 0 to node 6 with capacity 11,\nan edge from node 0 to node 9 with capacity 2,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 7 with capacity 4,\nan edge from node 1 to node 4 with capacity 6,\nan edge from node 1 to node 6 with capacity 12,\nan edge from node 1 to node 9 with capacity 18,\nan edge from node 2 to node 10 with capacity 6,\nan edge from node 2 to node 3 with capacity 5,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 9 with capacity 16,\nan edge from node 3 to node 4 with capacity 13,\nan edge from node 4 to node 7 with capacity 20,\nan edge from node 4 to node 8 with capacity 16,\nan edge from node 4 to node 6 with capacity 9,\nan edge from node 5 to node 7 with capacity 10,\nan edge from node 5 to node 6 with capacity 8,\nan edge from node 5 to node 9 with capacity 6,\nan edge from node 5 to node 11 with capacity 18,\nan edge from node 6 to node 5 with capacity 14,\nan edge from node 6 to node 3 with capacity 14,\nan edge from node 6 to node 8 with capacity 11,\nan edge from node 6 to node 9 with capacity 18,\nan edge from node 7 to node 10 with capacity 4,\nan edge from node 7 to node 5 with capacity 6,\nan edge from node 7 to node 2 with capacity 10,\nan edge from node 7 to node 3 with capacity 1,\nan edge from node 7 to node 0 with capacity 16,\nan edge from node 7 to node 4 with capacity 13,\nan edge from node 7 to node 6 with capacity 17,\nan edge from node 8 to node 1 with capacity 17,\nan edge from node 8 to node 0 with capacity 16,\nan edge from node 8 to node 9 with capacity 4,\nan edge from node 9 to node 5 with capacity 17,\nan edge from node 9 to node 3 with capacity 4,\nan edge from node 10 to node 3 with capacity 19,\nan edge from node 10 to node 8 with capacity 1,\nan edge from node 10 to node 4 with capacity 3,\nan edge from node 11 to node 2 with capacity 10,\nan edge from node 11 to node 8 with capacity 9,\nan edge from node 11 to node 6 with capacity 10.\nQ: What is the maximum flow from node 1 to node 0?\nA:", "answer": "The maximum flow from node 1 to node 0 is 32.", "difficulty": "hard", "doc_id": "201"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 8 with capacity 4,\nan edge from node 0 to node 4 with capacity 5,\nan edge from node 0 to node 3 with capacity 6,\nan edge from node 1 to node 6 with capacity 4,\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 2 to node 6 with capacity 2,\nan edge from node 3 to node 8 with capacity 1,\nan edge from node 3 to node 1 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 4 to node 0 with capacity 4,\nan edge from node 4 to node 7 with capacity 8,\nan edge from node 5 to node 4 with capacity 8,\nan edge from node 5 to node 0 with capacity 10,\nan edge from node 5 to node 7 with capacity 4,\nan edge from node 6 to node 5 with capacity 2,\nan edge from node 6 to node 3 with capacity 3,\nan edge from node 6 to node 0 with capacity 3,\nan edge from node 7 to node 3 with capacity 4,\nan edge from node 8 to node 4 with capacity 7,\nan edge from node 8 to node 1 with capacity 1,\nan edge from node 8 to node 0 with capacity 10.\nQ: What is the maximum flow from node 5 to node 3?\nA:", "answer": "The maximum flow from node 5 to node 3 is 11.", "difficulty": "easy", "doc_id": "202"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 2 with capacity 20,\nan edge from node 0 to node 5 with capacity 7,\nan edge from node 0 to node 7 with capacity 4,\nan edge from node 0 to node 3 with capacity 19,\nan edge from node 0 to node 4 with capacity 16,\nan edge from node 1 to node 2 with capacity 7,\nan edge from node 1 to node 3 with capacity 12,\nan edge from node 1 to node 0 with capacity 13,\nan edge from node 1 to node 4 with capacity 2,\nan edge from node 1 to node 9 with capacity 20,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 2 to node 9 with capacity 1,\nan edge from node 3 to node 10 with capacity 16,\nan edge from node 3 to node 0 with capacity 8,\nan edge from node 3 to node 9 with capacity 14,\nan edge from node 4 to node 2 with capacity 15,\nan edge from node 4 to node 6 with capacity 9,\nan edge from node 4 to node 7 with capacity 1,\nan edge from node 4 to node 8 with capacity 10,\nan edge from node 4 to node 0 with capacity 9,\nan edge from node 4 to node 1 with capacity 19,\nan edge from node 5 to node 8 with capacity 1,\nan edge from node 5 to node 1 with capacity 10,\nan edge from node 7 to node 1 with capacity 3,\nan edge from node 8 to node 6 with capacity 7,\nan edge from node 8 to node 3 with capacity 12,\nan edge from node 8 to node 1 with capacity 11,\nan edge from node 8 to node 4 with capacity 5,\nan edge from node 10 to node 6 with capacity 19,\nan edge from node 10 to node 8 with capacity 2.\nQ: What is the maximum flow from node 4 to node 9?\nA:", "answer": "The maximum flow from node 4 to node 9 is 35.", "difficulty": "hard", "doc_id": "203"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 7 with capacity 5,\nan edge from node 1 to node 8 with capacity 18,\nan edge from node 1 to node 0 with capacity 3,\nan edge from node 2 to node 10 with capacity 11,\nan edge from node 3 to node 4 with capacity 12,\nan edge from node 3 to node 6 with capacity 17,\nan edge from node 4 to node 8 with capacity 13,\nan edge from node 4 to node 6 with capacity 14,\nan edge from node 5 to node 7 with capacity 18,\nan edge from node 6 to node 3 with capacity 9,\nan edge from node 6 to node 8 with capacity 10,\nan edge from node 6 to node 11 with capacity 2,\nan edge from node 6 to node 9 with capacity 20,\nan edge from node 7 to node 4 with capacity 1,\nan edge from node 7 to node 11 with capacity 18,\nan edge from node 7 to node 6 with capacity 2,\nan edge from node 8 to node 7 with capacity 10,\nan edge from node 8 to node 10 with capacity 5,\nan edge from node 8 to node 6 with capacity 3,\nan edge from node 8 to node 0 with capacity 9,\nan edge from node 9 to node 3 with capacity 20,\nan edge from node 9 to node 4 with capacity 5,\nan edge from node 9 to node 1 with capacity 1,\nan edge from node 9 to node 7 with capacity 15,\nan edge from node 9 to node 10 with capacity 16,\nan edge from node 9 to node 2 with capacity 15,\nan edge from node 10 to node 3 with capacity 13,\nan edge from node 10 to node 4 with capacity 11,\nan edge from node 10 to node 1 with capacity 12,\nan edge from node 10 to node 7 with capacity 14,\nan edge from node 10 to node 6 with capacity 17,\nan edge from node 10 to node 9 with capacity 19,\nan edge from node 11 to node 1 with capacity 5,\nan edge from node 11 to node 6 with capacity 16.\nQ: What is the maximum flow from node 4 to node 2?\nA:", "answer": "The maximum flow from node 4 to node 2 is 15.", "difficulty": "hard", "doc_id": "204"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 4 with capacity 12,\nan edge from node 0 to node 13 with capacity 20,\nan edge from node 1 to node 4 with capacity 6,\nan edge from node 1 to node 8 with capacity 2,\nan edge from node 1 to node 9 with capacity 9,\nan edge from node 1 to node 11 with capacity 1,\nan edge from node 2 to node 3 with capacity 17,\nan edge from node 2 to node 8 with capacity 8,\nan edge from node 2 to node 9 with capacity 18,\nan edge from node 2 to node 7 with capacity 9,\nan edge from node 3 to node 1 with capacity 6,\nan edge from node 3 to node 5 with capacity 7,\nan edge from node 3 to node 10 with capacity 7,\nan edge from node 4 to node 3 with capacity 19,\nan edge from node 4 to node 8 with capacity 3,\nan edge from node 4 to node 2 with capacity 4,\nan edge from node 4 to node 1 with capacity 15,\nan edge from node 4 to node 0 with capacity 15,\nan edge from node 4 to node 11 with capacity 10,\nan edge from node 5 to node 4 with capacity 9,\nan edge from node 5 to node 6 with capacity 8,\nan edge from node 5 to node 7 with capacity 14,\nan edge from node 5 to node 0 with capacity 16,\nan edge from node 5 to node 10 with capacity 10,\nan edge from node 5 to node 12 with capacity 10,\nan edge from node 6 to node 8 with capacity 2,\nan edge from node 6 to node 7 with capacity 3,\nan edge from node 6 to node 5 with capacity 10,\nan edge from node 6 to node 10 with capacity 19,\nan edge from node 7 to node 13 with capacity 3,\nan edge from node 7 to node 0 with capacity 7,\nan edge from node 8 to node 1 with capacity 19,\nan edge from node 8 to node 9 with capacity 15,\nan edge from node 8 to node 13 with capacity 2,\nan edge from node 8 to node 0 with capacity 3,\nan edge from node 8 to node 12 with capacity 13,\nan edge from node 9 to node 13 with capacity 20,\nan edge from node 9 to node 0 with capacity 3,\nan edge from node 9 to node 11 with capacity 12,\nan edge from node 9 to node 10 with capacity 12,\nan edge from node 10 to node 9 with capacity 3,\nan edge from node 11 to node 9 with capacity 1,\nan edge from node 11 to node 7 with capacity 2,\nan edge from node 11 to node 13 with capacity 16,\nan edge from node 12 to node 3 with capacity 5,\nan edge from node 12 to node 5 with capacity 2,\nan edge from node 13 to node 4 with capacity 20,\nan edge from node 13 to node 0 with capacity 17,\nan edge from node 13 to node 11 with capacity 6.\nQ: What is the maximum flow from node 13 to node 4?\nA:", "answer": "The maximum flow from node 13 to node 4 is 32.", "difficulty": "hard", "doc_id": "205"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 3 with capacity 17,\nan edge from node 0 to node 1 with capacity 16,\nan edge from node 1 to node 3 with capacity 7,\nan edge from node 1 to node 4 with capacity 12,\nan edge from node 1 to node 5 with capacity 4,\nan edge from node 1 to node 15 with capacity 12,\nan edge from node 2 to node 7 with capacity 2,\nan edge from node 2 to node 8 with capacity 19,\nan edge from node 2 to node 16 with capacity 6,\nan edge from node 3 to node 2 with capacity 19,\nan edge from node 3 to node 10 with capacity 16,\nan edge from node 4 to node 13 with capacity 8,\nan edge from node 4 to node 6 with capacity 3,\nan edge from node 4 to node 14 with capacity 15,\nan edge from node 4 to node 9 with capacity 4,\nan edge from node 4 to node 15 with capacity 12,\nan edge from node 5 to node 11 with capacity 17,\nan edge from node 5 to node 3 with capacity 18,\nan edge from node 5 to node 16 with capacity 3,\nan edge from node 6 to node 12 with capacity 3,\nan edge from node 6 to node 1 with capacity 6,\nan edge from node 6 to node 5 with capacity 2,\nan edge from node 7 to node 2 with capacity 6,\nan edge from node 7 to node 6 with capacity 12,\nan edge from node 7 to node 8 with capacity 18,\nan edge from node 8 to node 4 with capacity 14,\nan edge from node 8 to node 14 with capacity 17,\nan edge from node 8 to node 1 with capacity 20,\nan edge from node 9 to node 12 with capacity 20,\nan edge from node 9 to node 3 with capacity 15,\nan edge from node 9 to node 1 with capacity 3,\nan edge from node 10 to node 3 with capacity 19,\nan edge from node 10 to node 16 with capacity 9,\nan edge from node 10 to node 14 with capacity 6,\nan edge from node 11 to node 0 with capacity 3,\nan edge from node 11 to node 7 with capacity 10,\nan edge from node 11 to node 16 with capacity 12,\nan edge from node 11 to node 5 with capacity 12,\nan edge from node 11 to node 9 with capacity 20,\nan edge from node 12 to node 16 with capacity 15,\nan edge from node 12 to node 5 with capacity 18,\nan edge from node 13 to node 3 with capacity 6,\nan edge from node 13 to node 14 with capacity 1,\nan edge from node 13 to node 5 with capacity 5,\nan edge from node 14 to node 7 with capacity 3,\nan edge from node 14 to node 10 with capacity 2,\nan edge from node 14 to node 16 with capacity 5,\nan edge from node 15 to node 12 with capacity 3,\nan edge from node 15 to node 4 with capacity 2,\nan edge from node 15 to node 10 with capacity 17,\nan edge from node 15 to node 16 with capacity 5.\nQ: What is the maximum flow from node 11 to node 8?\nA:", "answer": "The maximum flow from node 11 to node 8 is 32.", "difficulty": "hard", "doc_id": "206"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 7 with capacity 4,\nan edge from node 0 to node 4 with capacity 9,\nan edge from node 1 to node 3 with capacity 5,\nan edge from node 1 to node 8 with capacity 5,\nan edge from node 2 to node 1 with capacity 3,\nan edge from node 2 to node 6 with capacity 8,\nan edge from node 2 to node 8 with capacity 7,\nan edge from node 3 to node 6 with capacity 8,\nan edge from node 3 to node 8 with capacity 6,\nan edge from node 4 to node 1 with capacity 1,\nan edge from node 4 to node 8 with capacity 3,\nan edge from node 5 to node 7 with capacity 10,\nan edge from node 5 to node 4 with capacity 5,\nan edge from node 6 to node 3 with capacity 2,\nan edge from node 6 to node 0 with capacity 1,\nan edge from node 6 to node 5 with capacity 2,\nan edge from node 6 to node 2 with capacity 10,\nan edge from node 7 to node 0 with capacity 7,\nan edge from node 7 to node 8 with capacity 1,\nan edge from node 8 to node 3 with capacity 6,\nan edge from node 8 to node 4 with capacity 10.\nQ: What is the maximum flow from node 3 to node 4?\nA:", "answer": "The maximum flow from node 3 to node 4 is 13.", "difficulty": "easy", "doc_id": "207"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 12 with capacity 4,\nan edge from node 0 to node 14 with capacity 11,\nan edge from node 0 to node 13 with capacity 10,\nan edge from node 2 to node 15 with capacity 13,\nan edge from node 2 to node 16 with capacity 1,\nan edge from node 3 to node 1 with capacity 11,\nan edge from node 3 to node 4 with capacity 10,\nan edge from node 3 to node 10 with capacity 9,\nan edge from node 4 to node 11 with capacity 8,\nan edge from node 4 to node 14 with capacity 20,\nan edge from node 4 to node 6 with capacity 14,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 4 to node 10 with capacity 14,\nan edge from node 4 to node 13 with capacity 2,\nan edge from node 5 to node 2 with capacity 17,\nan edge from node 5 to node 6 with capacity 17,\nan edge from node 5 to node 13 with capacity 16,\nan edge from node 6 to node 11 with capacity 9,\nan edge from node 6 to node 9 with capacity 14,\nan edge from node 6 to node 13 with capacity 16,\nan edge from node 7 to node 14 with capacity 12,\nan edge from node 7 to node 0 with capacity 6,\nan edge from node 8 to node 12 with capacity 8,\nan edge from node 8 to node 14 with capacity 2,\nan edge from node 8 to node 7 with capacity 17,\nan edge from node 8 to node 0 with capacity 5,\nan edge from node 8 to node 4 with capacity 19,\nan edge from node 9 to node 8 with capacity 9,\nan edge from node 9 to node 2 with capacity 9,\nan edge from node 9 to node 7 with capacity 6,\nan edge from node 9 to node 10 with capacity 14,\nan edge from node 9 to node 13 with capacity 14,\nan edge from node 10 to node 8 with capacity 13,\nan edge from node 10 to node 12 with capacity 13,\nan edge from node 10 to node 15 with capacity 7,\nan edge from node 10 to node 2 with capacity 11,\nan edge from node 10 to node 16 with capacity 14,\nan edge from node 10 to node 5 with capacity 5,\nan edge from node 11 to node 9 with capacity 18,\nan edge from node 11 to node 6 with capacity 8,\nan edge from node 11 to node 16 with capacity 11,\nan edge from node 11 to node 10 with capacity 17,\nan edge from node 12 to node 8 with capacity 12,\nan edge from node 12 to node 2 with capacity 3,\nan edge from node 12 to node 11 with capacity 12,\nan edge from node 12 to node 7 with capacity 7,\nan edge from node 12 to node 6 with capacity 2,\nan edge from node 12 to node 5 with capacity 2,\nan edge from node 12 to node 10 with capacity 8,\nan edge from node 13 to node 15 with capacity 17,\nan edge from node 13 to node 7 with capacity 19,\nan edge from node 13 to node 9 with capacity 5,\nan edge from node 13 to node 4 with capacity 18,\nan edge from node 14 to node 1 with capacity 10,\nan edge from node 14 to node 15 with capacity 20,\nan edge from node 14 to node 7 with capacity 16,\nan edge from node 14 to node 6 with capacity 6,\nan edge from node 14 to node 13 with capacity 6,\nan edge from node 15 to node 12 with capacity 17,\nan edge from node 15 to node 2 with capacity 5,\nan edge from node 15 to node 6 with capacity 9,\nan edge from node 15 to node 0 with capacity 7,\nan edge from node 16 to node 14 with capacity 2,\nan edge from node 16 to node 13 with capacity 14.\nQ: What is the maximum flow from node 7 to node 4?\nA:", "answer": "The maximum flow from node 7 to node 4 is 18.", "difficulty": "hard", "doc_id": "208"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 2 with capacity 5,\nan edge from node 0 to node 1 with capacity 2,\nan edge from node 0 to node 5 with capacity 7,\nan edge from node 1 to node 4 with capacity 8,\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 2 to node 3 with capacity 3,\nan edge from node 2 to node 1 with capacity 4,\nan edge from node 2 to node 5 with capacity 7,\nan edge from node 2 to node 0 with capacity 1,\nan edge from node 3 to node 1 with capacity 6,\nan edge from node 3 to node 5 with capacity 7,\nan edge from node 3 to node 4 with capacity 9,\nan edge from node 4 to node 0 with capacity 4,\nan edge from node 5 to node 2 with capacity 10,\nan edge from node 5 to node 1 with capacity 1.\nQ: What is the maximum flow from node 5 to node 0?\nA:", "answer": "The maximum flow from node 5 to node 0 is 9.", "difficulty": "easy", "doc_id": "209"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 11 with capacity 13,\nan edge from node 0 to node 16 with capacity 16,\nan edge from node 0 to node 4 with capacity 20,\nan edge from node 1 to node 6 with capacity 13,\nan edge from node 1 to node 11 with capacity 2,\nan edge from node 1 to node 8 with capacity 19,\nan edge from node 2 to node 0 with capacity 19,\nan edge from node 2 to node 6 with capacity 17,\nan edge from node 2 to node 15 with capacity 11,\nan edge from node 2 to node 14 with capacity 8,\nan edge from node 2 to node 3 with capacity 18,\nan edge from node 2 to node 7 with capacity 18,\nan edge from node 2 to node 13 with capacity 13,\nan edge from node 3 to node 14 with capacity 14,\nan edge from node 3 to node 16 with capacity 5,\nan edge from node 3 to node 4 with capacity 12,\nan edge from node 3 to node 9 with capacity 3,\nan edge from node 3 to node 10 with capacity 17,\nan edge from node 4 to node 12 with capacity 1,\nan edge from node 4 to node 2 with capacity 13,\nan edge from node 4 to node 10 with capacity 1,\nan edge from node 5 to node 12 with capacity 14,\nan edge from node 5 to node 13 with capacity 7,\nan edge from node 6 to node 3 with capacity 12,\nan edge from node 6 to node 7 with capacity 14,\nan edge from node 6 to node 12 with capacity 7,\nan edge from node 6 to node 10 with capacity 9,\nan edge from node 7 to node 15 with capacity 15,\nan edge from node 8 to node 6 with capacity 18,\nan edge from node 8 to node 3 with capacity 19,\nan edge from node 8 to node 1 with capacity 17,\nan edge from node 8 to node 16 with capacity 7,\nan edge from node 8 to node 13 with capacity 13,\nan edge from node 8 to node 9 with capacity 14,\nan edge from node 8 to node 10 with capacity 10,\nan edge from node 9 to node 5 with capacity 5,\nan edge from node 9 to node 11 with capacity 11,\nan edge from node 9 to node 12 with capacity 7,\nan edge from node 10 to node 5 with capacity 15,\nan edge from node 10 to node 14 with capacity 6,\nan edge from node 10 to node 3 with capacity 17,\nan edge from node 10 to node 2 with capacity 17,\nan edge from node 11 to node 6 with capacity 2,\nan edge from node 11 to node 1 with capacity 17,\nan edge from node 11 to node 7 with capacity 20,\nan edge from node 11 to node 13 with capacity 1,\nan edge from node 12 to node 14 with capacity 13,\nan edge from node 12 to node 4 with capacity 20,\nan edge from node 13 to node 5 with capacity 7,\nan edge from node 13 to node 3 with capacity 14,\nan edge from node 13 to node 8 with capacity 17,\nan edge from node 13 to node 10 with capacity 11,\nan edge from node 14 to node 3 with capacity 8,\nan edge from node 14 to node 8 with capacity 14,\nan edge from node 14 to node 16 with capacity 16,\nan edge from node 14 to node 9 with capacity 12,\nan edge from node 14 to node 10 with capacity 15,\nan edge from node 15 to node 6 with capacity 6,\nan edge from node 15 to node 5 with capacity 3,\nan edge from node 15 to node 1 with capacity 7,\nan edge from node 15 to node 8 with capacity 13,\nan edge from node 15 to node 16 with capacity 16,\nan edge from node 15 to node 12 with capacity 18,\nan edge from node 16 to node 14 with capacity 11,\nan edge from node 16 to node 11 with capacity 18,\nan edge from node 16 to node 4 with capacity 2.\nQ: What is the maximum flow from node 4 to node 1?\nA:", "answer": "The maximum flow from node 4 to node 1 is 15.", "difficulty": "hard", "doc_id": "210"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 4 with capacity 9,\nan edge from node 1 to node 4 with capacity 2,\nan edge from node 2 to node 1 with capacity 2,\nan edge from node 2 to node 0 with capacity 4,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 0 with capacity 5,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 4 to node 3 with capacity 9.\nQ: What is the maximum flow from node 2 to node 3?\nA:", "answer": "The maximum flow from node 2 to node 3 is 9.", "difficulty": "easy", "doc_id": "211"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 1 to node 7 with capacity 3,\nan edge from node 2 to node 3 with capacity 7,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 3 to node 0 with capacity 5,\nan edge from node 3 to node 1 with capacity 4,\nan edge from node 3 to node 8 with capacity 6,\nan edge from node 4 to node 6 with capacity 10,\nan edge from node 5 to node 1 with capacity 6,\nan edge from node 5 to node 7 with capacity 3,\nan edge from node 7 to node 2 with capacity 7,\nan edge from node 8 to node 6 with capacity 5.\nQ: What is the maximum flow from node 5 to node 6?\nA:", "answer": "The maximum flow from node 5 to node 6 is 6.", "difficulty": "easy", "doc_id": "212"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 10 with capacity 12,\nan edge from node 0 to node 7 with capacity 17,\nan edge from node 0 to node 11 with capacity 17,\nan edge from node 1 to node 8 with capacity 20,\nan edge from node 2 to node 8 with capacity 3,\nan edge from node 2 to node 1 with capacity 1,\nan edge from node 2 to node 7 with capacity 18,\nan edge from node 2 to node 11 with capacity 8,\nan edge from node 3 to node 0 with capacity 9,\nan edge from node 3 to node 4 with capacity 11,\nan edge from node 3 to node 6 with capacity 9,\nan edge from node 4 to node 8 with capacity 10,\nan edge from node 4 to node 2 with capacity 19,\nan edge from node 4 to node 5 with capacity 16,\nan edge from node 5 to node 10 with capacity 12,\nan edge from node 5 to node 8 with capacity 5,\nan edge from node 5 to node 1 with capacity 2,\nan edge from node 5 to node 11 with capacity 6,\nan edge from node 5 to node 3 with capacity 16,\nan edge from node 6 to node 9 with capacity 14,\nan edge from node 6 to node 8 with capacity 17,\nan edge from node 7 to node 6 with capacity 15,\nan edge from node 8 to node 2 with capacity 3,\nan edge from node 8 to node 5 with capacity 20,\nan edge from node 8 to node 4 with capacity 18,\nan edge from node 8 to node 7 with capacity 6,\nan edge from node 8 to node 3 with capacity 6,\nan edge from node 9 to node 0 with capacity 16,\nan edge from node 9 to node 10 with capacity 8,\nan edge from node 9 to node 1 with capacity 20,\nan edge from node 9 to node 11 with capacity 16,\nan edge from node 10 to node 0 with capacity 16,\nan edge from node 10 to node 2 with capacity 6,\nan edge from node 10 to node 5 with capacity 5,\nan edge from node 10 to node 1 with capacity 7.\nQ: What is the maximum flow from node 0 to node 1?\nA:", "answer": "The maximum flow from node 0 to node 1 is 24.", "difficulty": "hard", "doc_id": "213"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 8 with capacity 13,\nan edge from node 0 to node 9 with capacity 9,\nan edge from node 0 to node 12 with capacity 17,\nan edge from node 0 to node 4 with capacity 5,\nan edge from node 0 to node 7 with capacity 19,\nan edge from node 1 to node 11 with capacity 16,\nan edge from node 2 to node 10 with capacity 8,\nan edge from node 2 to node 0 with capacity 15,\nan edge from node 3 to node 8 with capacity 10,\nan edge from node 3 to node 5 with capacity 18,\nan edge from node 4 to node 12 with capacity 5,\nan edge from node 5 to node 6 with capacity 3,\nan edge from node 5 to node 1 with capacity 19,\nan edge from node 7 to node 2 with capacity 7,\nan edge from node 7 to node 12 with capacity 17,\nan edge from node 7 to node 4 with capacity 1,\nan edge from node 7 to node 5 with capacity 20,\nan edge from node 7 to node 3 with capacity 17,\nan edge from node 7 to node 0 with capacity 17,\nan edge from node 8 to node 1 with capacity 10,\nan edge from node 8 to node 10 with capacity 17,\nan edge from node 8 to node 0 with capacity 3,\nan edge from node 9 to node 5 with capacity 1,\nan edge from node 9 to node 1 with capacity 11,\nan edge from node 9 to node 10 with capacity 5,\nan edge from node 9 to node 11 with capacity 17,\nan edge from node 9 to node 7 with capacity 12,\nan edge from node 10 to node 9 with capacity 14,\nan edge from node 10 to node 5 with capacity 7,\nan edge from node 10 to node 11 with capacity 20,\nan edge from node 11 to node 4 with capacity 6,\nan edge from node 11 to node 6 with capacity 13,\nan edge from node 12 to node 4 with capacity 9,\nan edge from node 12 to node 10 with capacity 15.\nQ: What is the maximum flow from node 2 to node 0?\nA:", "answer": "The maximum flow from node 2 to node 0 is 23.", "difficulty": "hard", "doc_id": "214"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 1 with capacity 14,\nan edge from node 0 to node 6 with capacity 7,\nan edge from node 0 to node 9 with capacity 8,\nan edge from node 0 to node 8 with capacity 13,\nan edge from node 1 to node 11 with capacity 11,\nan edge from node 1 to node 6 with capacity 20,\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 2 to node 11 with capacity 18,\nan edge from node 2 to node 5 with capacity 14,\nan edge from node 2 to node 7 with capacity 11,\nan edge from node 2 to node 0 with capacity 4,\nan edge from node 3 to node 1 with capacity 12,\nan edge from node 3 to node 11 with capacity 10,\nan edge from node 3 to node 12 with capacity 9,\nan edge from node 3 to node 6 with capacity 19,\nan edge from node 3 to node 0 with capacity 10,\nan edge from node 3 to node 2 with capacity 9,\nan edge from node 4 to node 5 with capacity 15,\nan edge from node 4 to node 12 with capacity 4,\nan edge from node 4 to node 0 with capacity 8,\nan edge from node 5 to node 3 with capacity 17,\nan edge from node 5 to node 12 with capacity 3,\nan edge from node 5 to node 6 with capacity 2,\nan edge from node 5 to node 7 with capacity 1,\nan edge from node 5 to node 4 with capacity 18,\nan edge from node 5 to node 10 with capacity 9,\nan edge from node 6 to node 3 with capacity 12,\nan edge from node 6 to node 1 with capacity 9,\nan edge from node 6 to node 13 with capacity 18,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 6 to node 4 with capacity 8,\nan edge from node 6 to node 2 with capacity 13,\nan edge from node 7 to node 1 with capacity 1,\nan edge from node 7 to node 0 with capacity 7,\nan edge from node 7 to node 14 with capacity 8,\nan edge from node 8 to node 13 with capacity 14,\nan edge from node 8 to node 0 with capacity 19,\nan edge from node 8 to node 4 with capacity 18,\nan edge from node 9 to node 5 with capacity 6,\nan edge from node 9 to node 6 with capacity 12,\nan edge from node 9 to node 7 with capacity 1,\nan edge from node 9 to node 4 with capacity 5,\nan edge from node 9 to node 10 with capacity 5,\nan edge from node 10 to node 14 with capacity 16,\nan edge from node 10 to node 4 with capacity 18,\nan edge from node 11 to node 3 with capacity 3,\nan edge from node 11 to node 1 with capacity 1,\nan edge from node 11 to node 12 with capacity 15,\nan edge from node 11 to node 4 with capacity 9,\nan edge from node 11 to node 10 with capacity 18,\nan edge from node 12 to node 13 with capacity 4,\nan edge from node 12 to node 4 with capacity 12,\nan edge from node 12 to node 2 with capacity 5,\nan edge from node 13 to node 1 with capacity 15,\nan edge from node 13 to node 12 with capacity 11,\nan edge from node 13 to node 14 with capacity 9,\nan edge from node 13 to node 9 with capacity 6,\nan edge from node 13 to node 8 with capacity 5,\nan edge from node 13 to node 10 with capacity 15,\nan edge from node 14 to node 1 with capacity 7,\nan edge from node 14 to node 11 with capacity 15,\nan edge from node 14 to node 6 with capacity 19.\nQ: What is the maximum flow from node 2 to node 6?\nA:", "answer": "The maximum flow from node 2 to node 6 is 47.", "difficulty": "hard", "doc_id": "215"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 3 with capacity 18,\nan edge from node 0 to node 4 with capacity 15,\nan edge from node 0 to node 10 with capacity 18,\nan edge from node 0 to node 11 with capacity 8,\nan edge from node 1 to node 12 with capacity 5,\nan edge from node 1 to node 14 with capacity 3,\nan edge from node 1 to node 7 with capacity 4,\nan edge from node 2 to node 9 with capacity 11,\nan edge from node 2 to node 7 with capacity 14,\nan edge from node 2 to node 15 with capacity 10,\nan edge from node 2 to node 11 with capacity 18,\nan edge from node 3 to node 1 with capacity 11,\nan edge from node 3 to node 12 with capacity 1,\nan edge from node 3 to node 10 with capacity 5,\nan edge from node 3 to node 13 with capacity 6,\nan edge from node 3 to node 5 with capacity 12,\nan edge from node 4 to node 3 with capacity 12,\nan edge from node 4 to node 0 with capacity 6,\nan edge from node 4 to node 7 with capacity 7,\nan edge from node 4 to node 16 with capacity 12,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 9 with capacity 6,\nan edge from node 5 to node 14 with capacity 6,\nan edge from node 5 to node 8 with capacity 11,\nan edge from node 5 to node 13 with capacity 12,\nan edge from node 5 to node 15 with capacity 19,\nan edge from node 6 to node 8 with capacity 6,\nan edge from node 6 to node 15 with capacity 16,\nan edge from node 6 to node 5 with capacity 11,\nan edge from node 7 to node 0 with capacity 12,\nan edge from node 7 to node 15 with capacity 4,\nan edge from node 8 to node 1 with capacity 18,\nan edge from node 8 to node 9 with capacity 6,\nan edge from node 8 to node 4 with capacity 12,\nan edge from node 8 to node 17 with capacity 4,\nan edge from node 8 to node 15 with capacity 6,\nan edge from node 9 to node 12 with capacity 7,\nan edge from node 9 to node 0 with capacity 7,\nan edge from node 9 to node 10 with capacity 9,\nan edge from node 9 to node 16 with capacity 12,\nan edge from node 9 to node 15 with capacity 6,\nan edge from node 9 to node 2 with capacity 13,\nan edge from node 10 to node 1 with capacity 17,\nan edge from node 10 to node 0 with capacity 12,\nan edge from node 10 to node 4 with capacity 8,\nan edge from node 10 to node 17 with capacity 9,\nan edge from node 10 to node 8 with capacity 10,\nan edge from node 10 to node 13 with capacity 4,\nan edge from node 11 to node 1 with capacity 10,\nan edge from node 11 to node 0 with capacity 16,\nan edge from node 11 to node 17 with capacity 10,\nan edge from node 11 to node 8 with capacity 11,\nan edge from node 11 to node 2 with capacity 5,\nan edge from node 11 to node 5 with capacity 17,\nan edge from node 12 to node 16 with capacity 5,\nan edge from node 12 to node 2 with capacity 19,\nan edge from node 13 to node 6 with capacity 12,\nan edge from node 13 to node 16 with capacity 12,\nan edge from node 13 to node 11 with capacity 1,\nan edge from node 14 to node 0 with capacity 20,\nan edge from node 14 to node 11 with capacity 4,\nan edge from node 15 to node 9 with capacity 8,\nan edge from node 15 to node 7 with capacity 20,\nan edge from node 15 to node 16 with capacity 1,\nan edge from node 15 to node 8 with capacity 2,\nan edge from node 16 to node 12 with capacity 2,\nan edge from node 16 to node 10 with capacity 15,\nan edge from node 16 to node 8 with capacity 11,\nan edge from node 17 to node 3 with capacity 12,\nan edge from node 17 to node 11 with capacity 6.\nQ: What is the maximum flow from node 3 to node 0?\nA:", "answer": "The maximum flow from node 3 to node 0 is 35.", "difficulty": "hard", "doc_id": "216"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 5 with capacity 2,\nan edge from node 0 to node 9 with capacity 14,\nan edge from node 0 to node 6 with capacity 1,\nan edge from node 0 to node 11 with capacity 4,\nan edge from node 1 to node 16 with capacity 14,\nan edge from node 1 to node 2 with capacity 12,\nan edge from node 1 to node 9 with capacity 19,\nan edge from node 1 to node 14 with capacity 3,\nan edge from node 1 to node 12 with capacity 1,\nan edge from node 1 to node 6 with capacity 3,\nan edge from node 1 to node 8 with capacity 13,\nan edge from node 1 to node 19 with capacity 11,\nan edge from node 2 to node 3 with capacity 7,\nan edge from node 2 to node 6 with capacity 8,\nan edge from node 2 to node 8 with capacity 8,\nan edge from node 3 to node 6 with capacity 2,\nan edge from node 4 to node 5 with capacity 10,\nan edge from node 4 to node 2 with capacity 8,\nan edge from node 4 to node 10 with capacity 5,\nan edge from node 4 to node 12 with capacity 6,\nan edge from node 4 to node 17 with capacity 5,\nan edge from node 4 to node 1 with capacity 5,\nan edge from node 5 to node 4 with capacity 11,\nan edge from node 5 to node 12 with capacity 8,\nan edge from node 5 to node 17 with capacity 10,\nan edge from node 5 to node 0 with capacity 20,\nan edge from node 6 to node 5 with capacity 3,\nan edge from node 6 to node 4 with capacity 1,\nan edge from node 6 to node 18 with capacity 1,\nan edge from node 6 to node 0 with capacity 13,\nan edge from node 6 to node 15 with capacity 20,\nan edge from node 7 to node 4 with capacity 12,\nan edge from node 7 to node 17 with capacity 16,\nan edge from node 7 to node 8 with capacity 5,\nan edge from node 8 to node 4 with capacity 10,\nan edge from node 8 to node 17 with capacity 9,\nan edge from node 8 to node 18 with capacity 2,\nan edge from node 8 to node 19 with capacity 17,\nan edge from node 9 to node 4 with capacity 13,\nan edge from node 9 to node 3 with capacity 7,\nan edge from node 9 to node 14 with capacity 10,\nan edge from node 9 to node 17 with capacity 8,\nan edge from node 9 to node 15 with capacity 8,\nan edge from node 10 to node 5 with capacity 9,\nan edge from node 10 to node 4 with capacity 20,\nan edge from node 10 to node 3 with capacity 12,\nan edge from node 10 to node 14 with capacity 5,\nan edge from node 10 to node 12 with capacity 7,\nan edge from node 10 to node 1 with capacity 12,\nan edge from node 10 to node 8 with capacity 6,\nan edge from node 10 to node 15 with capacity 16,\nan edge from node 11 to node 2 with capacity 18,\nan edge from node 11 to node 14 with capacity 9,\nan edge from node 11 to node 12 with capacity 13,\nan edge from node 11 to node 0 with capacity 3,\nan edge from node 11 to node 15 with capacity 1,\nan edge from node 11 to node 19 with capacity 7,\nan edge from node 12 to node 14 with capacity 13,\nan edge from node 12 to node 15 with capacity 19,\nan edge from node 13 to node 7 with capacity 9,\nan edge from node 13 to node 9 with capacity 17,\nan edge from node 13 to node 14 with capacity 6,\nan edge from node 13 to node 10 with capacity 8,\nan edge from node 13 to node 6 with capacity 14,\nan edge from node 13 to node 17 with capacity 6,\nan edge from node 13 to node 15 with capacity 14,\nan edge from node 14 to node 5 with capacity 2,\nan edge from node 14 to node 9 with capacity 11,\nan edge from node 14 to node 10 with capacity 4,\nan edge from node 14 to node 12 with capacity 13,\nan edge from node 14 to node 18 with capacity 18,\nan edge from node 14 to node 0 with capacity 16,\nan edge from node 14 to node 19 with capacity 3,\nan edge from node 15 to node 16 with capacity 16,\nan edge from node 15 to node 13 with capacity 2,\nan edge from node 15 to node 0 with capacity 13,\nan edge from node 15 to node 1 with capacity 14,\nan edge from node 16 to node 7 with capacity 19,\nan edge from node 16 to node 10 with capacity 13,\nan edge from node 16 to node 17 with capacity 13,\nan edge from node 16 to node 0 with capacity 13,\nan edge from node 16 to node 11 with capacity 10,\nan edge from node 17 to node 2 with capacity 6,\nan edge from node 17 to node 13 with capacity 3,\nan edge from node 17 to node 1 with capacity 10,\nan edge from node 17 to node 11 with capacity 6,\nan edge from node 18 to node 13 with capacity 6,\nan edge from node 18 to node 0 with capacity 9,\nan edge from node 18 to node 15 with capacity 13,\nan edge from node 19 to node 7 with capacity 11,\nan edge from node 19 to node 2 with capacity 19,\nan edge from node 19 to node 14 with capacity 8,\nan edge from node 19 to node 17 with capacity 19,\nan edge from node 19 to node 1 with capacity 16,\nan edge from node 19 to node 8 with capacity 1.\nQ: What is the maximum flow from node 9 to node 6?\nA:", "answer": "The maximum flow from node 9 to node 6 is 25.", "difficulty": "hard", "doc_id": "217"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 4 with capacity 9,\nan edge from node 1 to node 8 with capacity 2,\nan edge from node 1 to node 5 with capacity 7,\nan edge from node 2 to node 3 with capacity 4,\nan edge from node 2 to node 0 with capacity 10,\nan edge from node 2 to node 9 with capacity 1,\nan edge from node 3 to node 1 with capacity 3,\nan edge from node 3 to node 7 with capacity 3,\nan edge from node 3 to node 5 with capacity 6,\nan edge from node 4 to node 1 with capacity 3,\nan edge from node 4 to node 0 with capacity 7,\nan edge from node 4 to node 8 with capacity 2,\nan edge from node 5 to node 4 with capacity 10,\nan edge from node 6 to node 1 with capacity 10,\nan edge from node 7 to node 2 with capacity 3,\nan edge from node 8 to node 0 with capacity 1,\nan edge from node 8 to node 6 with capacity 6.\nQ: What is the maximum flow from node 6 to node 0?\nA:", "answer": "The maximum flow from node 6 to node 0 is 8.", "difficulty": "easy", "doc_id": "218"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 8 with capacity 1,\nan edge from node 0 to node 6 with capacity 10,\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 2 to node 7 with capacity 4,\nan edge from node 2 to node 6 with capacity 4,\nan edge from node 3 to node 9 with capacity 10,\nan edge from node 4 to node 8 with capacity 8,\nan edge from node 4 to node 5 with capacity 4,\nan edge from node 5 to node 4 with capacity 8,\nan edge from node 5 to node 7 with capacity 6,\nan edge from node 5 to node 3 with capacity 5,\nan edge from node 7 to node 9 with capacity 4,\nan edge from node 7 to node 6 with capacity 1,\nan edge from node 7 to node 3 with capacity 7,\nan edge from node 8 to node 3 with capacity 6,\nan edge from node 9 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 6?\nA:", "answer": "The maximum flow from node 0 to node 6 is 11.", "difficulty": "easy", "doc_id": "219"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 10 with capacity 14,\nan edge from node 0 to node 8 with capacity 13,\nan edge from node 1 to node 0 with capacity 1,\nan edge from node 1 to node 7 with capacity 2,\nan edge from node 3 to node 11 with capacity 18,\nan edge from node 3 to node 5 with capacity 16,\nan edge from node 3 to node 10 with capacity 15,\nan edge from node 3 to node 9 with capacity 15,\nan edge from node 3 to node 8 with capacity 5,\nan edge from node 4 to node 1 with capacity 7,\nan edge from node 4 to node 9 with capacity 15,\nan edge from node 4 to node 7 with capacity 8,\nan edge from node 5 to node 0 with capacity 9,\nan edge from node 5 to node 1 with capacity 4,\nan edge from node 5 to node 10 with capacity 8,\nan edge from node 6 to node 0 with capacity 17,\nan edge from node 6 to node 12 with capacity 17,\nan edge from node 6 to node 4 with capacity 7,\nan edge from node 6 to node 10 with capacity 9,\nan edge from node 6 to node 9 with capacity 1,\nan edge from node 6 to node 8 with capacity 17,\nan edge from node 7 to node 12 with capacity 12,\nan edge from node 7 to node 8 with capacity 18,\nan edge from node 8 to node 5 with capacity 11,\nan edge from node 8 to node 6 with capacity 1,\nan edge from node 8 to node 1 with capacity 6,\nan edge from node 8 to node 4 with capacity 4,\nan edge from node 8 to node 13 with capacity 9,\nan edge from node 9 to node 6 with capacity 11,\nan edge from node 9 to node 12 with capacity 1,\nan edge from node 9 to node 8 with capacity 17,\nan edge from node 10 to node 11 with capacity 17,\nan edge from node 10 to node 3 with capacity 8,\nan edge from node 10 to node 9 with capacity 19,\nan edge from node 11 to node 4 with capacity 1,\nan edge from node 11 to node 10 with capacity 19,\nan edge from node 11 to node 2 with capacity 8,\nan edge from node 12 to node 11 with capacity 14,\nan edge from node 12 to node 10 with capacity 16,\nan edge from node 12 to node 9 with capacity 17,\nan edge from node 12 to node 7 with capacity 17,\nan edge from node 12 to node 13 with capacity 19,\nan edge from node 12 to node 8 with capacity 7,\nan edge from node 13 to node 0 with capacity 15,\nan edge from node 13 to node 5 with capacity 11,\nan edge from node 13 to node 10 with capacity 7,\nan edge from node 13 to node 2 with capacity 1,\nan edge from node 13 to node 8 with capacity 11.\nQ: What is the maximum flow from node 6 to node 4?\nA:", "answer": "The maximum flow from node 6 to node 4 is 12.", "difficulty": "hard", "doc_id": "220"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 3 with capacity 2,\nan edge from node 0 to node 13 with capacity 8,\nan edge from node 0 to node 2 with capacity 20,\nan edge from node 0 to node 10 with capacity 17,\nan edge from node 1 to node 9 with capacity 4,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 2 to node 4 with capacity 16,\nan edge from node 2 to node 14 with capacity 14,\nan edge from node 2 to node 1 with capacity 11,\nan edge from node 2 to node 5 with capacity 18,\nan edge from node 3 to node 2 with capacity 18,\nan edge from node 3 to node 0 with capacity 14,\nan edge from node 3 to node 7 with capacity 6,\nan edge from node 3 to node 9 with capacity 12,\nan edge from node 3 to node 10 with capacity 17,\nan edge from node 4 to node 7 with capacity 1,\nan edge from node 4 to node 11 with capacity 15,\nan edge from node 4 to node 8 with capacity 4,\nan edge from node 5 to node 4 with capacity 17,\nan edge from node 5 to node 11 with capacity 11,\nan edge from node 5 to node 14 with capacity 16,\nan edge from node 5 to node 8 with capacity 5,\nan edge from node 6 to node 8 with capacity 20,\nan edge from node 6 to node 10 with capacity 4,\nan edge from node 6 to node 1 with capacity 13,\nan edge from node 7 to node 8 with capacity 7,\nan edge from node 8 to node 13 with capacity 3,\nan edge from node 8 to node 14 with capacity 5,\nan edge from node 9 to node 3 with capacity 19,\nan edge from node 9 to node 6 with capacity 16,\nan edge from node 9 to node 0 with capacity 14,\nan edge from node 9 to node 5 with capacity 6,\nan edge from node 10 to node 3 with capacity 5,\nan edge from node 10 to node 2 with capacity 3,\nan edge from node 11 to node 3 with capacity 18,\nan edge from node 11 to node 13 with capacity 6,\nan edge from node 11 to node 6 with capacity 13,\nan edge from node 11 to node 0 with capacity 6,\nan edge from node 11 to node 12 with capacity 7,\nan edge from node 11 to node 8 with capacity 3,\nan edge from node 12 to node 2 with capacity 19,\nan edge from node 12 to node 7 with capacity 8,\nan edge from node 12 to node 8 with capacity 20,\nan edge from node 13 to node 0 with capacity 4,\nan edge from node 14 to node 6 with capacity 10,\nan edge from node 14 to node 0 with capacity 9,\nan edge from node 14 to node 11 with capacity 19,\nan edge from node 14 to node 9 with capacity 3,\nan edge from node 14 to node 5 with capacity 12.\nQ: What is the maximum flow from node 11 to node 13?\nA:", "answer": "The maximum flow from node 11 to node 13 is 17.", "difficulty": "hard", "doc_id": "221"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 8 with capacity 3,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 2 to node 5 with capacity 7,\nan edge from node 2 to node 4 with capacity 4,\nan edge from node 2 to node 3 with capacity 6,\nan edge from node 3 to node 5 with capacity 5,\nan edge from node 3 to node 6 with capacity 10,\nan edge from node 4 to node 2 with capacity 2,\nan edge from node 5 to node 8 with capacity 7,\nan edge from node 8 to node 4 with capacity 8,\nan edge from node 8 to node 6 with capacity 6.\nQ: What is the maximum flow from node 2 to node 6?\nA:", "answer": "The maximum flow from node 2 to node 6 is 12.", "difficulty": "easy", "doc_id": "222"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 4 with capacity 3,\nan edge from node 1 to node 2 with capacity 9,\nan edge from node 1 to node 0 with capacity 8,\nan edge from node 2 to node 1 with capacity 9,\nan edge from node 2 to node 0 with capacity 5,\nan edge from node 3 to node 5 with capacity 9,\nan edge from node 4 to node 2 with capacity 6,\nan edge from node 5 to node 4 with capacity 2,\nan edge from node 5 to node 2 with capacity 9.\nQ: What is the maximum flow from node 2 to node 4?\nA:", "answer": "The maximum flow from node 2 to node 4 is 13.", "difficulty": "easy", "doc_id": "223"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 8 with capacity 17,\nan edge from node 0 to node 3 with capacity 8,\nan edge from node 0 to node 12 with capacity 1,\nan edge from node 0 to node 2 with capacity 15,\nan edge from node 0 to node 15 with capacity 10,\nan edge from node 1 to node 10 with capacity 9,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 1 to node 12 with capacity 18,\nan edge from node 1 to node 0 with capacity 19,\nan edge from node 1 to node 11 with capacity 10,\nan edge from node 1 to node 2 with capacity 10,\nan edge from node 2 to node 11 with capacity 4,\nan edge from node 2 to node 14 with capacity 13,\nan edge from node 3 to node 5 with capacity 9,\nan edge from node 3 to node 10 with capacity 3,\nan edge from node 3 to node 12 with capacity 4,\nan edge from node 3 to node 13 with capacity 6,\nan edge from node 3 to node 7 with capacity 3,\nan edge from node 3 to node 11 with capacity 10,\nan edge from node 4 to node 5 with capacity 13,\nan edge from node 4 to node 10 with capacity 14,\nan edge from node 4 to node 11 with capacity 19,\nan edge from node 4 to node 2 with capacity 1,\nan edge from node 4 to node 14 with capacity 13,\nan edge from node 5 to node 2 with capacity 11,\nan edge from node 6 to node 10 with capacity 11,\nan edge from node 7 to node 10 with capacity 12,\nan edge from node 7 to node 3 with capacity 13,\nan edge from node 7 to node 1 with capacity 19,\nan edge from node 7 to node 11 with capacity 11,\nan edge from node 7 to node 14 with capacity 9,\nan edge from node 8 to node 12 with capacity 3,\nan edge from node 8 to node 13 with capacity 20,\nan edge from node 8 to node 2 with capacity 17,\nan edge from node 8 to node 9 with capacity 6,\nan edge from node 8 to node 14 with capacity 13,\nan edge from node 9 to node 12 with capacity 4,\nan edge from node 9 to node 7 with capacity 13,\nan edge from node 9 to node 2 with capacity 9,\nan edge from node 10 to node 6 with capacity 8,\nan edge from node 11 to node 5 with capacity 16,\nan edge from node 12 to node 3 with capacity 12,\nan edge from node 12 to node 4 with capacity 9,\nan edge from node 13 to node 4 with capacity 19,\nan edge from node 13 to node 7 with capacity 9,\nan edge from node 13 to node 1 with capacity 3,\nan edge from node 13 to node 14 with capacity 8,\nan edge from node 14 to node 10 with capacity 8,\nan edge from node 14 to node 13 with capacity 11,\nan edge from node 14 to node 2 with capacity 10,\nan edge from node 15 to node 10 with capacity 9,\nan edge from node 15 to node 12 with capacity 2,\nan edge from node 15 to node 13 with capacity 15,\nan edge from node 15 to node 7 with capacity 12,\nan edge from node 15 to node 11 with capacity 5.\nQ: What is the maximum flow from node 7 to node 13?\nA:", "answer": "The maximum flow from node 7 to node 13 is 36.", "difficulty": "hard", "doc_id": "224"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 9,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 3 to node 0 with capacity 6,\nan edge from node 4 to node 1 with capacity 8,\nan edge from node 4 to node 2 with capacity 2.\nQ: What is the maximum flow from node 4 to node 0?\nA:", "answer": "The maximum flow from node 4 to node 0 is 10.", "difficulty": "easy", "doc_id": "225"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 7 with capacity 3,\nan edge from node 0 to node 14 with capacity 3,\nan edge from node 0 to node 13 with capacity 10,\nan edge from node 0 to node 9 with capacity 17,\nan edge from node 1 to node 6 with capacity 5,\nan edge from node 1 to node 10 with capacity 3,\nan edge from node 1 to node 16 with capacity 3,\nan edge from node 2 to node 14 with capacity 4,\nan edge from node 2 to node 1 with capacity 17,\nan edge from node 2 to node 5 with capacity 9,\nan edge from node 2 to node 17 with capacity 18,\nan edge from node 2 to node 15 with capacity 4,\nan edge from node 2 to node 12 with capacity 12,\nan edge from node 2 to node 8 with capacity 13,\nan edge from node 2 to node 0 with capacity 12,\nan edge from node 3 to node 14 with capacity 3,\nan edge from node 3 to node 1 with capacity 18,\nan edge from node 3 to node 12 with capacity 13,\nan edge from node 3 to node 8 with capacity 4,\nan edge from node 3 to node 16 with capacity 10,\nan edge from node 4 to node 6 with capacity 8,\nan edge from node 4 to node 14 with capacity 20,\nan edge from node 4 to node 3 with capacity 1,\nan edge from node 4 to node 17 with capacity 19,\nan edge from node 4 to node 9 with capacity 3,\nan edge from node 4 to node 16 with capacity 6,\nan edge from node 5 to node 6 with capacity 13,\nan edge from node 5 to node 14 with capacity 11,\nan edge from node 5 to node 15 with capacity 10,\nan edge from node 5 to node 4 with capacity 15,\nan edge from node 5 to node 2 with capacity 17,\nan edge from node 5 to node 0 with capacity 3,\nan edge from node 6 to node 7 with capacity 15,\nan edge from node 6 to node 13 with capacity 11,\nan edge from node 6 to node 1 with capacity 12,\nan edge from node 6 to node 5 with capacity 10,\nan edge from node 6 to node 17 with capacity 20,\nan edge from node 7 to node 3 with capacity 14,\nan edge from node 7 to node 11 with capacity 16,\nan edge from node 7 to node 8 with capacity 6,\nan edge from node 8 to node 13 with capacity 16,\nan edge from node 8 to node 12 with capacity 1,\nan edge from node 8 to node 0 with capacity 11,\nan edge from node 8 to node 16 with capacity 16,\nan edge from node 9 to node 1 with capacity 1,\nan edge from node 9 to node 17 with capacity 9,\nan edge from node 9 to node 10 with capacity 1,\nan edge from node 9 to node 12 with capacity 11,\nan edge from node 9 to node 4 with capacity 6,\nan edge from node 10 to node 1 with capacity 1,\nan edge from node 10 to node 11 with capacity 12,\nan edge from node 10 to node 4 with capacity 7,\nan edge from node 10 to node 0 with capacity 17,\nan edge from node 11 to node 14 with capacity 5,\nan edge from node 11 to node 2 with capacity 1,\nan edge from node 12 to node 14 with capacity 16,\nan edge from node 12 to node 13 with capacity 6,\nan edge from node 12 to node 15 with capacity 16,\nan edge from node 12 to node 2 with capacity 9,\nan edge from node 12 to node 8 with capacity 7,\nan edge from node 13 to node 14 with capacity 18,\nan edge from node 13 to node 2 with capacity 1,\nan edge from node 13 to node 8 with capacity 12,\nan edge from node 13 to node 0 with capacity 12,\nan edge from node 13 to node 16 with capacity 3,\nan edge from node 14 to node 17 with capacity 16,\nan edge from node 14 to node 12 with capacity 4,\nan edge from node 15 to node 1 with capacity 15,\nan edge from node 15 to node 10 with capacity 7,\nan edge from node 15 to node 12 with capacity 1,\nan edge from node 15 to node 0 with capacity 7,\nan edge from node 16 to node 5 with capacity 17,\nan edge from node 16 to node 10 with capacity 15,\nan edge from node 16 to node 9 with capacity 13,\nan edge from node 16 to node 11 with capacity 17,\nan edge from node 16 to node 15 with capacity 13,\nan edge from node 16 to node 2 with capacity 10,\nan edge from node 16 to node 8 with capacity 17,\nan edge from node 17 to node 3 with capacity 2,\nan edge from node 17 to node 9 with capacity 9,\nan edge from node 17 to node 12 with capacity 17,\nan edge from node 17 to node 2 with capacity 19,\nan edge from node 17 to node 16 with capacity 1.\nQ: What is the maximum flow from node 4 to node 15?\nA:", "answer": "The maximum flow from node 4 to node 15 is 43.", "difficulty": "hard", "doc_id": "226"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 11 with capacity 20,\nan edge from node 0 to node 14 with capacity 20,\nan edge from node 0 to node 10 with capacity 17,\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 0 to node 6 with capacity 7,\nan edge from node 1 to node 14 with capacity 18,\nan edge from node 1 to node 10 with capacity 19,\nan edge from node 1 to node 4 with capacity 18,\nan edge from node 1 to node 6 with capacity 15,\nan edge from node 1 to node 2 with capacity 14,\nan edge from node 1 to node 12 with capacity 9,\nan edge from node 2 to node 7 with capacity 17,\nan edge from node 2 to node 8 with capacity 8,\nan edge from node 2 to node 1 with capacity 7,\nan edge from node 2 to node 9 with capacity 14,\nan edge from node 3 to node 13 with capacity 15,\nan edge from node 3 to node 14 with capacity 17,\nan edge from node 3 to node 4 with capacity 20,\nan edge from node 3 to node 0 with capacity 3,\nan edge from node 3 to node 5 with capacity 10,\nan edge from node 3 to node 12 with capacity 1,\nan edge from node 4 to node 8 with capacity 18,\nan edge from node 4 to node 5 with capacity 1,\nan edge from node 5 to node 11 with capacity 2,\nan edge from node 5 to node 13 with capacity 9,\nan edge from node 5 to node 3 with capacity 20,\nan edge from node 5 to node 0 with capacity 9,\nan edge from node 5 to node 1 with capacity 11,\nan edge from node 6 to node 10 with capacity 12,\nan edge from node 6 to node 0 with capacity 11,\nan edge from node 6 to node 1 with capacity 17,\nan edge from node 7 to node 13 with capacity 13,\nan edge from node 7 to node 10 with capacity 13,\nan edge from node 7 to node 4 with capacity 20,\nan edge from node 7 to node 9 with capacity 3,\nan edge from node 8 to node 11 with capacity 10,\nan edge from node 8 to node 10 with capacity 7,\nan edge from node 8 to node 4 with capacity 20,\nan edge from node 8 to node 1 with capacity 19,\nan edge from node 8 to node 9 with capacity 8,\nan edge from node 9 to node 11 with capacity 10,\nan edge from node 9 to node 13 with capacity 19,\nan edge from node 9 to node 14 with capacity 1,\nan edge from node 9 to node 7 with capacity 19,\nan edge from node 9 to node 8 with capacity 2,\nan edge from node 10 to node 13 with capacity 8,\nan edge from node 10 to node 8 with capacity 17,\nan edge from node 10 to node 0 with capacity 3,\nan edge from node 10 to node 12 with capacity 5,\nan edge from node 11 to node 4 with capacity 13,\nan edge from node 11 to node 3 with capacity 13,\nan edge from node 11 to node 1 with capacity 20,\nan edge from node 11 to node 12 with capacity 16,\nan edge from node 12 to node 13 with capacity 11,\nan edge from node 12 to node 6 with capacity 13,\nan edge from node 12 to node 2 with capacity 12,\nan edge from node 12 to node 5 with capacity 7,\nan edge from node 13 to node 14 with capacity 11,\nan edge from node 13 to node 5 with capacity 15,\nan edge from node 14 to node 13 with capacity 5,\nan edge from node 14 to node 0 with capacity 5,\nan edge from node 14 to node 5 with capacity 17,\nan edge from node 14 to node 12 with capacity 14.\nQ: What is the maximum flow from node 4 to node 10?\nA:", "answer": "The maximum flow from node 4 to node 10 is 19.", "difficulty": "hard", "doc_id": "227"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 7 with capacity 12,\nan edge from node 0 to node 18 with capacity 17,\nan edge from node 0 to node 8 with capacity 10,\nan edge from node 0 to node 15 with capacity 5,\nan edge from node 0 to node 2 with capacity 17,\nan edge from node 0 to node 14 with capacity 4,\nan edge from node 1 to node 7 with capacity 2,\nan edge from node 1 to node 10 with capacity 7,\nan edge from node 1 to node 18 with capacity 7,\nan edge from node 1 to node 15 with capacity 9,\nan edge from node 1 to node 14 with capacity 4,\nan edge from node 2 to node 9 with capacity 18,\nan edge from node 2 to node 7 with capacity 16,\nan edge from node 2 to node 0 with capacity 13,\nan edge from node 2 to node 18 with capacity 1,\nan edge from node 3 to node 19 with capacity 9,\nan edge from node 3 to node 8 with capacity 20,\nan edge from node 4 to node 0 with capacity 2,\nan edge from node 4 to node 15 with capacity 9,\nan edge from node 4 to node 14 with capacity 11,\nan edge from node 5 to node 12 with capacity 1,\nan edge from node 5 to node 19 with capacity 14,\nan edge from node 5 to node 2 with capacity 20,\nan edge from node 6 to node 3 with capacity 4,\nan edge from node 6 to node 7 with capacity 2,\nan edge from node 6 to node 12 with capacity 3,\nan edge from node 6 to node 19 with capacity 15,\nan edge from node 6 to node 10 with capacity 11,\nan edge from node 6 to node 4 with capacity 4,\nan edge from node 6 to node 14 with capacity 20,\nan edge from node 7 to node 19 with capacity 18,\nan edge from node 7 to node 0 with capacity 1,\nan edge from node 7 to node 5 with capacity 6,\nan edge from node 7 to node 2 with capacity 1,\nan edge from node 8 to node 7 with capacity 7,\nan edge from node 8 to node 12 with capacity 19,\nan edge from node 8 to node 6 with capacity 4,\nan edge from node 8 to node 10 with capacity 10,\nan edge from node 8 to node 0 with capacity 4,\nan edge from node 8 to node 5 with capacity 2,\nan edge from node 8 to node 1 with capacity 9,\nan edge from node 8 to node 14 with capacity 11,\nan edge from node 9 to node 3 with capacity 1,\nan edge from node 9 to node 19 with capacity 7,\nan edge from node 9 to node 1 with capacity 5,\nan edge from node 9 to node 14 with capacity 1,\nan edge from node 10 to node 19 with capacity 4,\nan edge from node 10 to node 6 with capacity 6,\nan edge from node 10 to node 5 with capacity 15,\nan edge from node 10 to node 15 with capacity 11,\nan edge from node 10 to node 13 with capacity 17,\nan edge from node 10 to node 11 with capacity 5,\nan edge from node 11 to node 3 with capacity 18,\nan edge from node 11 to node 7 with capacity 19,\nan edge from node 11 to node 19 with capacity 8,\nan edge from node 11 to node 18 with capacity 4,\nan edge from node 11 to node 8 with capacity 19,\nan edge from node 11 to node 5 with capacity 3,\nan edge from node 11 to node 13 with capacity 13,\nan edge from node 11 to node 14 with capacity 11,\nan edge from node 12 to node 7 with capacity 4,\nan edge from node 12 to node 19 with capacity 3,\nan edge from node 12 to node 16 with capacity 17,\nan edge from node 12 to node 17 with capacity 13,\nan edge from node 13 to node 12 with capacity 17,\nan edge from node 13 to node 10 with capacity 16,\nan edge from node 13 to node 18 with capacity 15,\nan edge from node 13 to node 15 with capacity 3,\nan edge from node 13 to node 2 with capacity 6,\nan edge from node 13 to node 16 with capacity 6,\nan edge from node 13 to node 14 with capacity 1,\nan edge from node 14 to node 12 with capacity 11,\nan edge from node 14 to node 6 with capacity 6,\nan edge from node 14 to node 0 with capacity 9,\nan edge from node 14 to node 2 with capacity 11,\nan edge from node 14 to node 4 with capacity 3,\nan edge from node 14 to node 17 with capacity 18,\nan edge from node 15 to node 6 with capacity 5,\nan edge from node 15 to node 11 with capacity 2,\nan edge from node 15 to node 14 with capacity 1,\nan edge from node 16 to node 10 with capacity 4,\nan edge from node 16 to node 18 with capacity 16,\nan edge from node 16 to node 15 with capacity 16,\nan edge from node 16 to node 14 with capacity 8,\nan edge from node 17 to node 9 with capacity 3,\nan edge from node 17 to node 10 with capacity 8,\nan edge from node 17 to node 0 with capacity 18,\nan edge from node 17 to node 18 with capacity 4,\nan edge from node 17 to node 5 with capacity 20,\nan edge from node 17 to node 15 with capacity 4,\nan edge from node 17 to node 1 with capacity 16,\nan edge from node 17 to node 11 with capacity 3,\nan edge from node 17 to node 14 with capacity 6,\nan edge from node 18 to node 9 with capacity 15,\nan edge from node 18 to node 3 with capacity 17,\nan edge from node 18 to node 10 with capacity 12,\nan edge from node 18 to node 8 with capacity 15,\nan edge from node 19 to node 3 with capacity 13,\nan edge from node 19 to node 2 with capacity 14,\nan edge from node 19 to node 11 with capacity 14.\nQ: What is the maximum flow from node 16 to node 10?\nA:", "answer": "The maximum flow from node 16 to node 10 is 36.", "difficulty": "hard", "doc_id": "228"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 3 with capacity 4,\nan edge from node 0 to node 5 with capacity 6,\nan edge from node 1 to node 6 with capacity 9,\nan edge from node 1 to node 2 with capacity 17,\nan edge from node 1 to node 10 with capacity 3,\nan edge from node 2 to node 3 with capacity 12,\nan edge from node 2 to node 11 with capacity 17,\nan edge from node 2 to node 4 with capacity 14,\nan edge from node 3 to node 2 with capacity 9,\nan edge from node 3 to node 1 with capacity 5,\nan edge from node 3 to node 8 with capacity 8,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 7 with capacity 12,\nan edge from node 4 to node 2 with capacity 10,\nan edge from node 4 to node 3 with capacity 12,\nan edge from node 4 to node 5 with capacity 20,\nan edge from node 5 to node 9 with capacity 17,\nan edge from node 5 to node 2 with capacity 5,\nan edge from node 5 to node 3 with capacity 6,\nan edge from node 5 to node 8 with capacity 17,\nan edge from node 5 to node 10 with capacity 10,\nan edge from node 6 to node 7 with capacity 20,\nan edge from node 6 to node 0 with capacity 19,\nan edge from node 6 to node 10 with capacity 10,\nan edge from node 7 to node 2 with capacity 20,\nan edge from node 8 to node 5 with capacity 10,\nan edge from node 8 to node 4 with capacity 12,\nan edge from node 9 to node 7 with capacity 13,\nan edge from node 9 to node 6 with capacity 6,\nan edge from node 9 to node 0 with capacity 17,\nan edge from node 10 to node 1 with capacity 16,\nan edge from node 10 to node 5 with capacity 19,\nan edge from node 11 to node 0 with capacity 13.\nQ: What is the maximum flow from node 4 to node 2?\nA:", "answer": "The maximum flow from node 4 to node 2 is 54.", "difficulty": "hard", "doc_id": "229"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 2 with capacity 3,\nan edge from node 0 to node 6 with capacity 12,\nan edge from node 1 to node 5 with capacity 18,\nan edge from node 2 to node 5 with capacity 13,\nan edge from node 3 to node 0 with capacity 3,\nan edge from node 3 to node 10 with capacity 1,\nan edge from node 4 to node 10 with capacity 12,\nan edge from node 5 to node 2 with capacity 11,\nan edge from node 5 to node 8 with capacity 5,\nan edge from node 5 to node 6 with capacity 4,\nan edge from node 5 to node 10 with capacity 11,\nan edge from node 5 to node 1 with capacity 20,\nan edge from node 5 to node 11 with capacity 14,\nan edge from node 6 to node 0 with capacity 9,\nan edge from node 6 to node 8 with capacity 14,\nan edge from node 6 to node 4 with capacity 6,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 7 to node 8 with capacity 16,\nan edge from node 8 to node 0 with capacity 4,\nan edge from node 8 to node 2 with capacity 12,\nan edge from node 8 to node 6 with capacity 4,\nan edge from node 8 to node 1 with capacity 9,\nan edge from node 8 to node 4 with capacity 10,\nan edge from node 8 to node 11 with capacity 2,\nan edge from node 8 to node 5 with capacity 20,\nan edge from node 9 to node 11 with capacity 14,\nan edge from node 10 to node 1 with capacity 1,\nan edge from node 10 to node 4 with capacity 1,\nan edge from node 11 to node 6 with capacity 10,\nan edge from node 11 to node 3 with capacity 4,\nan edge from node 11 to node 1 with capacity 17.\nQ: What is the maximum flow from node 0 to node 3?\nA:", "answer": "The maximum flow from node 0 to node 3 is 4.", "difficulty": "hard", "doc_id": "230"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 2 to node 5 with capacity 9,\nan edge from node 2 to node 0 with capacity 6,\nan edge from node 2 to node 3 with capacity 5,\nan edge from node 3 to node 1 with capacity 8,\nan edge from node 3 to node 0 with capacity 5,\nan edge from node 4 to node 5 with capacity 4,\nan edge from node 4 to node 0 with capacity 3,\nan edge from node 4 to node 3 with capacity 9,\nan edge from node 5 to node 2 with capacity 9,\nan edge from node 5 to node 3 with capacity 2.\nQ: What is the maximum flow from node 5 to node 0?\nA:", "answer": "The maximum flow from node 5 to node 0 is 11.", "difficulty": "easy", "doc_id": "231"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 1 with capacity 7,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 0 to node 3 with capacity 5,\nan edge from node 1 to node 0 with capacity 4,\nan edge from node 1 to node 7 with capacity 10,\nan edge from node 2 to node 7 with capacity 2,\nan edge from node 2 to node 3 with capacity 2,\nan edge from node 3 to node 1 with capacity 1,\nan edge from node 3 to node 2 with capacity 4,\nan edge from node 3 to node 0 with capacity 4,\nan edge from node 3 to node 5 with capacity 9,\nan edge from node 4 to node 5 with capacity 10,\nan edge from node 5 to node 0 with capacity 2,\nan edge from node 5 to node 3 with capacity 2,\nan edge from node 6 to node 1 with capacity 8,\nan edge from node 6 to node 0 with capacity 9,\nan edge from node 7 to node 9 with capacity 4,\nan edge from node 8 to node 0 with capacity 10,\nan edge from node 8 to node 7 with capacity 6,\nan edge from node 8 to node 5 with capacity 9,\nan edge from node 8 to node 6 with capacity 10,\nan edge from node 9 to node 2 with capacity 6,\nan edge from node 9 to node 0 with capacity 4,\nan edge from node 9 to node 4 with capacity 1,\nan edge from node 9 to node 5 with capacity 4,\nan edge from node 9 to node 8 with capacity 2.\nQ: What is the maximum flow from node 4 to node 6?\nA:", "answer": "The maximum flow from node 4 to node 6 is 2.", "difficulty": "easy", "doc_id": "232"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 4 with capacity 7,\nan edge from node 0 to node 3 with capacity 6,\nan edge from node 2 to node 7 with capacity 8,\nan edge from node 2 to node 8 with capacity 2,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 8 with capacity 9,\nan edge from node 3 to node 1 with capacity 3,\nan edge from node 5 to node 2 with capacity 6,\nan edge from node 5 to node 4 with capacity 7,\nan edge from node 5 to node 3 with capacity 5,\nan edge from node 6 to node 3 with capacity 10,\nan edge from node 7 to node 2 with capacity 6,\nan edge from node 7 to node 4 with capacity 1,\nan edge from node 7 to node 1 with capacity 6,\nan edge from node 8 to node 4 with capacity 7,\nan edge from node 8 to node 0 with capacity 10.\nQ: What is the maximum flow from node 5 to node 4?\nA:", "answer": "The maximum flow from node 5 to node 4 is 15.", "difficulty": "easy", "doc_id": "233"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 16 with capacity 2,\nan edge from node 0 to node 15 with capacity 10,\nan edge from node 0 to node 8 with capacity 10,\nan edge from node 1 to node 16 with capacity 2,\nan edge from node 1 to node 14 with capacity 1,\nan edge from node 1 to node 2 with capacity 11,\nan edge from node 1 to node 11 with capacity 19,\nan edge from node 1 to node 12 with capacity 3,\nan edge from node 1 to node 4 with capacity 18,\nan edge from node 2 to node 10 with capacity 14,\nan edge from node 2 to node 9 with capacity 10,\nan edge from node 2 to node 4 with capacity 4,\nan edge from node 3 to node 5 with capacity 13,\nan edge from node 3 to node 11 with capacity 12,\nan edge from node 3 to node 0 with capacity 7,\nan edge from node 3 to node 15 with capacity 6,\nan edge from node 3 to node 12 with capacity 18,\nan edge from node 4 to node 2 with capacity 16,\nan edge from node 4 to node 15 with capacity 18,\nan edge from node 5 to node 7 with capacity 2,\nan edge from node 5 to node 16 with capacity 20,\nan edge from node 5 to node 12 with capacity 7,\nan edge from node 6 to node 7 with capacity 15,\nan edge from node 6 to node 3 with capacity 10,\nan edge from node 7 to node 10 with capacity 7,\nan edge from node 7 to node 6 with capacity 6,\nan edge from node 8 to node 7 with capacity 3,\nan edge from node 8 to node 14 with capacity 11,\nan edge from node 8 to node 9 with capacity 4,\nan edge from node 8 to node 2 with capacity 10,\nan edge from node 8 to node 1 with capacity 12,\nan edge from node 9 to node 8 with capacity 5,\nan edge from node 10 to node 5 with capacity 17,\nan edge from node 10 to node 13 with capacity 1,\nan edge from node 10 to node 12 with capacity 10,\nan edge from node 10 to node 1 with capacity 4,\nan edge from node 11 to node 5 with capacity 15,\nan edge from node 11 to node 2 with capacity 12,\nan edge from node 11 to node 1 with capacity 18,\nan edge from node 12 to node 7 with capacity 12,\nan edge from node 12 to node 10 with capacity 8,\nan edge from node 12 to node 5 with capacity 4,\nan edge from node 12 to node 9 with capacity 17,\nan edge from node 12 to node 0 with capacity 20,\nan edge from node 12 to node 8 with capacity 17,\nan edge from node 12 to node 4 with capacity 17,\nan edge from node 13 to node 3 with capacity 9,\nan edge from node 13 to node 2 with capacity 13,\nan edge from node 13 to node 11 with capacity 7,\nan edge from node 13 to node 1 with capacity 7,\nan edge from node 14 to node 16 with capacity 19,\nan edge from node 14 to node 5 with capacity 7,\nan edge from node 14 to node 11 with capacity 4,\nan edge from node 14 to node 6 with capacity 4,\nan edge from node 14 to node 8 with capacity 3,\nan edge from node 15 to node 10 with capacity 10,\nan edge from node 15 to node 3 with capacity 5,\nan edge from node 15 to node 13 with capacity 14,\nan edge from node 15 to node 2 with capacity 15,\nan edge from node 15 to node 12 with capacity 6,\nan edge from node 16 to node 14 with capacity 17,\nan edge from node 16 to node 8 with capacity 11,\nan edge from node 16 to node 4 with capacity 3.\nQ: What is the maximum flow from node 12 to node 8?\nA:", "answer": "The maximum flow from node 12 to node 8 is 46.", "difficulty": "hard", "doc_id": "234"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 6 with capacity 9,\nan edge from node 0 to node 10 with capacity 5,\nan edge from node 0 to node 8 with capacity 7,\nan edge from node 0 to node 13 with capacity 11,\nan edge from node 0 to node 17 with capacity 19,\nan edge from node 0 to node 15 with capacity 6,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 11 with capacity 14,\nan edge from node 1 to node 4 with capacity 14,\nan edge from node 2 to node 16 with capacity 2,\nan edge from node 2 to node 17 with capacity 9,\nan edge from node 2 to node 12 with capacity 9,\nan edge from node 2 to node 14 with capacity 3,\nan edge from node 3 to node 11 with capacity 6,\nan edge from node 3 to node 9 with capacity 15,\nan edge from node 3 to node 17 with capacity 17,\nan edge from node 3 to node 0 with capacity 7,\nan edge from node 3 to node 1 with capacity 4,\nan edge from node 4 to node 11 with capacity 6,\nan edge from node 4 to node 9 with capacity 1,\nan edge from node 5 to node 6 with capacity 18,\nan edge from node 5 to node 11 with capacity 19,\nan edge from node 5 to node 12 with capacity 8,\nan edge from node 5 to node 14 with capacity 1,\nan edge from node 6 to node 8 with capacity 20,\nan edge from node 6 to node 9 with capacity 10,\nan edge from node 6 to node 17 with capacity 4,\nan edge from node 6 to node 3 with capacity 13,\nan edge from node 6 to node 4 with capacity 9,\nan edge from node 7 to node 11 with capacity 20,\nan edge from node 7 to node 13 with capacity 5,\nan edge from node 7 to node 17 with capacity 17,\nan edge from node 7 to node 1 with capacity 12,\nan edge from node 8 to node 6 with capacity 17,\nan edge from node 8 to node 11 with capacity 10,\nan edge from node 8 to node 9 with capacity 5,\nan edge from node 8 to node 3 with capacity 14,\nan edge from node 8 to node 15 with capacity 19,\nan edge from node 8 to node 12 with capacity 13,\nan edge from node 8 to node 5 with capacity 15,\nan edge from node 8 to node 1 with capacity 2,\nan edge from node 9 to node 12 with capacity 3,\nan edge from node 9 to node 4 with capacity 13,\nan edge from node 10 to node 7 with capacity 10,\nan edge from node 10 to node 9 with capacity 2,\nan edge from node 10 to node 5 with capacity 14,\nan edge from node 10 to node 1 with capacity 1,\nan edge from node 10 to node 14 with capacity 13,\nan edge from node 11 to node 2 with capacity 12,\nan edge from node 11 to node 15 with capacity 12,\nan edge from node 11 to node 14 with capacity 14,\nan edge from node 12 to node 6 with capacity 17,\nan edge from node 12 to node 17 with capacity 18,\nan edge from node 12 to node 0 with capacity 6,\nan edge from node 12 to node 5 with capacity 19,\nan edge from node 12 to node 4 with capacity 3,\nan edge from node 12 to node 14 with capacity 12,\nan edge from node 13 to node 17 with capacity 8,\nan edge from node 13 to node 12 with capacity 3,\nan edge from node 13 to node 5 with capacity 12,\nan edge from node 14 to node 7 with capacity 15,\nan edge from node 14 to node 17 with capacity 1,\nan edge from node 14 to node 3 with capacity 16,\nan edge from node 14 to node 15 with capacity 6,\nan edge from node 15 to node 13 with capacity 20,\nan edge from node 15 to node 2 with capacity 3,\nan edge from node 15 to node 4 with capacity 14,\nan edge from node 16 to node 7 with capacity 5,\nan edge from node 16 to node 3 with capacity 14,\nan edge from node 16 to node 0 with capacity 10,\nan edge from node 16 to node 4 with capacity 3,\nan edge from node 17 to node 11 with capacity 13.\nQ: What is the maximum flow from node 10 to node 9?\nA:", "answer": "The maximum flow from node 10 to node 9 is 33.", "difficulty": "hard", "doc_id": "235"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 1 to node 3 with capacity 13,\nan edge from node 1 to node 0 with capacity 5,\nan edge from node 1 to node 13 with capacity 18,\nan edge from node 1 to node 10 with capacity 11,\nan edge from node 1 to node 5 with capacity 16,\nan edge from node 1 to node 2 with capacity 18,\nan edge from node 2 to node 8 with capacity 20,\nan edge from node 2 to node 1 with capacity 15,\nan edge from node 2 to node 5 with capacity 1,\nan edge from node 3 to node 6 with capacity 20,\nan edge from node 3 to node 11 with capacity 19,\nan edge from node 4 to node 12 with capacity 14,\nan edge from node 4 to node 1 with capacity 14,\nan edge from node 5 to node 14 with capacity 4,\nan edge from node 5 to node 9 with capacity 20,\nan edge from node 5 to node 7 with capacity 1,\nan edge from node 5 to node 0 with capacity 16,\nan edge from node 5 to node 2 with capacity 17,\nan edge from node 6 to node 9 with capacity 1,\nan edge from node 8 to node 7 with capacity 15,\nan edge from node 8 to node 3 with capacity 2,\nan edge from node 8 to node 0 with capacity 3,\nan edge from node 9 to node 12 with capacity 12,\nan edge from node 9 to node 5 with capacity 13,\nan edge from node 10 to node 6 with capacity 9,\nan edge from node 10 to node 0 with capacity 5,\nan edge from node 10 to node 12 with capacity 7,\nan edge from node 10 to node 11 with capacity 14,\nan edge from node 11 to node 6 with capacity 20,\nan edge from node 11 to node 12 with capacity 17,\nan edge from node 12 to node 7 with capacity 12,\nan edge from node 12 to node 0 with capacity 11,\nan edge from node 12 to node 13 with capacity 10,\nan edge from node 12 to node 1 with capacity 15,\nan edge from node 12 to node 4 with capacity 20,\nan edge from node 13 to node 6 with capacity 9,\nan edge from node 13 to node 8 with capacity 17,\nan edge from node 13 to node 0 with capacity 16,\nan edge from node 13 to node 10 with capacity 16,\nan edge from node 13 to node 2 with capacity 13,\nan edge from node 13 to node 11 with capacity 3,\nan edge from node 14 to node 3 with capacity 1,\nan edge from node 14 to node 4 with capacity 13.\nQ: What is the maximum flow from node 9 to node 7?\nA:", "answer": "The maximum flow from node 9 to node 7 is 25.", "difficulty": "hard", "doc_id": "236"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 1 to node 4 with capacity 3,\nan edge from node 2 to node 0 with capacity 8,\nan edge from node 3 to node 7 with capacity 2,\nan edge from node 3 to node 2 with capacity 5,\nan edge from node 5 to node 7 with capacity 5,\nan edge from node 5 to node 3 with capacity 3,\nan edge from node 6 to node 8 with capacity 8,\nan edge from node 7 to node 5 with capacity 4,\nan edge from node 7 to node 0 with capacity 7,\nan edge from node 8 to node 1 with capacity 5,\nan edge from node 8 to node 4 with capacity 6,\nan edge from node 8 to node 0 with capacity 3.\nQ: What is the maximum flow from node 5 to node 0?\nA:", "answer": "The maximum flow from node 5 to node 0 is 8.", "difficulty": "easy", "doc_id": "237"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 17 with capacity 20,\nan edge from node 0 to node 18 with capacity 16,\nan edge from node 0 to node 8 with capacity 6,\nan edge from node 0 to node 6 with capacity 6,\nan edge from node 0 to node 7 with capacity 10,\nan edge from node 1 to node 4 with capacity 20,\nan edge from node 1 to node 10 with capacity 1,\nan edge from node 1 to node 9 with capacity 2,\nan edge from node 1 to node 2 with capacity 14,\nan edge from node 1 to node 7 with capacity 13,\nan edge from node 2 to node 1 with capacity 8,\nan edge from node 2 to node 14 with capacity 17,\nan edge from node 2 to node 15 with capacity 7,\nan edge from node 3 to node 1 with capacity 16,\nan edge from node 3 to node 14 with capacity 5,\nan edge from node 3 to node 11 with capacity 3,\nan edge from node 4 to node 1 with capacity 4,\nan edge from node 4 to node 14 with capacity 17,\nan edge from node 4 to node 3 with capacity 12,\nan edge from node 4 to node 15 with capacity 5,\nan edge from node 5 to node 16 with capacity 9,\nan edge from node 5 to node 12 with capacity 13,\nan edge from node 6 to node 17 with capacity 11,\nan edge from node 6 to node 16 with capacity 20,\nan edge from node 6 to node 14 with capacity 14,\nan edge from node 6 to node 10 with capacity 5,\nan edge from node 6 to node 9 with capacity 3,\nan edge from node 6 to node 18 with capacity 11,\nan edge from node 6 to node 11 with capacity 12,\nan edge from node 6 to node 12 with capacity 5,\nan edge from node 7 to node 17 with capacity 16,\nan edge from node 7 to node 16 with capacity 14,\nan edge from node 7 to node 4 with capacity 3,\nan edge from node 7 to node 14 with capacity 7,\nan edge from node 7 to node 8 with capacity 13,\nan edge from node 7 to node 6 with capacity 3,\nan edge from node 7 to node 15 with capacity 2,\nan edge from node 8 to node 17 with capacity 15,\nan edge from node 8 to node 10 with capacity 11,\nan edge from node 8 to node 3 with capacity 1,\nan edge from node 8 to node 2 with capacity 10,\nan edge from node 8 to node 13 with capacity 13,\nan edge from node 8 to node 12 with capacity 2,\nan edge from node 8 to node 15 with capacity 11,\nan edge from node 9 to node 17 with capacity 7,\nan edge from node 9 to node 16 with capacity 12,\nan edge from node 9 to node 4 with capacity 13,\nan edge from node 9 to node 7 with capacity 10,\nan edge from node 10 to node 1 with capacity 5,\nan edge from node 10 to node 17 with capacity 11,\nan edge from node 10 to node 9 with capacity 13,\nan edge from node 10 to node 13 with capacity 20,\nan edge from node 10 to node 7 with capacity 5,\nan edge from node 10 to node 15 with capacity 17,\nan edge from node 11 to node 8 with capacity 8,\nan edge from node 11 to node 5 with capacity 6,\nan edge from node 12 to node 1 with capacity 17,\nan edge from node 12 to node 3 with capacity 15,\nan edge from node 12 to node 2 with capacity 7,\nan edge from node 13 to node 10 with capacity 8,\nan edge from node 13 to node 3 with capacity 17,\nan edge from node 13 to node 2 with capacity 11,\nan edge from node 13 to node 11 with capacity 14,\nan edge from node 13 to node 7 with capacity 9,\nan edge from node 13 to node 0 with capacity 19,\nan edge from node 14 to node 3 with capacity 9,\nan edge from node 14 to node 2 with capacity 2,\nan edge from node 14 to node 8 with capacity 17,\nan edge from node 14 to node 5 with capacity 10,\nan edge from node 14 to node 12 with capacity 16,\nan edge from node 15 to node 17 with capacity 5,\nan edge from node 15 to node 10 with capacity 6,\nan edge from node 15 to node 9 with capacity 20,\nan edge from node 15 to node 3 with capacity 12,\nan edge from node 15 to node 11 with capacity 7,\nan edge from node 15 to node 12 with capacity 7,\nan edge from node 15 to node 0 with capacity 9,\nan edge from node 16 to node 17 with capacity 12,\nan edge from node 16 to node 14 with capacity 18,\nan edge from node 16 to node 3 with capacity 4,\nan edge from node 16 to node 18 with capacity 2,\nan edge from node 16 to node 2 with capacity 5,\nan edge from node 16 to node 8 with capacity 4,\nan edge from node 16 to node 0 with capacity 1,\nan edge from node 17 to node 6 with capacity 5,\nan edge from node 17 to node 11 with capacity 10,\nan edge from node 17 to node 5 with capacity 17,\nan edge from node 17 to node 15 with capacity 8,\nan edge from node 18 to node 1 with capacity 9,\nan edge from node 18 to node 16 with capacity 11,\nan edge from node 18 to node 14 with capacity 6,\nan edge from node 18 to node 10 with capacity 11,\nan edge from node 18 to node 8 with capacity 13,\nan edge from node 18 to node 6 with capacity 17.\nQ: What is the maximum flow from node 5 to node 1?\nA:", "answer": "The maximum flow from node 5 to node 1 is 22.", "difficulty": "hard", "doc_id": "238"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 0 to node 6 with capacity 10,\nan edge from node 0 to node 5 with capacity 8,\nan edge from node 1 to node 2 with capacity 1,\nan edge from node 1 to node 0 with capacity 6,\nan edge from node 1 to node 6 with capacity 7,\nan edge from node 2 to node 3 with capacity 7,\nan edge from node 2 to node 5 with capacity 4,\nan edge from node 3 to node 5 with capacity 1,\nan edge from node 3 to node 4 with capacity 7,\nan edge from node 4 to node 2 with capacity 5,\nan edge from node 4 to node 1 with capacity 6,\nan edge from node 4 to node 0 with capacity 4,\nan edge from node 4 to node 6 with capacity 9,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 5 to node 6 with capacity 6,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 6 to node 4 with capacity 2.\nQ: What is the maximum flow from node 1 to node 2?\nA:", "answer": "The maximum flow from node 1 to node 2 is 6.", "difficulty": "easy", "doc_id": "239"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 1 with capacity 5,\nan edge from node 0 to node 4 with capacity 1,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 5 with capacity 7,\nan edge from node 2 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 6,\nan edge from node 3 to node 2 with capacity 2,\nan edge from node 3 to node 0 with capacity 10,\nan edge from node 4 to node 5 with capacity 7.\nQ: What is the maximum flow from node 3 to node 4?\nA:", "answer": "The maximum flow from node 3 to node 4 is 8.", "difficulty": "easy", "doc_id": "240"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 9 with capacity 10,\nan edge from node 0 to node 1 with capacity 7,\nan edge from node 1 to node 4 with capacity 1,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 3,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 5 to node 8 with capacity 3,\nan edge from node 5 to node 3 with capacity 2,\nan edge from node 6 to node 5 with capacity 5,\nan edge from node 6 to node 2 with capacity 5,\nan edge from node 7 to node 2 with capacity 10,\nan edge from node 7 to node 8 with capacity 10,\nan edge from node 8 to node 5 with capacity 3,\nan edge from node 8 to node 2 with capacity 4,\nan edge from node 8 to node 7 with capacity 5,\nan edge from node 8 to node 3 with capacity 5,\nan edge from node 9 to node 5 with capacity 2,\nan edge from node 9 to node 1 with capacity 10,\nan edge from node 9 to node 3 with capacity 10.\nQ: What is the maximum flow from node 0 to node 4?\nA:", "answer": "The maximum flow from node 0 to node 4 is 4.", "difficulty": "easy", "doc_id": "241"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 8 with capacity 9,\nan edge from node 1 to node 7 with capacity 7,\nan edge from node 2 to node 6 with capacity 2,\nan edge from node 2 to node 0 with capacity 9,\nan edge from node 4 to node 3 with capacity 9,\nan edge from node 6 to node 8 with capacity 7,\nan edge from node 6 to node 0 with capacity 10,\nan edge from node 7 to node 1 with capacity 6,\nan edge from node 7 to node 2 with capacity 5,\nan edge from node 8 to node 5 with capacity 4,\nan edge from node 8 to node 3 with capacity 9.\nQ: What is the maximum flow from node 2 to node 5?\nA:", "answer": "The maximum flow from node 2 to node 5 is 4.", "difficulty": "easy", "doc_id": "242"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 13 with capacity 14,\nan edge from node 0 to node 7 with capacity 15,\nan edge from node 1 to node 4 with capacity 11,\nan edge from node 1 to node 16 with capacity 18,\nan edge from node 2 to node 11 with capacity 19,\nan edge from node 2 to node 7 with capacity 5,\nan edge from node 2 to node 9 with capacity 6,\nan edge from node 2 to node 0 with capacity 4,\nan edge from node 3 to node 13 with capacity 2,\nan edge from node 3 to node 4 with capacity 3,\nan edge from node 3 to node 9 with capacity 18,\nan edge from node 3 to node 0 with capacity 15,\nan edge from node 3 to node 16 with capacity 17,\nan edge from node 3 to node 5 with capacity 17,\nan edge from node 3 to node 12 with capacity 15,\nan edge from node 4 to node 13 with capacity 10,\nan edge from node 4 to node 1 with capacity 1,\nan edge from node 4 to node 9 with capacity 20,\nan edge from node 4 to node 5 with capacity 16,\nan edge from node 4 to node 14 with capacity 8,\nan edge from node 5 to node 13 with capacity 16,\nan edge from node 5 to node 1 with capacity 4,\nan edge from node 5 to node 10 with capacity 9,\nan edge from node 5 to node 9 with capacity 6,\nan edge from node 5 to node 14 with capacity 12,\nan edge from node 5 to node 12 with capacity 18,\nan edge from node 6 to node 11 with capacity 8,\nan edge from node 6 to node 8 with capacity 11,\nan edge from node 6 to node 3 with capacity 5,\nan edge from node 6 to node 0 with capacity 7,\nan edge from node 6 to node 15 with capacity 9,\nan edge from node 6 to node 12 with capacity 20,\nan edge from node 7 to node 11 with capacity 13,\nan edge from node 7 to node 9 with capacity 17,\nan edge from node 7 to node 3 with capacity 5,\nan edge from node 7 to node 15 with capacity 13,\nan edge from node 7 to node 16 with capacity 16,\nan edge from node 7 to node 5 with capacity 3,\nan edge from node 7 to node 14 with capacity 11,\nan edge from node 8 to node 11 with capacity 17,\nan edge from node 8 to node 9 with capacity 16,\nan edge from node 8 to node 6 with capacity 14,\nan edge from node 8 to node 2 with capacity 12,\nan edge from node 8 to node 3 with capacity 15,\nan edge from node 8 to node 14 with capacity 14,\nan edge from node 9 to node 13 with capacity 17,\nan edge from node 9 to node 4 with capacity 16,\nan edge from node 9 to node 0 with capacity 13,\nan edge from node 10 to node 1 with capacity 8,\nan edge from node 11 to node 8 with capacity 15,\nan edge from node 11 to node 15 with capacity 11,\nan edge from node 11 to node 14 with capacity 10,\nan edge from node 11 to node 12 with capacity 8,\nan edge from node 12 to node 11 with capacity 4,\nan edge from node 12 to node 9 with capacity 14,\nan edge from node 12 to node 3 with capacity 9,\nan edge from node 12 to node 14 with capacity 4,\nan edge from node 13 to node 10 with capacity 11,\nan edge from node 13 to node 4 with capacity 16,\nan edge from node 13 to node 15 with capacity 8,\nan edge from node 13 to node 12 with capacity 20,\nan edge from node 14 to node 10 with capacity 1,\nan edge from node 14 to node 15 with capacity 3,\nan edge from node 14 to node 16 with capacity 19,\nan edge from node 15 to node 4 with capacity 3,\nan edge from node 15 to node 6 with capacity 5,\nan edge from node 15 to node 3 with capacity 7,\nan edge from node 15 to node 0 with capacity 8,\nan edge from node 16 to node 7 with capacity 19,\nan edge from node 16 to node 6 with capacity 15,\nan edge from node 16 to node 3 with capacity 13,\nan edge from node 16 to node 0 with capacity 20,\nan edge from node 16 to node 15 with capacity 15,\nan edge from node 16 to node 5 with capacity 6.\nQ: What is the maximum flow from node 1 to node 10?\nA:", "answer": "The maximum flow from node 1 to node 10 is 21.", "difficulty": "hard", "doc_id": "243"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 3 with capacity 18,\nan edge from node 0 to node 6 with capacity 9,\nan edge from node 0 to node 8 with capacity 4,\nan edge from node 0 to node 1 with capacity 18,\nan edge from node 0 to node 7 with capacity 4,\nan edge from node 0 to node 4 with capacity 5,\nan edge from node 0 to node 10 with capacity 20,\nan edge from node 1 to node 2 with capacity 8,\nan edge from node 2 to node 6 with capacity 4,\nan edge from node 2 to node 8 with capacity 3,\nan edge from node 2 to node 1 with capacity 19,\nan edge from node 2 to node 9 with capacity 11,\nan edge from node 2 to node 10 with capacity 5,\nan edge from node 3 to node 8 with capacity 20,\nan edge from node 3 to node 7 with capacity 16,\nan edge from node 3 to node 4 with capacity 8,\nan edge from node 4 to node 2 with capacity 18,\nan edge from node 4 to node 7 with capacity 2,\nan edge from node 5 to node 2 with capacity 5,\nan edge from node 5 to node 9 with capacity 15,\nan edge from node 6 to node 7 with capacity 20,\nan edge from node 6 to node 4 with capacity 3,\nan edge from node 6 to node 10 with capacity 19,\nan edge from node 7 to node 3 with capacity 10,\nan edge from node 7 to node 4 with capacity 9,\nan edge from node 8 to node 2 with capacity 11,\nan edge from node 8 to node 4 with capacity 16,\nan edge from node 9 to node 0 with capacity 19,\nan edge from node 9 to node 3 with capacity 17,\nan edge from node 9 to node 8 with capacity 12,\nan edge from node 9 to node 2 with capacity 3,\nan edge from node 10 to node 9 with capacity 11.\nQ: What is the maximum flow from node 6 to node 4?\nA:", "answer": "The maximum flow from node 6 to node 4 is 33.", "difficulty": "hard", "doc_id": "244"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 5 with capacity 9,\nan edge from node 0 to node 6 with capacity 3,\nan edge from node 1 to node 3 with capacity 10,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 2 to node 6 with capacity 3,\nan edge from node 3 to node 6 with capacity 9,\nan edge from node 5 to node 1 with capacity 8,\nan edge from node 6 to node 5 with capacity 4,\nan edge from node 6 to node 3 with capacity 6,\nan edge from node 6 to node 7 with capacity 2.\nQ: What is the maximum flow from node 0 to node 3?\nA:", "answer": "The maximum flow from node 0 to node 3 is 11.", "difficulty": "easy", "doc_id": "245"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 10 with capacity 8,\nan edge from node 0 to node 3 with capacity 20,\nan edge from node 0 to node 12 with capacity 12,\nan edge from node 0 to node 2 with capacity 1,\nan edge from node 0 to node 6 with capacity 9,\nan edge from node 1 to node 3 with capacity 15,\nan edge from node 1 to node 5 with capacity 20,\nan edge from node 1 to node 8 with capacity 9,\nan edge from node 1 to node 0 with capacity 1,\nan edge from node 1 to node 6 with capacity 12,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 2 to node 11 with capacity 17,\nan edge from node 2 to node 1 with capacity 11,\nan edge from node 2 to node 8 with capacity 17,\nan edge from node 2 to node 14 with capacity 7,\nan edge from node 2 to node 6 with capacity 3,\nan edge from node 3 to node 15 with capacity 9,\nan edge from node 3 to node 11 with capacity 15,\nan edge from node 3 to node 6 with capacity 19,\nan edge from node 4 to node 10 with capacity 18,\nan edge from node 4 to node 3 with capacity 14,\nan edge from node 4 to node 11 with capacity 15,\nan edge from node 4 to node 13 with capacity 14,\nan edge from node 4 to node 6 with capacity 3,\nan edge from node 5 to node 11 with capacity 18,\nan edge from node 5 to node 8 with capacity 9,\nan edge from node 5 to node 13 with capacity 2,\nan edge from node 5 to node 0 with capacity 17,\nan edge from node 6 to node 4 with capacity 19,\nan edge from node 6 to node 15 with capacity 11,\nan edge from node 6 to node 7 with capacity 1,\nan edge from node 6 to node 1 with capacity 15,\nan edge from node 7 to node 13 with capacity 14,\nan edge from node 8 to node 9 with capacity 5,\nan edge from node 8 to node 10 with capacity 12,\nan edge from node 8 to node 3 with capacity 9,\nan edge from node 8 to node 11 with capacity 9,\nan edge from node 8 to node 13 with capacity 17,\nan edge from node 8 to node 0 with capacity 5,\nan edge from node 8 to node 6 with capacity 5,\nan edge from node 9 to node 4 with capacity 3,\nan edge from node 9 to node 3 with capacity 13,\nan edge from node 9 to node 5 with capacity 7,\nan edge from node 9 to node 7 with capacity 18,\nan edge from node 10 to node 3 with capacity 7,\nan edge from node 10 to node 15 with capacity 19,\nan edge from node 10 to node 8 with capacity 10,\nan edge from node 11 to node 3 with capacity 20,\nan edge from node 11 to node 2 with capacity 12,\nan edge from node 11 to node 0 with capacity 16,\nan edge from node 12 to node 4 with capacity 16,\nan edge from node 12 to node 5 with capacity 11,\nan edge from node 12 to node 11 with capacity 15,\nan edge from node 12 to node 8 with capacity 3,\nan edge from node 12 to node 2 with capacity 8,\nan edge from node 12 to node 14 with capacity 13,\nan edge from node 13 to node 3 with capacity 6,\nan edge from node 13 to node 7 with capacity 5,\nan edge from node 13 to node 8 with capacity 2,\nan edge from node 14 to node 10 with capacity 12,\nan edge from node 14 to node 5 with capacity 17,\nan edge from node 14 to node 8 with capacity 19,\nan edge from node 15 to node 5 with capacity 8,\nan edge from node 15 to node 1 with capacity 5,\nan edge from node 15 to node 2 with capacity 9.\nQ: What is the maximum flow from node 5 to node 12?\nA:", "answer": "The maximum flow from node 5 to node 12 is 12.", "difficulty": "hard", "doc_id": "246"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 7 with capacity 5,\nan edge from node 0 to node 8 with capacity 16,\nan edge from node 1 to node 0 with capacity 3,\nan edge from node 2 to node 15 with capacity 9,\nan edge from node 2 to node 11 with capacity 15,\nan edge from node 2 to node 5 with capacity 16,\nan edge from node 3 to node 16 with capacity 14,\nan edge from node 3 to node 14 with capacity 18,\nan edge from node 3 to node 13 with capacity 17,\nan edge from node 3 to node 7 with capacity 18,\nan edge from node 3 to node 1 with capacity 15,\nan edge from node 4 to node 3 with capacity 20,\nan edge from node 4 to node 2 with capacity 18,\nan edge from node 4 to node 12 with capacity 16,\nan edge from node 4 to node 8 with capacity 10,\nan edge from node 5 to node 3 with capacity 3,\nan edge from node 5 to node 15 with capacity 20,\nan edge from node 5 to node 2 with capacity 9,\nan edge from node 5 to node 6 with capacity 1,\nan edge from node 6 to node 10 with capacity 20,\nan edge from node 6 to node 4 with capacity 18,\nan edge from node 7 to node 9 with capacity 7,\nan edge from node 7 to node 14 with capacity 6,\nan edge from node 7 to node 1 with capacity 11,\nan edge from node 8 to node 11 with capacity 7,\nan edge from node 8 to node 14 with capacity 11,\nan edge from node 9 to node 3 with capacity 16,\nan edge from node 9 to node 16 with capacity 8,\nan edge from node 9 to node 13 with capacity 12,\nan edge from node 9 to node 4 with capacity 20,\nan edge from node 10 to node 2 with capacity 8,\nan edge from node 10 to node 11 with capacity 11,\nan edge from node 10 to node 4 with capacity 6,\nan edge from node 10 to node 0 with capacity 6,\nan edge from node 10 to node 8 with capacity 13,\nan edge from node 11 to node 16 with capacity 15,\nan edge from node 11 to node 14 with capacity 6,\nan edge from node 12 to node 16 with capacity 16,\nan edge from node 13 to node 9 with capacity 10,\nan edge from node 13 to node 2 with capacity 17,\nan edge from node 13 to node 11 with capacity 16,\nan edge from node 13 to node 7 with capacity 9,\nan edge from node 14 to node 9 with capacity 14,\nan edge from node 14 to node 11 with capacity 5,\nan edge from node 14 to node 13 with capacity 8,\nan edge from node 14 to node 4 with capacity 3,\nan edge from node 15 to node 3 with capacity 5,\nan edge from node 15 to node 13 with capacity 10,\nan edge from node 15 to node 12 with capacity 18,\nan edge from node 16 to node 3 with capacity 19,\nan edge from node 16 to node 9 with capacity 16,\nan edge from node 16 to node 14 with capacity 13,\nan edge from node 16 to node 13 with capacity 9.\nQ: What is the maximum flow from node 2 to node 5?\nA:", "answer": "The maximum flow from node 2 to node 5 is 16.", "difficulty": "hard", "doc_id": "247"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 1 with capacity 15,\nan edge from node 0 to node 8 with capacity 19,\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 9 with capacity 5,\nan edge from node 0 to node 2 with capacity 5,\nan edge from node 1 to node 10 with capacity 4,\nan edge from node 2 to node 5 with capacity 4,\nan edge from node 3 to node 6 with capacity 6,\nan edge from node 3 to node 8 with capacity 20,\nan edge from node 3 to node 5 with capacity 8,\nan edge from node 4 to node 8 with capacity 14,\nan edge from node 4 to node 2 with capacity 18,\nan edge from node 5 to node 0 with capacity 13,\nan edge from node 5 to node 6 with capacity 13,\nan edge from node 6 to node 8 with capacity 1,\nan edge from node 6 to node 9 with capacity 18,\nan edge from node 7 to node 6 with capacity 15,\nan edge from node 7 to node 8 with capacity 9,\nan edge from node 7 to node 5 with capacity 12,\nan edge from node 8 to node 1 with capacity 20,\nan edge from node 9 to node 0 with capacity 7,\nan edge from node 9 to node 7 with capacity 17,\nan edge from node 9 to node 8 with capacity 15,\nan edge from node 9 to node 4 with capacity 8,\nan edge from node 9 to node 5 with capacity 15,\nan edge from node 9 to node 2 with capacity 19,\nan edge from node 9 to node 10 with capacity 19,\nan edge from node 10 to node 6 with capacity 12,\nan edge from node 10 to node 3 with capacity 3,\nan edge from node 10 to node 5 with capacity 20,\nan edge from node 10 to node 2 with capacity 10.\nQ: What is the maximum flow from node 3 to node 10?\nA:", "answer": "The maximum flow from node 3 to node 10 is 18.", "difficulty": "hard", "doc_id": "248"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 9 with capacity 5,\nan edge from node 0 to node 2 with capacity 17,\nan edge from node 0 to node 15 with capacity 7,\nan edge from node 0 to node 1 with capacity 9,\nan edge from node 0 to node 3 with capacity 19,\nan edge from node 0 to node 7 with capacity 18,\nan edge from node 0 to node 10 with capacity 8,\nan edge from node 1 to node 2 with capacity 12,\nan edge from node 1 to node 15 with capacity 20,\nan edge from node 2 to node 8 with capacity 7,\nan edge from node 2 to node 0 with capacity 17,\nan edge from node 2 to node 9 with capacity 6,\nan edge from node 2 to node 6 with capacity 15,\nan edge from node 2 to node 3 with capacity 17,\nan edge from node 2 to node 7 with capacity 13,\nan edge from node 2 to node 12 with capacity 12,\nan edge from node 3 to node 14 with capacity 13,\nan edge from node 3 to node 15 with capacity 10,\nan edge from node 3 to node 4 with capacity 5,\nan edge from node 3 to node 10 with capacity 10,\nan edge from node 4 to node 8 with capacity 11,\nan edge from node 4 to node 9 with capacity 2,\nan edge from node 4 to node 2 with capacity 15,\nan edge from node 4 to node 3 with capacity 6,\nan edge from node 4 to node 7 with capacity 5,\nan edge from node 5 to node 14 with capacity 13,\nan edge from node 5 to node 15 with capacity 15,\nan edge from node 5 to node 11 with capacity 17,\nan edge from node 5 to node 7 with capacity 7,\nan edge from node 5 to node 13 with capacity 20,\nan edge from node 6 to node 5 with capacity 6,\nan edge from node 6 to node 9 with capacity 1,\nan edge from node 6 to node 11 with capacity 20,\nan edge from node 6 to node 12 with capacity 16,\nan edge from node 7 to node 5 with capacity 15,\nan edge from node 7 to node 14 with capacity 7,\nan edge from node 8 to node 1 with capacity 15,\nan edge from node 8 to node 11 with capacity 6,\nan edge from node 8 to node 13 with capacity 12,\nan edge from node 9 to node 0 with capacity 3,\nan edge from node 9 to node 6 with capacity 16,\nan edge from node 9 to node 13 with capacity 5,\nan edge from node 9 to node 12 with capacity 17,\nan edge from node 10 to node 4 with capacity 19,\nan edge from node 10 to node 3 with capacity 14,\nan edge from node 10 to node 12 with capacity 12,\nan edge from node 11 to node 5 with capacity 6,\nan edge from node 11 to node 14 with capacity 1,\nan edge from node 11 to node 9 with capacity 6,\nan edge from node 11 to node 2 with capacity 19,\nan edge from node 11 to node 7 with capacity 6,\nan edge from node 11 to node 12 with capacity 15,\nan edge from node 12 to node 5 with capacity 11,\nan edge from node 12 to node 6 with capacity 8,\nan edge from node 12 to node 15 with capacity 7,\nan edge from node 12 to node 11 with capacity 1,\nan edge from node 12 to node 13 with capacity 15,\nan edge from node 13 to node 14 with capacity 7,\nan edge from node 13 to node 15 with capacity 10,\nan edge from node 13 to node 3 with capacity 2,\nan edge from node 13 to node 11 with capacity 16,\nan edge from node 13 to node 7 with capacity 4,\nan edge from node 13 to node 12 with capacity 16,\nan edge from node 14 to node 0 with capacity 2,\nan edge from node 14 to node 15 with capacity 2,\nan edge from node 14 to node 3 with capacity 6,\nan edge from node 14 to node 7 with capacity 3,\nan edge from node 14 to node 13 with capacity 12,\nan edge from node 14 to node 12 with capacity 15,\nan edge from node 15 to node 14 with capacity 9,\nan edge from node 15 to node 1 with capacity 16,\nan edge from node 15 to node 7 with capacity 16.\nQ: What is the maximum flow from node 12 to node 8?\nA:", "answer": "The maximum flow from node 12 to node 8 is 18.", "difficulty": "hard", "doc_id": "249"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 1 with capacity 15,\nan edge from node 0 to node 7 with capacity 19,\nan edge from node 0 to node 3 with capacity 18,\nan edge from node 0 to node 4 with capacity 14,\nan edge from node 1 to node 12 with capacity 5,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 0 with capacity 16,\nan edge from node 1 to node 4 with capacity 20,\nan edge from node 2 to node 3 with capacity 10,\nan edge from node 2 to node 0 with capacity 15,\nan edge from node 3 to node 11 with capacity 5,\nan edge from node 3 to node 8 with capacity 17,\nan edge from node 4 to node 12 with capacity 14,\nan edge from node 4 to node 11 with capacity 20,\nan edge from node 4 to node 0 with capacity 16,\nan edge from node 4 to node 14 with capacity 17,\nan edge from node 5 to node 6 with capacity 6,\nan edge from node 5 to node 9 with capacity 16,\nan edge from node 5 to node 7 with capacity 11,\nan edge from node 5 to node 0 with capacity 12,\nan edge from node 5 to node 4 with capacity 19,\nan edge from node 6 to node 1 with capacity 9,\nan edge from node 6 to node 11 with capacity 6,\nan edge from node 6 to node 10 with capacity 18,\nan edge from node 6 to node 0 with capacity 7,\nan edge from node 7 to node 9 with capacity 13,\nan edge from node 7 to node 5 with capacity 17,\nan edge from node 8 to node 9 with capacity 19,\nan edge from node 8 to node 0 with capacity 4,\nan edge from node 8 to node 14 with capacity 17,\nan edge from node 9 to node 12 with capacity 7,\nan edge from node 10 to node 9 with capacity 15,\nan edge from node 10 to node 5 with capacity 7,\nan edge from node 10 to node 3 with capacity 12,\nan edge from node 11 to node 12 with capacity 9,\nan edge from node 11 to node 9 with capacity 10,\nan edge from node 11 to node 5 with capacity 1,\nan edge from node 11 to node 8 with capacity 11,\nan edge from node 11 to node 3 with capacity 17,\nan edge from node 11 to node 2 with capacity 8,\nan edge from node 12 to node 1 with capacity 12,\nan edge from node 12 to node 13 with capacity 5,\nan edge from node 12 to node 8 with capacity 10,\nan edge from node 12 to node 0 with capacity 5,\nan edge from node 12 to node 4 with capacity 15,\nan edge from node 12 to node 14 with capacity 5,\nan edge from node 13 to node 12 with capacity 3,\nan edge from node 13 to node 11 with capacity 6,\nan edge from node 13 to node 8 with capacity 4,\nan edge from node 13 to node 3 with capacity 18,\nan edge from node 14 to node 1 with capacity 14,\nan edge from node 14 to node 12 with capacity 8,\nan edge from node 14 to node 6 with capacity 20,\nan edge from node 14 to node 9 with capacity 1,\nan edge from node 14 to node 4 with capacity 10.\nQ: What is the maximum flow from node 7 to node 9?\nA:", "answer": "The maximum flow from node 7 to node 9 is 30.", "difficulty": "hard", "doc_id": "250"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 3 with capacity 18,\nan edge from node 0 to node 2 with capacity 5,\nan edge from node 1 to node 4 with capacity 15,\nan edge from node 1 to node 3 with capacity 15,\nan edge from node 1 to node 2 with capacity 1,\nan edge from node 1 to node 7 with capacity 5,\nan edge from node 2 to node 11 with capacity 18,\nan edge from node 2 to node 6 with capacity 20,\nan edge from node 2 to node 10 with capacity 17,\nan edge from node 3 to node 8 with capacity 2,\nan edge from node 3 to node 10 with capacity 12,\nan edge from node 3 to node 5 with capacity 4,\nan edge from node 5 to node 4 with capacity 15,\nan edge from node 5 to node 10 with capacity 8,\nan edge from node 5 to node 9 with capacity 9,\nan edge from node 5 to node 1 with capacity 11,\nan edge from node 5 to node 0 with capacity 17,\nan edge from node 6 to node 11 with capacity 1,\nan edge from node 6 to node 7 with capacity 16,\nan edge from node 6 to node 1 with capacity 5,\nan edge from node 7 to node 3 with capacity 4,\nan edge from node 8 to node 4 with capacity 7,\nan edge from node 8 to node 3 with capacity 12,\nan edge from node 8 to node 2 with capacity 1,\nan edge from node 8 to node 9 with capacity 5,\nan edge from node 8 to node 0 with capacity 10,\nan edge from node 9 to node 11 with capacity 12,\nan edge from node 9 to node 4 with capacity 8,\nan edge from node 9 to node 8 with capacity 7,\nan edge from node 9 to node 10 with capacity 10,\nan edge from node 9 to node 7 with capacity 13,\nan edge from node 9 to node 0 with capacity 2,\nan edge from node 10 to node 3 with capacity 18,\nan edge from node 10 to node 8 with capacity 1,\nan edge from node 10 to node 7 with capacity 10,\nan edge from node 11 to node 6 with capacity 18,\nan edge from node 11 to node 10 with capacity 4,\nan edge from node 11 to node 9 with capacity 17,\nan edge from node 11 to node 1 with capacity 3.\nQ: What is the maximum flow from node 2 to node 1?\nA:", "answer": "The maximum flow from node 2 to node 1 is 12.", "difficulty": "hard", "doc_id": "251"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 5 with capacity 1,\nan edge from node 0 to node 3 with capacity 5,\nan edge from node 1 to node 6 with capacity 9,\nan edge from node 1 to node 7 with capacity 9,\nan edge from node 3 to node 0 with capacity 2,\nan edge from node 3 to node 9 with capacity 3,\nan edge from node 3 to node 1 with capacity 3,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 5 to node 6 with capacity 9,\nan edge from node 6 to node 9 with capacity 2,\nan edge from node 6 to node 1 with capacity 1,\nan edge from node 7 to node 5 with capacity 7,\nan edge from node 7 to node 4 with capacity 8,\nan edge from node 7 to node 3 with capacity 10,\nan edge from node 8 to node 6 with capacity 2,\nan edge from node 8 to node 3 with capacity 3,\nan edge from node 8 to node 1 with capacity 7,\nan edge from node 9 to node 3 with capacity 4,\nan edge from node 9 to node 7 with capacity 7.\nQ: What is the maximum flow from node 6 to node 0?\nA:", "answer": "The maximum flow from node 6 to node 0 is 2.", "difficulty": "easy", "doc_id": "252"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 6 with capacity 7,\nan edge from node 0 to node 9 with capacity 1,\nan edge from node 1 to node 4 with capacity 6,\nan edge from node 1 to node 2 with capacity 10,\nan edge from node 2 to node 3 with capacity 18,\nan edge from node 2 to node 10 with capacity 7,\nan edge from node 3 to node 0 with capacity 18,\nan edge from node 3 to node 7 with capacity 19,\nan edge from node 4 to node 6 with capacity 8,\nan edge from node 4 to node 2 with capacity 19,\nan edge from node 4 to node 10 with capacity 8,\nan edge from node 5 to node 6 with capacity 18,\nan edge from node 5 to node 3 with capacity 16,\nan edge from node 6 to node 7 with capacity 8,\nan edge from node 6 to node 2 with capacity 14,\nan edge from node 6 to node 10 with capacity 5,\nan edge from node 7 to node 4 with capacity 16,\nan edge from node 7 to node 6 with capacity 13,\nan edge from node 7 to node 8 with capacity 11,\nan edge from node 7 to node 9 with capacity 14,\nan edge from node 7 to node 10 with capacity 8,\nan edge from node 8 to node 2 with capacity 17,\nan edge from node 9 to node 6 with capacity 9,\nan edge from node 9 to node 2 with capacity 16,\nan edge from node 9 to node 3 with capacity 14,\nan edge from node 9 to node 10 with capacity 18,\nan edge from node 10 to node 7 with capacity 4,\nan edge from node 10 to node 8 with capacity 18,\nan edge from node 10 to node 9 with capacity 17,\nan edge from node 10 to node 5 with capacity 6,\nan edge from node 10 to node 1 with capacity 13.\nQ: What is the maximum flow from node 3 to node 9?\nA:", "answer": "The maximum flow from node 3 to node 9 is 27.", "difficulty": "hard", "doc_id": "253"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 14 with capacity 19,\nan edge from node 0 to node 13 with capacity 16,\nan edge from node 0 to node 18 with capacity 20,\nan edge from node 0 to node 12 with capacity 14,\nan edge from node 1 to node 0 with capacity 12,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 2 to node 15 with capacity 4,\nan edge from node 2 to node 3 with capacity 10,\nan edge from node 2 to node 13 with capacity 4,\nan edge from node 2 to node 1 with capacity 13,\nan edge from node 2 to node 7 with capacity 17,\nan edge from node 2 to node 12 with capacity 12,\nan edge from node 3 to node 2 with capacity 18,\nan edge from node 3 to node 17 with capacity 5,\nan edge from node 3 to node 6 with capacity 8,\nan edge from node 3 to node 0 with capacity 15,\nan edge from node 4 to node 13 with capacity 18,\nan edge from node 4 to node 10 with capacity 12,\nan edge from node 4 to node 7 with capacity 17,\nan edge from node 5 to node 4 with capacity 20,\nan edge from node 5 to node 13 with capacity 1,\nan edge from node 6 to node 4 with capacity 16,\nan edge from node 6 to node 14 with capacity 18,\nan edge from node 6 to node 9 with capacity 6,\nan edge from node 6 to node 5 with capacity 20,\nan edge from node 6 to node 12 with capacity 10,\nan edge from node 7 to node 4 with capacity 14,\nan edge from node 7 to node 15 with capacity 17,\nan edge from node 7 to node 11 with capacity 18,\nan edge from node 8 to node 2 with capacity 3,\nan edge from node 8 to node 16 with capacity 5,\nan edge from node 8 to node 4 with capacity 11,\nan edge from node 8 to node 3 with capacity 3,\nan edge from node 8 to node 9 with capacity 11,\nan edge from node 8 to node 18 with capacity 6,\nan edge from node 9 to node 2 with capacity 13,\nan edge from node 9 to node 17 with capacity 14,\nan edge from node 9 to node 8 with capacity 11,\nan edge from node 9 to node 10 with capacity 4,\nan edge from node 9 to node 1 with capacity 16,\nan edge from node 9 to node 11 with capacity 12,\nan edge from node 9 to node 7 with capacity 11,\nan edge from node 9 to node 18 with capacity 11,\nan edge from node 10 to node 5 with capacity 20,\nan edge from node 11 to node 6 with capacity 10,\nan edge from node 11 to node 15 with capacity 9,\nan edge from node 11 to node 10 with capacity 15,\nan edge from node 11 to node 1 with capacity 4,\nan edge from node 11 to node 5 with capacity 2,\nan edge from node 12 to node 4 with capacity 17,\nan edge from node 12 to node 13 with capacity 14,\nan edge from node 12 to node 11 with capacity 5,\nan edge from node 12 to node 18 with capacity 9,\nan edge from node 12 to node 5 with capacity 15,\nan edge from node 13 to node 8 with capacity 11,\nan edge from node 13 to node 15 with capacity 1,\nan edge from node 13 to node 11 with capacity 1,\nan edge from node 13 to node 7 with capacity 14,\nan edge from node 13 to node 18 with capacity 15,\nan edge from node 13 to node 12 with capacity 17,\nan edge from node 14 to node 17 with capacity 15,\nan edge from node 14 to node 13 with capacity 20,\nan edge from node 14 to node 11 with capacity 20,\nan edge from node 14 to node 18 with capacity 2,\nan edge from node 14 to node 5 with capacity 17,\nan edge from node 15 to node 2 with capacity 12,\nan edge from node 15 to node 16 with capacity 10,\nan edge from node 15 to node 6 with capacity 17,\nan edge from node 15 to node 8 with capacity 14,\nan edge from node 15 to node 0 with capacity 9,\nan edge from node 15 to node 3 with capacity 20,\nan edge from node 15 to node 7 with capacity 2,\nan edge from node 15 to node 9 with capacity 11,\nan edge from node 15 to node 18 with capacity 6,\nan edge from node 15 to node 5 with capacity 20,\nan edge from node 16 to node 0 with capacity 12,\nan edge from node 16 to node 3 with capacity 11,\nan edge from node 16 to node 1 with capacity 9,\nan edge from node 17 to node 2 with capacity 18,\nan edge from node 17 to node 3 with capacity 20,\nan edge from node 17 to node 9 with capacity 9,\nan edge from node 18 to node 2 with capacity 1,\nan edge from node 18 to node 4 with capacity 10,\nan edge from node 18 to node 10 with capacity 11,\nan edge from node 18 to node 1 with capacity 17.\nQ: What is the maximum flow from node 1 to node 13?\nA:", "answer": "The maximum flow from node 1 to node 13 is 12.", "difficulty": "hard", "doc_id": "254"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 1 with capacity 14,\nan edge from node 0 to node 10 with capacity 14,\nan edge from node 0 to node 6 with capacity 6,\nan edge from node 1 to node 9 with capacity 8,\nan edge from node 2 to node 6 with capacity 5,\nan edge from node 2 to node 3 with capacity 18,\nan edge from node 3 to node 5 with capacity 11,\nan edge from node 3 to node 1 with capacity 16,\nan edge from node 3 to node 0 with capacity 15,\nan edge from node 3 to node 4 with capacity 10,\nan edge from node 3 to node 2 with capacity 15,\nan edge from node 3 to node 7 with capacity 13,\nan edge from node 4 to node 5 with capacity 17,\nan edge from node 5 to node 9 with capacity 19,\nan edge from node 5 to node 2 with capacity 20,\nan edge from node 6 to node 1 with capacity 15,\nan edge from node 6 to node 11 with capacity 7,\nan edge from node 6 to node 3 with capacity 11,\nan edge from node 7 to node 5 with capacity 5,\nan edge from node 7 to node 1 with capacity 11,\nan edge from node 7 to node 10 with capacity 2,\nan edge from node 7 to node 8 with capacity 9,\nan edge from node 8 to node 0 with capacity 1,\nan edge from node 8 to node 3 with capacity 17,\nan edge from node 9 to node 1 with capacity 3,\nan edge from node 9 to node 8 with capacity 3,\nan edge from node 9 to node 3 with capacity 15,\nan edge from node 10 to node 5 with capacity 13,\nan edge from node 10 to node 1 with capacity 20,\nan edge from node 11 to node 5 with capacity 12,\nan edge from node 11 to node 1 with capacity 19,\nan edge from node 11 to node 2 with capacity 3,\nan edge from node 11 to node 8 with capacity 10,\nan edge from node 11 to node 7 with capacity 20.\nQ: What is the maximum flow from node 6 to node 9?\nA:", "answer": "The maximum flow from node 6 to node 9 is 26.", "difficulty": "hard", "doc_id": "255"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 5 with capacity 1,\nan edge from node 0 to node 2 with capacity 5,\nan edge from node 1 to node 3 with capacity 8,\nan edge from node 2 to node 1 with capacity 4,\nan edge from node 4 to node 3 with capacity 4,\nan edge from node 4 to node 5 with capacity 10,\nan edge from node 4 to node 1 with capacity 3,\nan edge from node 5 to node 3 with capacity 7.\nQ: What is the maximum flow from node 4 to node 3?\nA:", "answer": "The maximum flow from node 4 to node 3 is 14.", "difficulty": "easy", "doc_id": "256"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 9 with capacity 10,\nan edge from node 0 to node 1 with capacity 8,\nan edge from node 0 to node 2 with capacity 11,\nan edge from node 0 to node 4 with capacity 9,\nan edge from node 0 to node 8 with capacity 8,\nan edge from node 1 to node 7 with capacity 15,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 0 with capacity 2,\nan edge from node 1 to node 11 with capacity 6,\nan edge from node 2 to node 6 with capacity 2,\nan edge from node 2 to node 1 with capacity 1,\nan edge from node 2 to node 7 with capacity 3,\nan edge from node 2 to node 4 with capacity 14,\nan edge from node 2 to node 0 with capacity 20,\nan edge from node 3 to node 6 with capacity 11,\nan edge from node 3 to node 8 with capacity 3,\nan edge from node 4 to node 2 with capacity 5,\nan edge from node 4 to node 8 with capacity 20,\nan edge from node 4 to node 12 with capacity 13,\nan edge from node 5 to node 1 with capacity 20,\nan edge from node 5 to node 7 with capacity 11,\nan edge from node 5 to node 0 with capacity 8,\nan edge from node 5 to node 11 with capacity 20,\nan edge from node 6 to node 10 with capacity 20,\nan edge from node 6 to node 4 with capacity 18,\nan edge from node 6 to node 5 with capacity 20,\nan edge from node 6 to node 3 with capacity 6,\nan edge from node 7 to node 9 with capacity 20,\nan edge from node 7 to node 6 with capacity 19,\nan edge from node 7 to node 4 with capacity 17,\nan edge from node 7 to node 5 with capacity 8,\nan edge from node 8 to node 6 with capacity 2,\nan edge from node 8 to node 12 with capacity 13,\nan edge from node 9 to node 2 with capacity 20,\nan edge from node 9 to node 5 with capacity 1,\nan edge from node 9 to node 3 with capacity 10,\nan edge from node 10 to node 4 with capacity 4,\nan edge from node 10 to node 5 with capacity 4,\nan edge from node 10 to node 3 with capacity 16,\nan edge from node 11 to node 7 with capacity 20,\nan edge from node 11 to node 4 with capacity 14,\nan edge from node 12 to node 4 with capacity 10,\nan edge from node 12 to node 5 with capacity 9.\nQ: What is the maximum flow from node 3 to node 6?\nA:", "answer": "The maximum flow from node 3 to node 6 is 14.", "difficulty": "hard", "doc_id": "257"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 2 with capacity 3,\nan edge from node 1 to node 5 with capacity 4,\nan edge from node 1 to node 0 with capacity 3,\nan edge from node 1 to node 7 with capacity 3,\nan edge from node 2 to node 4 with capacity 3,\nan edge from node 3 to node 4 with capacity 9,\nan edge from node 3 to node 0 with capacity 3,\nan edge from node 3 to node 1 with capacity 4,\nan edge from node 5 to node 6 with capacity 7,\nan edge from node 6 to node 3 with capacity 1.\nQ: What is the maximum flow from node 6 to node 4?\nA:", "answer": "The maximum flow from node 6 to node 4 is 1.", "difficulty": "easy", "doc_id": "258"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 10 with capacity 10,\nan edge from node 0 to node 5 with capacity 14,\nan edge from node 0 to node 14 with capacity 3,\nan edge from node 0 to node 2 with capacity 20,\nan edge from node 0 to node 15 with capacity 16,\nan edge from node 0 to node 7 with capacity 6,\nan edge from node 0 to node 1 with capacity 9,\nan edge from node 1 to node 6 with capacity 10,\nan edge from node 1 to node 13 with capacity 3,\nan edge from node 1 to node 18 with capacity 12,\nan edge from node 1 to node 5 with capacity 9,\nan edge from node 1 to node 11 with capacity 11,\nan edge from node 1 to node 3 with capacity 17,\nan edge from node 1 to node 14 with capacity 9,\nan edge from node 2 to node 10 with capacity 20,\nan edge from node 2 to node 11 with capacity 14,\nan edge from node 2 to node 15 with capacity 8,\nan edge from node 2 to node 7 with capacity 8,\nan edge from node 3 to node 13 with capacity 18,\nan edge from node 3 to node 8 with capacity 11,\nan edge from node 3 to node 4 with capacity 3,\nan edge from node 3 to node 17 with capacity 6,\nan edge from node 3 to node 16 with capacity 3,\nan edge from node 4 to node 10 with capacity 4,\nan edge from node 4 to node 9 with capacity 5,\nan edge from node 4 to node 5 with capacity 10,\nan edge from node 4 to node 2 with capacity 18,\nan edge from node 4 to node 7 with capacity 6,\nan edge from node 4 to node 12 with capacity 6,\nan edge from node 5 to node 10 with capacity 4,\nan edge from node 5 to node 13 with capacity 2,\nan edge from node 5 to node 8 with capacity 1,\nan edge from node 5 to node 1 with capacity 18,\nan edge from node 6 to node 13 with capacity 7,\nan edge from node 6 to node 18 with capacity 16,\nan edge from node 6 to node 11 with capacity 1,\nan edge from node 6 to node 15 with capacity 6,\nan edge from node 6 to node 19 with capacity 12,\nan edge from node 7 to node 8 with capacity 12,\nan edge from node 7 to node 11 with capacity 19,\nan edge from node 7 to node 2 with capacity 10,\nan edge from node 7 to node 12 with capacity 4,\nan edge from node 8 to node 18 with capacity 19,\nan edge from node 8 to node 3 with capacity 13,\nan edge from node 8 to node 7 with capacity 5,\nan edge from node 9 to node 10 with capacity 2,\nan edge from node 9 to node 18 with capacity 2,\nan edge from node 9 to node 4 with capacity 4,\nan edge from node 9 to node 16 with capacity 2,\nan edge from node 9 to node 1 with capacity 11,\nan edge from node 10 to node 5 with capacity 5,\nan edge from node 11 to node 10 with capacity 17,\nan edge from node 11 to node 13 with capacity 1,\nan edge from node 11 to node 8 with capacity 17,\nan edge from node 11 to node 3 with capacity 11,\nan edge from node 11 to node 2 with capacity 18,\nan edge from node 12 to node 6 with capacity 7,\nan edge from node 12 to node 2 with capacity 8,\nan edge from node 12 to node 17 with capacity 3,\nan edge from node 12 to node 7 with capacity 3,\nan edge from node 13 to node 10 with capacity 6,\nan edge from node 13 to node 3 with capacity 2,\nan edge from node 13 to node 7 with capacity 12,\nan edge from node 14 to node 13 with capacity 9,\nan edge from node 14 to node 18 with capacity 10,\nan edge from node 14 to node 5 with capacity 16,\nan edge from node 14 to node 3 with capacity 20,\nan edge from node 14 to node 19 with capacity 17,\nan edge from node 15 to node 18 with capacity 19,\nan edge from node 15 to node 5 with capacity 19,\nan edge from node 15 to node 3 with capacity 17,\nan edge from node 15 to node 14 with capacity 5,\nan edge from node 15 to node 17 with capacity 15,\nan edge from node 16 to node 10 with capacity 18,\nan edge from node 16 to node 1 with capacity 20,\nan edge from node 17 to node 9 with capacity 13,\nan edge from node 17 to node 18 with capacity 14,\nan edge from node 17 to node 5 with capacity 11,\nan edge from node 17 to node 11 with capacity 15,\nan edge from node 17 to node 19 with capacity 6,\nan edge from node 17 to node 16 with capacity 1,\nan edge from node 17 to node 0 with capacity 10,\nan edge from node 18 to node 10 with capacity 14,\nan edge from node 18 to node 13 with capacity 17,\nan edge from node 18 to node 5 with capacity 4,\nan edge from node 18 to node 3 with capacity 4,\nan edge from node 18 to node 1 with capacity 13,\nan edge from node 19 to node 10 with capacity 3,\nan edge from node 19 to node 4 with capacity 11,\nan edge from node 19 to node 3 with capacity 8,\nan edge from node 19 to node 15 with capacity 1,\nan edge from node 19 to node 1 with capacity 14.\nQ: What is the maximum flow from node 10 to node 3?\nA:", "answer": "The maximum flow from node 10 to node 3 is 5.", "difficulty": "hard", "doc_id": "259"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 13 with capacity 12,\nan edge from node 0 to node 12 with capacity 10,\nan edge from node 0 to node 5 with capacity 1,\nan edge from node 0 to node 17 with capacity 16,\nan edge from node 0 to node 18 with capacity 20,\nan edge from node 1 to node 9 with capacity 8,\nan edge from node 1 to node 10 with capacity 9,\nan edge from node 2 to node 16 with capacity 16,\nan edge from node 2 to node 3 with capacity 15,\nan edge from node 2 to node 12 with capacity 4,\nan edge from node 2 to node 8 with capacity 6,\nan edge from node 2 to node 7 with capacity 18,\nan edge from node 2 to node 4 with capacity 19,\nan edge from node 3 to node 13 with capacity 4,\nan edge from node 3 to node 16 with capacity 17,\nan edge from node 3 to node 14 with capacity 7,\nan edge from node 3 to node 18 with capacity 15,\nan edge from node 3 to node 15 with capacity 4,\nan edge from node 3 to node 10 with capacity 7,\nan edge from node 4 to node 12 with capacity 15,\nan edge from node 4 to node 2 with capacity 5,\nan edge from node 4 to node 9 with capacity 5,\nan edge from node 4 to node 17 with capacity 2,\nan edge from node 5 to node 12 with capacity 4,\nan edge from node 5 to node 2 with capacity 8,\nan edge from node 5 to node 4 with capacity 3,\nan edge from node 6 to node 13 with capacity 1,\nan edge from node 6 to node 3 with capacity 20,\nan edge from node 6 to node 12 with capacity 15,\nan edge from node 6 to node 7 with capacity 17,\nan edge from node 6 to node 11 with capacity 15,\nan edge from node 7 to node 16 with capacity 1,\nan edge from node 7 to node 6 with capacity 13,\nan edge from node 7 to node 2 with capacity 17,\nan edge from node 7 to node 9 with capacity 14,\nan edge from node 7 to node 1 with capacity 5,\nan edge from node 7 to node 0 with capacity 13,\nan edge from node 7 to node 10 with capacity 18,\nan edge from node 8 to node 6 with capacity 3,\nan edge from node 8 to node 3 with capacity 4,\nan edge from node 8 to node 2 with capacity 14,\nan edge from node 8 to node 4 with capacity 19,\nan edge from node 9 to node 14 with capacity 9,\nan edge from node 9 to node 1 with capacity 8,\nan edge from node 9 to node 15 with capacity 5,\nan edge from node 10 to node 2 with capacity 8,\nan edge from node 10 to node 14 with capacity 6,\nan edge from node 10 to node 8 with capacity 20,\nan edge from node 10 to node 9 with capacity 8,\nan edge from node 10 to node 11 with capacity 10,\nan edge from node 11 to node 3 with capacity 3,\nan edge from node 11 to node 14 with capacity 20,\nan edge from node 11 to node 5 with capacity 17,\nan edge from node 12 to node 6 with capacity 15,\nan edge from node 12 to node 3 with capacity 3,\nan edge from node 12 to node 7 with capacity 9,\nan edge from node 12 to node 11 with capacity 13,\nan edge from node 12 to node 1 with capacity 15,\nan edge from node 12 to node 10 with capacity 2,\nan edge from node 13 to node 3 with capacity 1,\nan edge from node 13 to node 9 with capacity 15,\nan edge from node 13 to node 4 with capacity 2,\nan edge from node 13 to node 17 with capacity 10,\nan edge from node 14 to node 8 with capacity 8,\nan edge from node 14 to node 4 with capacity 19,\nan edge from node 14 to node 17 with capacity 6,\nan edge from node 15 to node 16 with capacity 15,\nan edge from node 15 to node 3 with capacity 4,\nan edge from node 15 to node 12 with capacity 4,\nan edge from node 15 to node 14 with capacity 7,\nan edge from node 15 to node 8 with capacity 15,\nan edge from node 15 to node 0 with capacity 13,\nan edge from node 16 to node 2 with capacity 17,\nan edge from node 16 to node 7 with capacity 16,\nan edge from node 16 to node 11 with capacity 20,\nan edge from node 16 to node 18 with capacity 17,\nan edge from node 16 to node 15 with capacity 10,\nan edge from node 17 to node 16 with capacity 1,\nan edge from node 17 to node 4 with capacity 16,\nan edge from node 17 to node 11 with capacity 18,\nan edge from node 17 to node 18 with capacity 10,\nan edge from node 17 to node 1 with capacity 13,\nan edge from node 18 to node 13 with capacity 14,\nan edge from node 18 to node 6 with capacity 4,\nan edge from node 18 to node 3 with capacity 15,\nan edge from node 18 to node 8 with capacity 8,\nan edge from node 18 to node 7 with capacity 17.\nQ: What is the maximum flow from node 3 to node 5?\nA:", "answer": "The maximum flow from node 3 to node 5 is 18.", "difficulty": "hard", "doc_id": "260"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 12 with capacity 10,\nan edge from node 0 to node 10 with capacity 17,\nan edge from node 0 to node 9 with capacity 17,\nan edge from node 1 to node 3 with capacity 2,\nan edge from node 1 to node 7 with capacity 12,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 12 with capacity 1,\nan edge from node 3 to node 13 with capacity 1,\nan edge from node 3 to node 11 with capacity 6,\nan edge from node 3 to node 7 with capacity 15,\nan edge from node 3 to node 5 with capacity 11,\nan edge from node 4 to node 1 with capacity 17,\nan edge from node 4 to node 13 with capacity 2,\nan edge from node 4 to node 9 with capacity 15,\nan edge from node 5 to node 12 with capacity 13,\nan edge from node 5 to node 1 with capacity 10,\nan edge from node 5 to node 3 with capacity 12,\nan edge from node 5 to node 0 with capacity 17,\nan edge from node 5 to node 13 with capacity 1,\nan edge from node 5 to node 11 with capacity 18,\nan edge from node 5 to node 7 with capacity 10,\nan edge from node 7 to node 3 with capacity 16,\nan edge from node 7 to node 13 with capacity 13,\nan edge from node 7 to node 14 with capacity 1,\nan edge from node 7 to node 8 with capacity 13,\nan edge from node 8 to node 12 with capacity 1,\nan edge from node 8 to node 1 with capacity 7,\nan edge from node 8 to node 3 with capacity 6,\nan edge from node 8 to node 0 with capacity 7,\nan edge from node 9 to node 6 with capacity 2,\nan edge from node 9 to node 1 with capacity 11,\nan edge from node 9 to node 3 with capacity 18,\nan edge from node 9 to node 0 with capacity 10,\nan edge from node 9 to node 4 with capacity 11,\nan edge from node 9 to node 7 with capacity 4,\nan edge from node 10 to node 1 with capacity 8,\nan edge from node 10 to node 8 with capacity 2,\nan edge from node 11 to node 6 with capacity 1,\nan edge from node 11 to node 3 with capacity 18,\nan edge from node 11 to node 13 with capacity 17,\nan edge from node 11 to node 7 with capacity 1,\nan edge from node 12 to node 0 with capacity 1,\nan edge from node 13 to node 1 with capacity 2,\nan edge from node 13 to node 5 with capacity 10.\nQ: What is the maximum flow from node 9 to node 12?\nA:", "answer": "The maximum flow from node 9 to node 12 is 25.", "difficulty": "hard", "doc_id": "261"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 4 with capacity 9,\nan edge from node 0 to node 1 with capacity 4,\nan edge from node 0 to node 3 with capacity 4,\nan edge from node 1 to node 0 with capacity 9,\nan edge from node 1 to node 2 with capacity 3,\nan edge from node 2 to node 0 with capacity 3,\nan edge from node 2 to node 4 with capacity 10,\nan edge from node 2 to node 3 with capacity 1,\nan edge from node 3 to node 1 with capacity 4,\nan edge from node 4 to node 2 with capacity 3,\nan edge from node 4 to node 3 with capacity 1.\nQ: What is the maximum flow from node 0 to node 3?\nA:", "answer": "The maximum flow from node 0 to node 3 is 6.", "difficulty": "easy", "doc_id": "262"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 14, and the edges are:\nan edge from node 0 to node 4 with capacity 11,\nan edge from node 0 to node 2 with capacity 18,\nan edge from node 0 to node 3 with capacity 13,\nan edge from node 0 to node 11 with capacity 14,\nan edge from node 0 to node 13 with capacity 2,\nan edge from node 1 to node 4 with capacity 6,\nan edge from node 1 to node 2 with capacity 7,\nan edge from node 1 to node 6 with capacity 15,\nan edge from node 1 to node 14 with capacity 2,\nan edge from node 2 to node 5 with capacity 18,\nan edge from node 2 to node 8 with capacity 10,\nan edge from node 2 to node 13 with capacity 14,\nan edge from node 3 to node 12 with capacity 10,\nan edge from node 3 to node 1 with capacity 18,\nan edge from node 3 to node 6 with capacity 12,\nan edge from node 3 to node 13 with capacity 13,\nan edge from node 4 to node 2 with capacity 14,\nan edge from node 4 to node 8 with capacity 7,\nan edge from node 4 to node 0 with capacity 1,\nan edge from node 4 to node 6 with capacity 7,\nan edge from node 4 to node 13 with capacity 13,\nan edge from node 5 to node 0 with capacity 1,\nan edge from node 5 to node 10 with capacity 18,\nan edge from node 6 to node 3 with capacity 18,\nan edge from node 6 to node 11 with capacity 10,\nan edge from node 7 to node 2 with capacity 4,\nan edge from node 7 to node 9 with capacity 9,\nan edge from node 7 to node 1 with capacity 13,\nan edge from node 7 to node 13 with capacity 19,\nan edge from node 8 to node 2 with capacity 2,\nan edge from node 8 to node 0 with capacity 14,\nan edge from node 8 to node 9 with capacity 15,\nan edge from node 8 to node 10 with capacity 5,\nan edge from node 8 to node 13 with capacity 6,\nan edge from node 9 to node 2 with capacity 4,\nan edge from node 9 to node 5 with capacity 18,\nan edge from node 9 to node 14 with capacity 14,\nan edge from node 10 to node 8 with capacity 18,\nan edge from node 10 to node 0 with capacity 12,\nan edge from node 10 to node 6 with capacity 7,\nan edge from node 11 to node 5 with capacity 16,\nan edge from node 11 to node 1 with capacity 19,\nan edge from node 12 to node 5 with capacity 7,\nan edge from node 12 to node 8 with capacity 11,\nan edge from node 12 to node 1 with capacity 13,\nan edge from node 13 to node 5 with capacity 13,\nan edge from node 14 to node 3 with capacity 20,\nan edge from node 14 to node 9 with capacity 16.\nQ: What is the maximum flow from node 0 to node 12?\nA:", "answer": "The maximum flow from node 0 to node 12 is 10.", "difficulty": "hard", "doc_id": "263"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 1 with capacity 5,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 4 with capacity 3,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 3 with capacity 2,\nan edge from node 2 to node 1 with capacity 4,\nan edge from node 3 to node 2 with capacity 10,\nan edge from node 4 to node 2 with capacity 1,\nan edge from node 4 to node 1 with capacity 3,\nan edge from node 5 to node 0 with capacity 6,\nan edge from node 5 to node 1 with capacity 2,\nan edge from node 5 to node 4 with capacity 1.\nQ: What is the maximum flow from node 3 to node 4?\nA:", "answer": "The maximum flow from node 3 to node 4 is 4.", "difficulty": "easy", "doc_id": "264"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 17, and the edges are:\nan edge from node 0 to node 9 with capacity 12,\nan edge from node 0 to node 7 with capacity 15,\nan edge from node 0 to node 16 with capacity 13,\nan edge from node 0 to node 13 with capacity 2,\nan edge from node 1 to node 7 with capacity 6,\nan edge from node 1 to node 12 with capacity 12,\nan edge from node 1 to node 11 with capacity 13,\nan edge from node 2 to node 10 with capacity 19,\nan edge from node 2 to node 0 with capacity 1,\nan edge from node 2 to node 17 with capacity 14,\nan edge from node 3 to node 9 with capacity 16,\nan edge from node 3 to node 0 with capacity 7,\nan edge from node 3 to node 2 with capacity 20,\nan edge from node 3 to node 8 with capacity 17,\nan edge from node 3 to node 15 with capacity 7,\nan edge from node 4 to node 3 with capacity 19,\nan edge from node 4 to node 7 with capacity 16,\nan edge from node 4 to node 2 with capacity 5,\nan edge from node 4 to node 16 with capacity 2,\nan edge from node 4 to node 17 with capacity 7,\nan edge from node 5 to node 9 with capacity 7,\nan edge from node 6 to node 3 with capacity 4,\nan edge from node 6 to node 1 with capacity 17,\nan edge from node 6 to node 15 with capacity 10,\nan edge from node 7 to node 9 with capacity 7,\nan edge from node 7 to node 4 with capacity 20,\nan edge from node 7 to node 10 with capacity 14,\nan edge from node 7 to node 13 with capacity 7,\nan edge from node 7 to node 17 with capacity 8,\nan edge from node 8 to node 3 with capacity 8,\nan edge from node 8 to node 4 with capacity 11,\nan edge from node 8 to node 10 with capacity 18,\nan edge from node 9 to node 3 with capacity 20,\nan edge from node 9 to node 15 with capacity 14,\nan edge from node 9 to node 13 with capacity 1,\nan edge from node 10 to node 6 with capacity 3,\nan edge from node 10 to node 2 with capacity 9,\nan edge from node 10 to node 12 with capacity 7,\nan edge from node 10 to node 5 with capacity 6,\nan edge from node 11 to node 9 with capacity 15,\nan edge from node 11 to node 2 with capacity 16,\nan edge from node 11 to node 8 with capacity 19,\nan edge from node 11 to node 16 with capacity 7,\nan edge from node 11 to node 5 with capacity 17,\nan edge from node 11 to node 17 with capacity 13,\nan edge from node 12 to node 4 with capacity 15,\nan edge from node 12 to node 13 with capacity 13,\nan edge from node 13 to node 3 with capacity 18,\nan edge from node 13 to node 14 with capacity 12,\nan edge from node 13 to node 6 with capacity 1,\nan edge from node 13 to node 0 with capacity 13,\nan edge from node 13 to node 8 with capacity 4,\nan edge from node 13 to node 11 with capacity 5,\nan edge from node 14 to node 7 with capacity 19,\nan edge from node 14 to node 8 with capacity 16,\nan edge from node 14 to node 5 with capacity 7,\nan edge from node 14 to node 15 with capacity 20,\nan edge from node 14 to node 17 with capacity 20,\nan edge from node 15 to node 6 with capacity 18,\nan edge from node 15 to node 8 with capacity 4,\nan edge from node 15 to node 12 with capacity 1,\nan edge from node 15 to node 5 with capacity 20,\nan edge from node 16 to node 9 with capacity 2,\nan edge from node 16 to node 1 with capacity 15,\nan edge from node 16 to node 14 with capacity 5,\nan edge from node 16 to node 12 with capacity 2,\nan edge from node 16 to node 17 with capacity 16,\nan edge from node 17 to node 4 with capacity 7,\nan edge from node 17 to node 0 with capacity 9,\nan edge from node 17 to node 16 with capacity 19,\nan edge from node 17 to node 15 with capacity 12.\nQ: What is the maximum flow from node 4 to node 16?\nA:", "answer": "The maximum flow from node 4 to node 16 is 41.", "difficulty": "hard", "doc_id": "265"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 11, and the edges are:\nan edge from node 0 to node 9 with capacity 7,\nan edge from node 0 to node 10 with capacity 15,\nan edge from node 1 to node 10 with capacity 7,\nan edge from node 1 to node 6 with capacity 6,\nan edge from node 1 to node 11 with capacity 15,\nan edge from node 1 to node 7 with capacity 16,\nan edge from node 2 to node 3 with capacity 17,\nan edge from node 2 to node 9 with capacity 19,\nan edge from node 2 to node 6 with capacity 18,\nan edge from node 2 to node 0 with capacity 18,\nan edge from node 2 to node 8 with capacity 13,\nan edge from node 3 to node 6 with capacity 20,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 4 to node 1 with capacity 11,\nan edge from node 5 to node 9 with capacity 5,\nan edge from node 5 to node 6 with capacity 4,\nan edge from node 5 to node 0 with capacity 12,\nan edge from node 5 to node 11 with capacity 6,\nan edge from node 5 to node 8 with capacity 11,\nan edge from node 6 to node 8 with capacity 4,\nan edge from node 6 to node 7 with capacity 8,\nan edge from node 6 to node 1 with capacity 18,\nan edge from node 7 to node 5 with capacity 7,\nan edge from node 7 to node 8 with capacity 16,\nan edge from node 9 to node 5 with capacity 16,\nan edge from node 9 to node 2 with capacity 16,\nan edge from node 9 to node 1 with capacity 4,\nan edge from node 10 to node 0 with capacity 12,\nan edge from node 10 to node 8 with capacity 10,\nan edge from node 11 to node 10 with capacity 12,\nan edge from node 11 to node 4 with capacity 2,\nan edge from node 11 to node 1 with capacity 6.\nQ: What is the maximum flow from node 0 to node 11?\nA:", "answer": "The maximum flow from node 0 to node 11 is 7.", "difficulty": "hard", "doc_id": "266"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 7 with capacity 5,\nan edge from node 0 to node 9 with capacity 1,\nan edge from node 1 to node 9 with capacity 7,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 2 to node 1 with capacity 1,\nan edge from node 2 to node 5 with capacity 4,\nan edge from node 2 to node 3 with capacity 2,\nan edge from node 3 to node 9 with capacity 2,\nan edge from node 4 to node 1 with capacity 3,\nan edge from node 4 to node 6 with capacity 1,\nan edge from node 4 to node 7 with capacity 5,\nan edge from node 4 to node 2 with capacity 9,\nan edge from node 5 to node 4 with capacity 9,\nan edge from node 5 to node 8 with capacity 1,\nan edge from node 6 to node 4 with capacity 3,\nan edge from node 6 to node 7 with capacity 3,\nan edge from node 6 to node 9 with capacity 3,\nan edge from node 8 to node 1 with capacity 8,\nan edge from node 8 to node 7 with capacity 5,\nan edge from node 8 to node 0 with capacity 4,\nan edge from node 8 to node 5 with capacity 10,\nan edge from node 8 to node 3 with capacity 6,\nan edge from node 9 to node 1 with capacity 5,\nan edge from node 9 to node 0 with capacity 8,\nan edge from node 9 to node 3 with capacity 9.\nQ: What is the maximum flow from node 9 to node 3?\nA:", "answer": "The maximum flow from node 9 to node 3 is 12.", "difficulty": "easy", "doc_id": "267"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 5,\nan edge from node 1 to node 0 with capacity 3,\nan edge from node 2 to node 1 with capacity 7,\nan edge from node 3 to node 0 with capacity 6,\nan edge from node 3 to node 2 with capacity 10,\nan edge from node 4 to node 0 with capacity 5,\nan edge from node 4 to node 3 with capacity 10.\nQ: What is the maximum flow from node 4 to node 0?\nA:", "answer": "The maximum flow from node 4 to node 0 is 14.", "difficulty": "easy", "doc_id": "268"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 4 with capacity 1,\nan edge from node 0 to node 8 with capacity 4,\nan edge from node 1 to node 5 with capacity 7,\nan edge from node 1 to node 9 with capacity 6,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 2 to node 8 with capacity 4,\nan edge from node 2 to node 9 with capacity 9,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 3 to node 2 with capacity 2,\nan edge from node 4 to node 6 with capacity 1,\nan edge from node 4 to node 1 with capacity 6,\nan edge from node 5 to node 6 with capacity 3,\nan edge from node 5 to node 1 with capacity 8,\nan edge from node 5 to node 2 with capacity 10,\nan edge from node 7 to node 4 with capacity 5,\nan edge from node 7 to node 6 with capacity 5,\nan edge from node 8 to node 0 with capacity 7,\nan edge from node 9 to node 7 with capacity 10,\nan edge from node 9 to node 8 with capacity 1.\nQ: What is the maximum flow from node 1 to node 6?\nA:", "answer": "The maximum flow from node 1 to node 6 is 9.", "difficulty": "easy", "doc_id": "269"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 5 with capacity 4,\nan edge from node 1 to node 0 with capacity 4,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 2 to node 3 with capacity 5,\nan edge from node 3 to node 1 with capacity 6,\nan edge from node 3 to node 5 with capacity 2,\nan edge from node 3 to node 2 with capacity 5,\nan edge from node 4 to node 3 with capacity 8,\nan edge from node 5 to node 2 with capacity 1.\nQ: What is the maximum flow from node 1 to node 2?\nA:", "answer": "The maximum flow from node 1 to node 2 is 6.", "difficulty": "easy", "doc_id": "270"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 1 to node 3 with capacity 12,\nan edge from node 1 to node 4 with capacity 16,\nan edge from node 1 to node 5 with capacity 7,\nan edge from node 1 to node 10 with capacity 5,\nan edge from node 1 to node 9 with capacity 12,\nan edge from node 1 to node 12 with capacity 18,\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 2 to node 5 with capacity 10,\nan edge from node 3 to node 4 with capacity 5,\nan edge from node 3 to node 9 with capacity 12,\nan edge from node 4 to node 2 with capacity 8,\nan edge from node 4 to node 12 with capacity 11,\nan edge from node 5 to node 7 with capacity 4,\nan edge from node 5 to node 9 with capacity 6,\nan edge from node 5 to node 8 with capacity 16,\nan edge from node 5 to node 0 with capacity 12,\nan edge from node 6 to node 7 with capacity 18,\nan edge from node 6 to node 2 with capacity 9,\nan edge from node 6 to node 0 with capacity 5,\nan edge from node 7 to node 9 with capacity 16,\nan edge from node 7 to node 12 with capacity 9,\nan edge from node 7 to node 0 with capacity 11,\nan edge from node 7 to node 11 with capacity 15,\nan edge from node 8 to node 1 with capacity 15,\nan edge from node 9 to node 0 with capacity 5,\nan edge from node 10 to node 6 with capacity 10,\nan edge from node 10 to node 11 with capacity 11,\nan edge from node 11 to node 3 with capacity 9,\nan edge from node 11 to node 5 with capacity 16,\nan edge from node 11 to node 10 with capacity 1,\nan edge from node 12 to node 3 with capacity 12,\nan edge from node 12 to node 4 with capacity 19,\nan edge from node 12 to node 5 with capacity 15,\nan edge from node 12 to node 8 with capacity 5.\nQ: What is the maximum flow from node 1 to node 9?\nA:", "answer": "The maximum flow from node 1 to node 9 is 39.", "difficulty": "hard", "doc_id": "271"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 7 with capacity 4,\nan edge from node 0 to node 4 with capacity 6,\nan edge from node 0 to node 5 with capacity 4,\nan edge from node 0 to node 2 with capacity 5,\nan edge from node 3 to node 7 with capacity 8,\nan edge from node 3 to node 5 with capacity 9,\nan edge from node 4 to node 0 with capacity 4,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 8 with capacity 4,\nan edge from node 4 to node 1 with capacity 7,\nan edge from node 6 to node 7 with capacity 4,\nan edge from node 6 to node 3 with capacity 7,\nan edge from node 6 to node 8 with capacity 6,\nan edge from node 6 to node 1 with capacity 7,\nan edge from node 7 to node 0 with capacity 3,\nan edge from node 8 to node 2 with capacity 2.\nQ: What is the maximum flow from node 6 to node 2?\nA:", "answer": "The maximum flow from node 6 to node 2 is 5.", "difficulty": "easy", "doc_id": "272"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 6 with capacity 5,\nan edge from node 1 to node 2 with capacity 4,\nan edge from node 1 to node 0 with capacity 3,\nan edge from node 1 to node 9 with capacity 6,\nan edge from node 2 to node 1 with capacity 4,\nan edge from node 2 to node 9 with capacity 1,\nan edge from node 3 to node 0 with capacity 9,\nan edge from node 3 to node 1 with capacity 1,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 6 with capacity 7,\nan edge from node 4 to node 8 with capacity 6,\nan edge from node 4 to node 3 with capacity 1,\nan edge from node 4 to node 9 with capacity 5,\nan edge from node 6 to node 7 with capacity 2,\nan edge from node 7 to node 4 with capacity 8,\nan edge from node 9 to node 6 with capacity 5.\nQ: What is the maximum flow from node 0 to node 9?\nA:", "answer": "The maximum flow from node 0 to node 9 is 2.", "difficulty": "easy", "doc_id": "273"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 18, and the edges are:\nan edge from node 0 to node 5 with capacity 6,\nan edge from node 0 to node 3 with capacity 20,\nan edge from node 0 to node 9 with capacity 3,\nan edge from node 0 to node 14 with capacity 8,\nan edge from node 0 to node 12 with capacity 18,\nan edge from node 0 to node 2 with capacity 8,\nan edge from node 1 to node 5 with capacity 4,\nan edge from node 1 to node 16 with capacity 15,\nan edge from node 2 to node 11 with capacity 10,\nan edge from node 2 to node 3 with capacity 9,\nan edge from node 2 to node 18 with capacity 4,\nan edge from node 2 to node 7 with capacity 1,\nan edge from node 2 to node 9 with capacity 6,\nan edge from node 2 to node 15 with capacity 19,\nan edge from node 2 to node 4 with capacity 11,\nan edge from node 2 to node 16 with capacity 15,\nan edge from node 2 to node 1 with capacity 18,\nan edge from node 3 to node 16 with capacity 12,\nan edge from node 4 to node 11 with capacity 10,\nan edge from node 4 to node 5 with capacity 6,\nan edge from node 4 to node 3 with capacity 18,\nan edge from node 4 to node 9 with capacity 1,\nan edge from node 4 to node 15 with capacity 15,\nan edge from node 4 to node 14 with capacity 13,\nan edge from node 4 to node 17 with capacity 14,\nan edge from node 4 to node 12 with capacity 4,\nan edge from node 4 to node 8 with capacity 11,\nan edge from node 5 to node 11 with capacity 9,\nan edge from node 5 to node 18 with capacity 17,\nan edge from node 5 to node 7 with capacity 3,\nan edge from node 5 to node 4 with capacity 17,\nan edge from node 6 to node 13 with capacity 10,\nan edge from node 6 to node 10 with capacity 18,\nan edge from node 6 to node 16 with capacity 14,\nan edge from node 6 to node 1 with capacity 15,\nan edge from node 7 to node 13 with capacity 14,\nan edge from node 7 to node 9 with capacity 1,\nan edge from node 7 to node 4 with capacity 13,\nan edge from node 7 to node 2 with capacity 15,\nan edge from node 8 to node 11 with capacity 19,\nan edge from node 8 to node 5 with capacity 2,\nan edge from node 8 to node 6 with capacity 3,\nan edge from node 8 to node 15 with capacity 11,\nan edge from node 8 to node 14 with capacity 13,\nan edge from node 8 to node 16 with capacity 20,\nan edge from node 9 to node 3 with capacity 7,\nan edge from node 9 to node 18 with capacity 9,\nan edge from node 9 to node 0 with capacity 12,\nan edge from node 9 to node 4 with capacity 9,\nan edge from node 9 to node 2 with capacity 4,\nan edge from node 9 to node 1 with capacity 6,\nan edge from node 10 to node 13 with capacity 16,\nan edge from node 10 to node 5 with capacity 16,\nan edge from node 10 to node 3 with capacity 16,\nan edge from node 10 to node 18 with capacity 13,\nan edge from node 10 to node 0 with capacity 7,\nan edge from node 10 to node 6 with capacity 10,\nan edge from node 10 to node 4 with capacity 18,\nan edge from node 10 to node 2 with capacity 17,\nan edge from node 10 to node 8 with capacity 13,\nan edge from node 11 to node 14 with capacity 8,\nan edge from node 11 to node 12 with capacity 5,\nan edge from node 11 to node 8 with capacity 14,\nan edge from node 12 to node 6 with capacity 8,\nan edge from node 12 to node 14 with capacity 18,\nan edge from node 12 to node 2 with capacity 19,\nan edge from node 12 to node 10 with capacity 16,\nan edge from node 12 to node 16 with capacity 3,\nan edge from node 13 to node 11 with capacity 4,\nan edge from node 13 to node 5 with capacity 15,\nan edge from node 13 to node 9 with capacity 15,\nan edge from node 13 to node 12 with capacity 16,\nan edge from node 13 to node 2 with capacity 16,\nan edge from node 14 to node 5 with capacity 5,\nan edge from node 14 to node 9 with capacity 17,\nan edge from node 14 to node 6 with capacity 14,\nan edge from node 14 to node 4 with capacity 13,\nan edge from node 14 to node 17 with capacity 14,\nan edge from node 14 to node 12 with capacity 15,\nan edge from node 14 to node 2 with capacity 2,\nan edge from node 15 to node 0 with capacity 5,\nan edge from node 15 to node 9 with capacity 7,\nan edge from node 15 to node 17 with capacity 6,\nan edge from node 15 to node 8 with capacity 9,\nan edge from node 15 to node 1 with capacity 19,\nan edge from node 16 to node 5 with capacity 2,\nan edge from node 16 to node 6 with capacity 2,\nan edge from node 16 to node 4 with capacity 3,\nan edge from node 17 to node 18 with capacity 4,\nan edge from node 17 to node 14 with capacity 20,\nan edge from node 17 to node 12 with capacity 7,\nan edge from node 17 to node 10 with capacity 10,\nan edge from node 18 to node 11 with capacity 16,\nan edge from node 18 to node 7 with capacity 1,\nan edge from node 18 to node 15 with capacity 9,\nan edge from node 18 to node 14 with capacity 15,\nan edge from node 18 to node 17 with capacity 16,\nan edge from node 18 to node 2 with capacity 14.\nQ: What is the maximum flow from node 14 to node 18?\nA:", "answer": "The maximum flow from node 14 to node 18 is 47.", "difficulty": "hard", "doc_id": "274"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 2 with capacity 10,\nan edge from node 0 to node 4 with capacity 2,\nan edge from node 1 to node 5 with capacity 4,\nan edge from node 1 to node 3 with capacity 1,\nan edge from node 3 to node 2 with capacity 4,\nan edge from node 3 to node 4 with capacity 9,\nan edge from node 4 to node 2 with capacity 3,\nan edge from node 4 to node 3 with capacity 5,\nan edge from node 5 to node 3 with capacity 9.\nQ: What is the maximum flow from node 1 to node 2?\nA:", "answer": "The maximum flow from node 1 to node 2 is 5.", "difficulty": "easy", "doc_id": "275"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 5 with capacity 7,\nan edge from node 0 to node 1 with capacity 3,\nan edge from node 1 to node 5 with capacity 5,\nan edge from node 1 to node 2 with capacity 10,\nan edge from node 1 to node 4 with capacity 1,\nan edge from node 1 to node 3 with capacity 5,\nan edge from node 2 to node 0 with capacity 1,\nan edge from node 3 to node 1 with capacity 5,\nan edge from node 4 to node 1 with capacity 6,\nan edge from node 5 to node 2 with capacity 3,\nan edge from node 5 to node 4 with capacity 6.\nQ: What is the maximum flow from node 2 to node 4?\nA:", "answer": "The maximum flow from node 2 to node 4 is 1.", "difficulty": "easy", "doc_id": "276"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 11 with capacity 17,\nan edge from node 0 to node 1 with capacity 8,\nan edge from node 0 to node 14 with capacity 12,\nan edge from node 1 to node 3 with capacity 11,\nan edge from node 1 to node 14 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 1 to node 9 with capacity 9,\nan edge from node 2 to node 1 with capacity 16,\nan edge from node 2 to node 3 with capacity 20,\nan edge from node 2 to node 14 with capacity 18,\nan edge from node 2 to node 4 with capacity 3,\nan edge from node 2 to node 8 with capacity 18,\nan edge from node 2 to node 9 with capacity 7,\nan edge from node 2 to node 10 with capacity 13,\nan edge from node 3 to node 12 with capacity 14,\nan edge from node 3 to node 1 with capacity 17,\nan edge from node 3 to node 14 with capacity 8,\nan edge from node 3 to node 7 with capacity 13,\nan edge from node 3 to node 6 with capacity 18,\nan edge from node 3 to node 13 with capacity 4,\nan edge from node 4 to node 0 with capacity 19,\nan edge from node 4 to node 1 with capacity 12,\nan edge from node 4 to node 6 with capacity 14,\nan edge from node 4 to node 5 with capacity 17,\nan edge from node 4 to node 13 with capacity 14,\nan edge from node 5 to node 15 with capacity 7,\nan edge from node 5 to node 8 with capacity 5,\nan edge from node 5 to node 13 with capacity 3,\nan edge from node 6 to node 14 with capacity 7,\nan edge from node 7 to node 1 with capacity 12,\nan edge from node 7 to node 16 with capacity 9,\nan edge from node 7 to node 13 with capacity 2,\nan edge from node 8 to node 11 with capacity 8,\nan edge from node 8 to node 4 with capacity 18,\nan edge from node 8 to node 10 with capacity 1,\nan edge from node 8 to node 16 with capacity 7,\nan edge from node 9 to node 11 with capacity 8,\nan edge from node 9 to node 1 with capacity 17,\nan edge from node 9 to node 4 with capacity 17,\nan edge from node 9 to node 6 with capacity 15,\nan edge from node 9 to node 5 with capacity 17,\nan edge from node 9 to node 13 with capacity 1,\nan edge from node 10 to node 0 with capacity 13,\nan edge from node 10 to node 2 with capacity 17,\nan edge from node 10 to node 7 with capacity 17,\nan edge from node 10 to node 6 with capacity 6,\nan edge from node 11 to node 3 with capacity 1,\nan edge from node 11 to node 4 with capacity 2,\nan edge from node 11 to node 8 with capacity 12,\nan edge from node 11 to node 10 with capacity 17,\nan edge from node 12 to node 11 with capacity 2,\nan edge from node 12 to node 15 with capacity 15,\nan edge from node 12 to node 2 with capacity 8,\nan edge from node 12 to node 14 with capacity 5,\nan edge from node 12 to node 4 with capacity 7,\nan edge from node 12 to node 9 with capacity 2,\nan edge from node 13 to node 15 with capacity 16,\nan edge from node 13 to node 14 with capacity 14,\nan edge from node 13 to node 8 with capacity 5,\nan edge from node 14 to node 2 with capacity 6,\nan edge from node 14 to node 3 with capacity 18,\nan edge from node 14 to node 6 with capacity 13,\nan edge from node 14 to node 9 with capacity 19,\nan edge from node 14 to node 16 with capacity 20,\nan edge from node 15 to node 12 with capacity 9,\nan edge from node 15 to node 1 with capacity 2,\nan edge from node 15 to node 14 with capacity 1,\nan edge from node 15 to node 10 with capacity 14,\nan edge from node 15 to node 13 with capacity 10,\nan edge from node 16 to node 0 with capacity 15,\nan edge from node 16 to node 7 with capacity 12,\nan edge from node 16 to node 6 with capacity 16,\nan edge from node 16 to node 8 with capacity 6.\nQ: What is the maximum flow from node 4 to node 13?\nA:", "answer": "The maximum flow from node 4 to node 13 is 34.", "difficulty": "hard", "doc_id": "277"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 13, and the edges are:\nan edge from node 0 to node 11 with capacity 12,\nan edge from node 0 to node 3 with capacity 9,\nan edge from node 0 to node 9 with capacity 2,\nan edge from node 1 to node 10 with capacity 4,\nan edge from node 1 to node 5 with capacity 3,\nan edge from node 1 to node 4 with capacity 8,\nan edge from node 2 to node 11 with capacity 3,\nan edge from node 2 to node 1 with capacity 1,\nan edge from node 2 to node 13 with capacity 20,\nan edge from node 2 to node 7 with capacity 14,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 2 with capacity 7,\nan edge from node 3 to node 10 with capacity 9,\nan edge from node 3 to node 0 with capacity 6,\nan edge from node 3 to node 13 with capacity 2,\nan edge from node 3 to node 7 with capacity 8,\nan edge from node 3 to node 12 with capacity 4,\nan edge from node 3 to node 9 with capacity 20,\nan edge from node 4 to node 11 with capacity 11,\nan edge from node 4 to node 10 with capacity 1,\nan edge from node 4 to node 12 with capacity 8,\nan edge from node 4 to node 3 with capacity 16,\nan edge from node 5 to node 2 with capacity 11,\nan edge from node 5 to node 10 with capacity 20,\nan edge from node 5 to node 8 with capacity 12,\nan edge from node 6 to node 1 with capacity 19,\nan edge from node 6 to node 13 with capacity 4,\nan edge from node 7 to node 11 with capacity 3,\nan edge from node 7 to node 12 with capacity 2,\nan edge from node 8 to node 1 with capacity 9,\nan edge from node 8 to node 5 with capacity 18,\nan edge from node 8 to node 12 with capacity 17,\nan edge from node 8 to node 3 with capacity 3,\nan edge from node 8 to node 9 with capacity 11,\nan edge from node 9 to node 7 with capacity 18,\nan edge from node 9 to node 3 with capacity 10,\nan edge from node 10 to node 2 with capacity 1,\nan edge from node 10 to node 3 with capacity 4,\nan edge from node 10 to node 9 with capacity 8,\nan edge from node 11 to node 13 with capacity 20,\nan edge from node 11 to node 4 with capacity 6,\nan edge from node 12 to node 9 with capacity 5,\nan edge from node 13 to node 11 with capacity 10,\nan edge from node 13 to node 2 with capacity 4,\nan edge from node 13 to node 0 with capacity 10,\nan edge from node 13 to node 8 with capacity 7,\nan edge from node 13 to node 12 with capacity 15,\nan edge from node 13 to node 3 with capacity 6,\nan edge from node 13 to node 4 with capacity 16.\nQ: What is the maximum flow from node 8 to node 5?\nA:", "answer": "The maximum flow from node 8 to node 5 is 21.", "difficulty": "hard", "doc_id": "278"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 8, and the edges are:\nan edge from node 0 to node 3 with capacity 6,\nan edge from node 0 to node 8 with capacity 9,\nan edge from node 1 to node 8 with capacity 5,\nan edge from node 2 to node 5 with capacity 1,\nan edge from node 2 to node 8 with capacity 6,\nan edge from node 3 to node 6 with capacity 7,\nan edge from node 3 to node 7 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 5,\nan edge from node 4 to node 8 with capacity 8,\nan edge from node 4 to node 0 with capacity 10,\nan edge from node 5 to node 3 with capacity 3,\nan edge from node 5 to node 1 with capacity 3,\nan edge from node 5 to node 0 with capacity 10,\nan edge from node 7 to node 2 with capacity 10,\nan edge from node 8 to node 0 with capacity 3.\nQ: What is the maximum flow from node 2 to node 8?\nA:", "answer": "The maximum flow from node 2 to node 8 is 7.", "difficulty": "easy", "doc_id": "279"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 0 to node 1 with capacity 1,\nan edge from node 0 to node 7 with capacity 3,\nan edge from node 1 to node 3 with capacity 10,\nan edge from node 1 to node 5 with capacity 4,\nan edge from node 2 to node 6 with capacity 4,\nan edge from node 3 to node 4 with capacity 10,\nan edge from node 3 to node 7 with capacity 3,\nan edge from node 4 to node 5 with capacity 6,\nan edge from node 4 to node 1 with capacity 5,\nan edge from node 4 to node 7 with capacity 2,\nan edge from node 5 to node 2 with capacity 5,\nan edge from node 6 to node 5 with capacity 2.\nQ: What is the maximum flow from node 0 to node 7?\nA:", "answer": "The maximum flow from node 0 to node 7 is 4.", "difficulty": "easy", "doc_id": "280"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 1 with capacity 9,\nan edge from node 1 to node 7 with capacity 6,\nan edge from node 1 to node 10 with capacity 1,\nan edge from node 1 to node 17 with capacity 13,\nan edge from node 2 to node 3 with capacity 11,\nan edge from node 2 to node 8 with capacity 7,\nan edge from node 3 to node 5 with capacity 9,\nan edge from node 3 to node 12 with capacity 19,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 3 to node 9 with capacity 4,\nan edge from node 4 to node 18 with capacity 3,\nan edge from node 4 to node 6 with capacity 9,\nan edge from node 4 to node 12 with capacity 7,\nan edge from node 4 to node 8 with capacity 10,\nan edge from node 4 to node 17 with capacity 8,\nan edge from node 5 to node 19 with capacity 6,\nan edge from node 5 to node 3 with capacity 15,\nan edge from node 5 to node 4 with capacity 11,\nan edge from node 5 to node 17 with capacity 17,\nan edge from node 6 to node 5 with capacity 1,\nan edge from node 6 to node 8 with capacity 7,\nan edge from node 7 to node 1 with capacity 14,\nan edge from node 7 to node 14 with capacity 20,\nan edge from node 7 to node 5 with capacity 7,\nan edge from node 7 to node 16 with capacity 16,\nan edge from node 8 to node 15 with capacity 6,\nan edge from node 8 to node 18 with capacity 1,\nan edge from node 8 to node 11 with capacity 9,\nan edge from node 8 to node 2 with capacity 15,\nan edge from node 8 to node 9 with capacity 8,\nan edge from node 8 to node 17 with capacity 6,\nan edge from node 9 to node 14 with capacity 14,\nan edge from node 9 to node 6 with capacity 16,\nan edge from node 9 to node 0 with capacity 6,\nan edge from node 9 to node 7 with capacity 14,\nan edge from node 10 to node 1 with capacity 11,\nan edge from node 10 to node 5 with capacity 14,\nan edge from node 10 to node 19 with capacity 13,\nan edge from node 10 to node 11 with capacity 14,\nan edge from node 10 to node 3 with capacity 6,\nan edge from node 10 to node 16 with capacity 17,\nan edge from node 10 to node 4 with capacity 18,\nan edge from node 10 to node 8 with capacity 20,\nan edge from node 11 to node 13 with capacity 3,\nan edge from node 11 to node 16 with capacity 9,\nan edge from node 11 to node 7 with capacity 9,\nan edge from node 11 to node 17 with capacity 13,\nan edge from node 12 to node 1 with capacity 2,\nan edge from node 12 to node 14 with capacity 16,\nan edge from node 12 to node 15 with capacity 13,\nan edge from node 12 to node 19 with capacity 15,\nan edge from node 12 to node 18 with capacity 4,\nan edge from node 12 to node 6 with capacity 14,\nan edge from node 12 to node 9 with capacity 9,\nan edge from node 13 to node 5 with capacity 17,\nan edge from node 13 to node 6 with capacity 6,\nan edge from node 13 to node 12 with capacity 15,\nan edge from node 13 to node 3 with capacity 2,\nan edge from node 13 to node 4 with capacity 10,\nan edge from node 13 to node 2 with capacity 19,\nan edge from node 13 to node 0 with capacity 10,\nan edge from node 14 to node 6 with capacity 9,\nan edge from node 14 to node 12 with capacity 17,\nan edge from node 14 to node 3 with capacity 12,\nan edge from node 14 to node 0 with capacity 4,\nan edge from node 15 to node 1 with capacity 11,\nan edge from node 15 to node 14 with capacity 10,\nan edge from node 15 to node 19 with capacity 7,\nan edge from node 15 to node 13 with capacity 10,\nan edge from node 15 to node 3 with capacity 13,\nan edge from node 15 to node 9 with capacity 18,\nan edge from node 16 to node 15 with capacity 11,\nan edge from node 16 to node 11 with capacity 11,\nan edge from node 16 to node 2 with capacity 15,\nan edge from node 16 to node 0 with capacity 14,\nan edge from node 16 to node 17 with capacity 17,\nan edge from node 17 to node 18 with capacity 12,\nan edge from node 17 to node 6 with capacity 16,\nan edge from node 17 to node 13 with capacity 19,\nan edge from node 17 to node 8 with capacity 5,\nan edge from node 17 to node 0 with capacity 11,\nan edge from node 17 to node 7 with capacity 14,\nan edge from node 18 to node 14 with capacity 16,\nan edge from node 18 to node 11 with capacity 5,\nan edge from node 18 to node 16 with capacity 11,\nan edge from node 18 to node 0 with capacity 11,\nan edge from node 18 to node 10 with capacity 19,\nan edge from node 19 to node 1 with capacity 12,\nan edge from node 19 to node 14 with capacity 10,\nan edge from node 19 to node 12 with capacity 19,\nan edge from node 19 to node 3 with capacity 2,\nan edge from node 19 to node 2 with capacity 6,\nan edge from node 19 to node 8 with capacity 10,\nan edge from node 19 to node 10 with capacity 14,\nan edge from node 19 to node 9 with capacity 10.\nQ: What is the maximum flow from node 17 to node 14?\nA:", "answer": "The maximum flow from node 17 to node 14 is 65.", "difficulty": "hard", "doc_id": "281"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 9, and the edges are:\nan edge from node 0 to node 8 with capacity 9,\nan edge from node 1 to node 8 with capacity 3,\nan edge from node 1 to node 4 with capacity 4,\nan edge from node 2 to node 6 with capacity 9,\nan edge from node 2 to node 8 with capacity 8,\nan edge from node 2 to node 7 with capacity 2,\nan edge from node 3 to node 5 with capacity 2,\nan edge from node 3 to node 0 with capacity 6,\nan edge from node 3 to node 1 with capacity 3,\nan edge from node 4 to node 0 with capacity 7,\nan edge from node 5 to node 2 with capacity 7,\nan edge from node 5 to node 0 with capacity 9,\nan edge from node 6 to node 3 with capacity 6,\nan edge from node 7 to node 1 with capacity 2,\nan edge from node 8 to node 2 with capacity 1,\nan edge from node 9 to node 5 with capacity 3,\nan edge from node 9 to node 4 with capacity 5,\nan edge from node 9 to node 3 with capacity 4,\nan edge from node 9 to node 7 with capacity 5.\nQ: What is the maximum flow from node 2 to node 0?\nA:", "answer": "The maximum flow from node 2 to node 0 is 8.", "difficulty": "easy", "doc_id": "282"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 3 with capacity 15,\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 0 to node 8 with capacity 4,\nan edge from node 0 to node 11 with capacity 17,\nan edge from node 0 to node 12 with capacity 1,\nan edge from node 1 to node 3 with capacity 16,\nan edge from node 1 to node 14 with capacity 14,\nan edge from node 1 to node 11 with capacity 5,\nan edge from node 2 to node 14 with capacity 14,\nan edge from node 2 to node 12 with capacity 8,\nan edge from node 2 to node 7 with capacity 1,\nan edge from node 3 to node 6 with capacity 16,\nan edge from node 3 to node 1 with capacity 20,\nan edge from node 3 to node 0 with capacity 9,\nan edge from node 3 to node 13 with capacity 15,\nan edge from node 4 to node 1 with capacity 9,\nan edge from node 4 to node 9 with capacity 13,\nan edge from node 4 to node 15 with capacity 2,\nan edge from node 4 to node 14 with capacity 20,\nan edge from node 4 to node 11 with capacity 5,\nan edge from node 4 to node 2 with capacity 17,\nan edge from node 5 to node 6 with capacity 20,\nan edge from node 5 to node 1 with capacity 20,\nan edge from node 5 to node 15 with capacity 19,\nan edge from node 5 to node 3 with capacity 16,\nan edge from node 5 to node 10 with capacity 17,\nan edge from node 6 to node 15 with capacity 11,\nan edge from node 6 to node 3 with capacity 9,\nan edge from node 6 to node 13 with capacity 1,\nan edge from node 7 to node 4 with capacity 11,\nan edge from node 7 to node 14 with capacity 12,\nan edge from node 7 to node 11 with capacity 3,\nan edge from node 7 to node 12 with capacity 18,\nan edge from node 7 to node 0 with capacity 5,\nan edge from node 7 to node 13 with capacity 3,\nan edge from node 8 to node 15 with capacity 10,\nan edge from node 8 to node 14 with capacity 2,\nan edge from node 9 to node 3 with capacity 2,\nan edge from node 9 to node 14 with capacity 17,\nan edge from node 9 to node 12 with capacity 15,\nan edge from node 9 to node 2 with capacity 5,\nan edge from node 10 to node 6 with capacity 2,\nan edge from node 10 to node 9 with capacity 9,\nan edge from node 10 to node 14 with capacity 4,\nan edge from node 10 to node 11 with capacity 18,\nan edge from node 10 to node 12 with capacity 11,\nan edge from node 10 to node 7 with capacity 3,\nan edge from node 11 to node 9 with capacity 15,\nan edge from node 11 to node 12 with capacity 3,\nan edge from node 12 to node 6 with capacity 6,\nan edge from node 12 to node 0 with capacity 14,\nan edge from node 12 to node 10 with capacity 11,\nan edge from node 13 to node 1 with capacity 11,\nan edge from node 13 to node 9 with capacity 20,\nan edge from node 13 to node 8 with capacity 19,\nan edge from node 13 to node 0 with capacity 14,\nan edge from node 13 to node 2 with capacity 15,\nan edge from node 14 to node 6 with capacity 20,\nan edge from node 14 to node 1 with capacity 4,\nan edge from node 14 to node 12 with capacity 1,\nan edge from node 15 to node 9 with capacity 17,\nan edge from node 15 to node 11 with capacity 1,\nan edge from node 15 to node 12 with capacity 9.\nQ: What is the maximum flow from node 14 to node 12?\nA:", "answer": "The maximum flow from node 14 to node 12 is 25.", "difficulty": "hard", "doc_id": "283"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 15, and the edges are:\nan edge from node 0 to node 8 with capacity 10,\nan edge from node 0 to node 3 with capacity 9,\nan edge from node 0 to node 13 with capacity 18,\nan edge from node 0 to node 14 with capacity 15,\nan edge from node 1 to node 8 with capacity 4,\nan edge from node 1 to node 12 with capacity 3,\nan edge from node 1 to node 7 with capacity 4,\nan edge from node 2 to node 8 with capacity 20,\nan edge from node 2 to node 4 with capacity 1,\nan edge from node 3 to node 9 with capacity 9,\nan edge from node 3 to node 15 with capacity 9,\nan edge from node 3 to node 12 with capacity 15,\nan edge from node 3 to node 7 with capacity 7,\nan edge from node 3 to node 4 with capacity 13,\nan edge from node 4 to node 0 with capacity 11,\nan edge from node 4 to node 10 with capacity 15,\nan edge from node 5 to node 0 with capacity 20,\nan edge from node 5 to node 15 with capacity 7,\nan edge from node 5 to node 12 with capacity 11,\nan edge from node 5 to node 6 with capacity 12,\nan edge from node 5 to node 11 with capacity 18,\nan edge from node 6 to node 2 with capacity 5,\nan edge from node 6 to node 1 with capacity 15,\nan edge from node 6 to node 12 with capacity 9,\nan edge from node 6 to node 7 with capacity 11,\nan edge from node 7 to node 10 with capacity 2,\nan edge from node 7 to node 12 with capacity 15,\nan edge from node 7 to node 13 with capacity 19,\nan edge from node 7 to node 6 with capacity 17,\nan edge from node 7 to node 4 with capacity 16,\nan edge from node 8 to node 9 with capacity 4,\nan edge from node 9 to node 3 with capacity 2,\nan edge from node 9 to node 15 with capacity 15,\nan edge from node 9 to node 10 with capacity 4,\nan edge from node 9 to node 13 with capacity 19,\nan edge from node 9 to node 11 with capacity 19,\nan edge from node 9 to node 14 with capacity 17,\nan edge from node 10 to node 11 with capacity 5,\nan edge from node 11 to node 5 with capacity 8,\nan edge from node 11 to node 6 with capacity 16,\nan edge from node 11 to node 4 with capacity 18,\nan edge from node 12 to node 9 with capacity 13,\nan edge from node 12 to node 2 with capacity 3,\nan edge from node 12 to node 15 with capacity 8,\nan edge from node 12 to node 1 with capacity 14,\nan edge from node 12 to node 4 with capacity 9,\nan edge from node 12 to node 11 with capacity 5,\nan edge from node 12 to node 14 with capacity 18,\nan edge from node 13 to node 0 with capacity 9,\nan edge from node 13 to node 10 with capacity 16,\nan edge from node 13 to node 14 with capacity 17,\nan edge from node 14 to node 3 with capacity 5,\nan edge from node 14 to node 15 with capacity 5,\nan edge from node 14 to node 5 with capacity 18,\nan edge from node 14 to node 6 with capacity 18,\nan edge from node 15 to node 5 with capacity 13.\nQ: What is the maximum flow from node 2 to node 1?\nA:", "answer": "The maximum flow from node 2 to node 1 is 5.", "difficulty": "hard", "doc_id": "284"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 7, and the edges are:\nan edge from node 1 to node 6 with capacity 10,\nan edge from node 3 to node 0 with capacity 6,\nan edge from node 4 to node 0 with capacity 6,\nan edge from node 5 to node 6 with capacity 5,\nan edge from node 5 to node 1 with capacity 1,\nan edge from node 6 to node 1 with capacity 2,\nan edge from node 7 to node 5 with capacity 7,\nan edge from node 7 to node 4 with capacity 5,\nan edge from node 7 to node 1 with capacity 10,\nan edge from node 7 to node 0 with capacity 1.\nQ: What is the maximum flow from node 7 to node 1?\nA:", "answer": "The maximum flow from node 7 to node 1 is 13.", "difficulty": "easy", "doc_id": "285"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 12, and the edges are:\nan edge from node 0 to node 2 with capacity 7,\nan edge from node 0 to node 11 with capacity 20,\nan edge from node 0 to node 12 with capacity 2,\nan edge from node 1 to node 5 with capacity 4,\nan edge from node 1 to node 6 with capacity 9,\nan edge from node 1 to node 8 with capacity 5,\nan edge from node 2 to node 4 with capacity 8,\nan edge from node 2 to node 9 with capacity 7,\nan edge from node 3 to node 7 with capacity 11,\nan edge from node 3 to node 6 with capacity 8,\nan edge from node 3 to node 8 with capacity 1,\nan edge from node 4 to node 7 with capacity 20,\nan edge from node 4 to node 8 with capacity 16,\nan edge from node 5 to node 2 with capacity 19,\nan edge from node 6 to node 7 with capacity 13,\nan edge from node 6 to node 4 with capacity 3,\nan edge from node 6 to node 0 with capacity 3,\nan edge from node 6 to node 12 with capacity 15,\nan edge from node 7 to node 11 with capacity 3,\nan edge from node 7 to node 10 with capacity 2,\nan edge from node 7 to node 0 with capacity 4,\nan edge from node 8 to node 1 with capacity 4,\nan edge from node 8 to node 2 with capacity 4,\nan edge from node 8 to node 11 with capacity 18,\nan edge from node 8 to node 9 with capacity 5,\nan edge from node 9 to node 2 with capacity 3,\nan edge from node 9 to node 3 with capacity 7,\nan edge from node 11 to node 2 with capacity 4,\nan edge from node 11 to node 10 with capacity 8,\nan edge from node 11 to node 4 with capacity 7,\nan edge from node 11 to node 6 with capacity 2,\nan edge from node 11 to node 8 with capacity 1,\nan edge from node 12 to node 10 with capacity 8,\nan edge from node 12 to node 7 with capacity 16,\nan edge from node 12 to node 4 with capacity 7,\nan edge from node 12 to node 8 with capacity 14.\nQ: What is the maximum flow from node 11 to node 5?\nA:", "answer": "The maximum flow from node 11 to node 5 is 4.", "difficulty": "hard", "doc_id": "286"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 16, and the edges are:\nan edge from node 0 to node 11 with capacity 10,\nan edge from node 0 to node 5 with capacity 2,\nan edge from node 0 to node 10 with capacity 3,\nan edge from node 0 to node 9 with capacity 11,\nan edge from node 1 to node 4 with capacity 18,\nan edge from node 1 to node 10 with capacity 18,\nan edge from node 1 to node 12 with capacity 5,\nan edge from node 1 to node 9 with capacity 14,\nan edge from node 1 to node 0 with capacity 16,\nan edge from node 2 to node 5 with capacity 18,\nan edge from node 2 to node 10 with capacity 19,\nan edge from node 3 to node 4 with capacity 17,\nan edge from node 3 to node 15 with capacity 14,\nan edge from node 3 to node 13 with capacity 13,\nan edge from node 3 to node 12 with capacity 16,\nan edge from node 3 to node 9 with capacity 2,\nan edge from node 4 to node 11 with capacity 6,\nan edge from node 4 to node 5 with capacity 12,\nan edge from node 4 to node 15 with capacity 19,\nan edge from node 4 to node 10 with capacity 2,\nan edge from node 4 to node 12 with capacity 13,\nan edge from node 4 to node 6 with capacity 3,\nan edge from node 4 to node 0 with capacity 16,\nan edge from node 5 to node 11 with capacity 15,\nan edge from node 5 to node 4 with capacity 2,\nan edge from node 5 to node 13 with capacity 5,\nan edge from node 5 to node 16 with capacity 5,\nan edge from node 6 to node 15 with capacity 16,\nan edge from node 6 to node 14 with capacity 14,\nan edge from node 7 to node 13 with capacity 10,\nan edge from node 8 to node 7 with capacity 13,\nan edge from node 8 to node 3 with capacity 2,\nan edge from node 8 to node 1 with capacity 4,\nan edge from node 9 to node 4 with capacity 18,\nan edge from node 9 to node 10 with capacity 19,\nan edge from node 9 to node 8 with capacity 18,\nan edge from node 10 to node 3 with capacity 10,\nan edge from node 10 to node 4 with capacity 1,\nan edge from node 10 to node 15 with capacity 16,\nan edge from node 10 to node 14 with capacity 6,\nan edge from node 10 to node 12 with capacity 7,\nan edge from node 11 to node 2 with capacity 13,\nan edge from node 11 to node 3 with capacity 1,\nan edge from node 11 to node 15 with capacity 14,\nan edge from node 12 to node 2 with capacity 5,\nan edge from node 12 to node 8 with capacity 7,\nan edge from node 12 to node 6 with capacity 14,\nan edge from node 12 to node 1 with capacity 1,\nan edge from node 13 to node 11 with capacity 15,\nan edge from node 13 to node 2 with capacity 12,\nan edge from node 13 to node 16 with capacity 5,\nan edge from node 13 to node 8 with capacity 10,\nan edge from node 13 to node 6 with capacity 2,\nan edge from node 13 to node 1 with capacity 20,\nan edge from node 14 to node 11 with capacity 10,\nan edge from node 14 to node 2 with capacity 8,\nan edge from node 14 to node 5 with capacity 4,\nan edge from node 14 to node 16 with capacity 20,\nan edge from node 14 to node 12 with capacity 14,\nan edge from node 14 to node 9 with capacity 15,\nan edge from node 14 to node 1 with capacity 17,\nan edge from node 15 to node 5 with capacity 16,\nan edge from node 15 to node 3 with capacity 3,\nan edge from node 15 to node 4 with capacity 4,\nan edge from node 15 to node 13 with capacity 16,\nan edge from node 15 to node 9 with capacity 1,\nan edge from node 16 to node 7 with capacity 12,\nan edge from node 16 to node 11 with capacity 15,\nan edge from node 16 to node 2 with capacity 19,\nan edge from node 16 to node 3 with capacity 3,\nan edge from node 16 to node 15 with capacity 3,\nan edge from node 16 to node 9 with capacity 15.\nQ: What is the maximum flow from node 13 to node 6?\nA:", "answer": "The maximum flow from node 13 to node 6 is 19.", "difficulty": "hard", "doc_id": "287"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 19, and the edges are:\nan edge from node 0 to node 13 with capacity 3,\nan edge from node 0 to node 16 with capacity 1,\nan edge from node 0 to node 14 with capacity 9,\nan edge from node 1 to node 10 with capacity 17,\nan edge from node 1 to node 15 with capacity 11,\nan edge from node 2 to node 0 with capacity 19,\nan edge from node 2 to node 12 with capacity 12,\nan edge from node 2 to node 4 with capacity 9,\nan edge from node 2 to node 8 with capacity 17,\nan edge from node 2 to node 17 with capacity 8,\nan edge from node 3 to node 9 with capacity 15,\nan edge from node 3 to node 12 with capacity 6,\nan edge from node 3 to node 14 with capacity 6,\nan edge from node 3 to node 17 with capacity 3,\nan edge from node 4 to node 8 with capacity 17,\nan edge from node 4 to node 6 with capacity 15,\nan edge from node 4 to node 11 with capacity 17,\nan edge from node 4 to node 17 with capacity 16,\nan edge from node 5 to node 10 with capacity 12,\nan edge from node 5 to node 3 with capacity 20,\nan edge from node 5 to node 14 with capacity 15,\nan edge from node 5 to node 7 with capacity 7,\nan edge from node 5 to node 2 with capacity 9,\nan edge from node 6 to node 13 with capacity 1,\nan edge from node 6 to node 16 with capacity 6,\nan edge from node 6 to node 14 with capacity 10,\nan edge from node 6 to node 18 with capacity 19,\nan edge from node 6 to node 2 with capacity 1,\nan edge from node 6 to node 1 with capacity 15,\nan edge from node 6 to node 5 with capacity 9,\nan edge from node 7 to node 16 with capacity 13,\nan edge from node 7 to node 8 with capacity 9,\nan edge from node 7 to node 11 with capacity 15,\nan edge from node 7 to node 17 with capacity 18,\nan edge from node 8 to node 13 with capacity 16,\nan edge from node 8 to node 12 with capacity 5,\nan edge from node 8 to node 15 with capacity 6,\nan edge from node 8 to node 7 with capacity 18,\nan edge from node 8 to node 1 with capacity 6,\nan edge from node 8 to node 11 with capacity 16,\nan edge from node 8 to node 17 with capacity 15,\nan edge from node 9 to node 15 with capacity 10,\nan edge from node 9 to node 8 with capacity 13,\nan edge from node 9 to node 5 with capacity 3,\nan edge from node 10 to node 3 with capacity 9,\nan edge from node 10 to node 14 with capacity 16,\nan edge from node 10 to node 4 with capacity 1,\nan edge from node 10 to node 6 with capacity 5,\nan edge from node 10 to node 17 with capacity 19,\nan edge from node 11 to node 9 with capacity 20,\nan edge from node 11 to node 19 with capacity 3,\nan edge from node 11 to node 13 with capacity 19,\nan edge from node 12 to node 16 with capacity 14,\nan edge from node 12 to node 6 with capacity 8,\nan edge from node 13 to node 9 with capacity 13,\nan edge from node 13 to node 12 with capacity 6,\nan edge from node 13 to node 18 with capacity 11,\nan edge from node 13 to node 7 with capacity 14,\nan edge from node 13 to node 6 with capacity 3,\nan edge from node 14 to node 3 with capacity 7,\nan edge from node 14 to node 4 with capacity 3,\nan edge from node 14 to node 6 with capacity 11,\nan edge from node 15 to node 19 with capacity 5,\nan edge from node 15 to node 16 with capacity 4,\nan edge from node 15 to node 14 with capacity 3,\nan edge from node 15 to node 4 with capacity 12,\nan edge from node 15 to node 7 with capacity 20,\nan edge from node 15 to node 6 with capacity 1,\nan edge from node 15 to node 1 with capacity 2,\nan edge from node 15 to node 11 with capacity 12,\nan edge from node 16 to node 14 with capacity 20,\nan edge from node 16 to node 15 with capacity 4,\nan edge from node 16 to node 11 with capacity 8,\nan edge from node 17 to node 13 with capacity 11,\nan edge from node 17 to node 16 with capacity 3,\nan edge from node 17 to node 4 with capacity 7,\nan edge from node 17 to node 8 with capacity 10,\nan edge from node 17 to node 6 with capacity 5,\nan edge from node 17 to node 5 with capacity 7,\nan edge from node 18 to node 10 with capacity 5,\nan edge from node 18 to node 9 with capacity 9,\nan edge from node 18 to node 13 with capacity 1,\nan edge from node 18 to node 12 with capacity 12,\nan edge from node 18 to node 16 with capacity 10,\nan edge from node 18 to node 4 with capacity 12,\nan edge from node 18 to node 8 with capacity 2,\nan edge from node 18 to node 2 with capacity 4,\nan edge from node 19 to node 9 with capacity 8,\nan edge from node 19 to node 12 with capacity 17,\nan edge from node 19 to node 15 with capacity 18,\nan edge from node 19 to node 4 with capacity 6,\nan edge from node 19 to node 6 with capacity 5,\nan edge from node 19 to node 17 with capacity 13.\nQ: What is the maximum flow from node 0 to node 8?\nA:", "answer": "The maximum flow from node 0 to node 8 is 13.", "difficulty": "hard", "doc_id": "288"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 10, and the edges are:\nan edge from node 0 to node 5 with capacity 19,\nan edge from node 0 to node 1 with capacity 13,\nan edge from node 1 to node 2 with capacity 8,\nan edge from node 1 to node 8 with capacity 13,\nan edge from node 1 to node 7 with capacity 13,\nan edge from node 1 to node 9 with capacity 9,\nan edge from node 2 to node 3 with capacity 2,\nan edge from node 2 to node 9 with capacity 4,\nan edge from node 2 to node 6 with capacity 8,\nan edge from node 3 to node 6 with capacity 13,\nan edge from node 4 to node 2 with capacity 10,\nan edge from node 4 to node 3 with capacity 16,\nan edge from node 4 to node 8 with capacity 8,\nan edge from node 4 to node 7 with capacity 4,\nan edge from node 5 to node 2 with capacity 17,\nan edge from node 5 to node 0 with capacity 12,\nan edge from node 5 to node 3 with capacity 4,\nan edge from node 5 to node 6 with capacity 15,\nan edge from node 7 to node 4 with capacity 5,\nan edge from node 7 to node 2 with capacity 14,\nan edge from node 7 to node 10 with capacity 7,\nan edge from node 7 to node 0 with capacity 13,\nan edge from node 7 to node 3 with capacity 12,\nan edge from node 7 to node 1 with capacity 6,\nan edge from node 7 to node 6 with capacity 4,\nan edge from node 8 to node 5 with capacity 13,\nan edge from node 8 to node 10 with capacity 16,\nan edge from node 8 to node 6 with capacity 9,\nan edge from node 9 to node 4 with capacity 20,\nan edge from node 9 to node 2 with capacity 15,\nan edge from node 9 to node 8 with capacity 18,\nan edge from node 9 to node 7 with capacity 16,\nan edge from node 10 to node 5 with capacity 17.\nQ: What is the maximum flow from node 4 to node 3?\nA:", "answer": "The maximum flow from node 4 to node 3 is 34.", "difficulty": "hard", "doc_id": "289"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 0 to node 4 with capacity 5,\nan edge from node 1 to node 0 with capacity 9,\nan edge from node 1 to node 6 with capacity 8,\nan edge from node 2 to node 4 with capacity 2,\nan edge from node 4 to node 1 with capacity 9,\nan edge from node 4 to node 2 with capacity 2,\nan edge from node 5 to node 1 with capacity 2,\nan edge from node 5 to node 2 with capacity 10,\nan edge from node 5 to node 6 with capacity 6,\nan edge from node 6 to node 3 with capacity 6.\nQ: What is the maximum flow from node 5 to node 6?\nA:", "answer": "The maximum flow from node 5 to node 6 is 10.", "difficulty": "easy", "doc_id": "290"}
{"question": "In a directed graph, the nodes are numbered from 0 to 3, and the edges are:\nan edge from node 1 to node 0 with capacity 10,\nan edge from node 0 to node 2 with capacity 6,\nan edge from node 2 to node 3 with capacity 4.\nQ: What is the maximum flow from node 1 to node 3?\nA: From the source (node 1), we can send 10 units of flow to node 0.\n10 units of flow arrive at node 0, then we can send 6 units of flow from node 0 to node 2.\n6 units of flow arrive at node 0, then we can send 4 units of flow from node 2 to node 3.\nThe total unit of flow arrives at node 3 is 4. So the maximum flow from node 1 to node 3 is 4.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 3 with capacity 3,\nan edge from node 0 to node 4 with capacity 10,\nan edge from node 1 to node 3 with capacity 3,\nan edge from node 1 to node 4 with capacity 5,\nan edge from node 2 to node 1 with capacity 10,\nan edge from node 3 to node 1 with capacity 10,\nan edge from node 3 to node 4 with capacity 1,\nan edge from node 4 to node 3 with capacity 2.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 4, and 3 units of flow to node 3.\n3 units of flow arrive at node 3, then we can send 3 units of flow from node 3 to node 1.\n3 units of flow arrive at node 1, then we can send 3 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 0 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 4, and the edges are:\nan edge from node 0 to node 1 with capacity 10,\nan edge from node 1 to node 2 with capacity 6,\nan edge from node 0 to node 2 with capacity 2,\nan edge from node 1 to node 3 with capacity 6,\nan edge from node 3 to node 2 with capacity 3,\nan edge from node 2 to node 4 with capacity 5,\nan edge from node 3 to node 4 with capacity 8.\nQ: What is the maximum flow from node 0 to node 4?\nA: From the source (node 0), we can send 10 units of flow to node 1, and 2 units of flow to node 2.\n10 units of flow arrive at node 1, then we can send 6 units of flow from node 3 to node 1, and 4 units of flow from node 3 to node 2.\n6 units of flow arrive at node 2, then we can send 5 units of flow from node 2 to node 4.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 4.\nThe total unit of flow arrives at node 4 is 11. So the maximum flow from node 0 to node 4 is 11.\n\nIn a directed graph, the nodes are numbered from 0 to 5, and the edges are:\nan edge from node 0 to node 4 with capacity 8,\nan edge from node 2 to node 0 with capacity 2,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 4 to node 3 with capacity 10,\nan edge from node 4 to node 1 with capacity 2,\nan edge from node 4 to node 5 with capacity 2,\nan edge from node 5 to node 4 with capacity 7.\nQ: What is the maximum flow from node 2 to node 1?\nA: From the source (node 1), we can send 2 units of flow to node 0, and 6 units of flow to node 4.\n2 units of flow arrive at node 0, then we can send 2 units of flow from node 0 to node 4.\n8 units of flow arrive at node 4, then we can send 2 units of flow from node 4 to node 1, and 6 units of flow from node 4 to node 3.\n6 units of flow arrive at node 3, then we can send 6 units of flow from node 3 to node 1.\nThe total unit of flow arrives at node 4 is 8. So the maximum flow from node 2 to node 1 is 8.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 3 with capacity 1,\nan edge from node 1 to node 4 with capacity 7,\nan edge from node 2 to node 6 with capacity 1,\nan edge from node 2 to node 4 with capacity 6,\nan edge from node 2 to node 1 with capacity 6,\nan edge from node 3 to node 4 with capacity 4,\nan edge from node 3 to node 1 with capacity 7,\nan edge from node 5 to node 1 with capacity 9,\nan edge from node 6 to node 4 with capacity 10.\nQ: What is the maximum flow from node 2 to node 4?\nA: From the source (node 2), we can send 1 units of flow to node 0, 6 units of flow to node 1, and 6 units of flow to node 4.\n1 unit of flow arrive at node 0, then we can send 1 unit of flow from node 0 to node 4.\n6 units of flow arrive at node 1, then we can send 6 units of flow from node 1 to node 4.\nThe total unit of flow arrives at node 4 is 13. So the maximum flow from node 2 to node 4 is 13.\n\nIn a directed graph, the nodes are numbered from 0 to 6, and the edges are:\nan edge from node 0 to node 2 with capacity 7,\nan edge from node 0 to node 5 with capacity 4,\nan edge from node 1 to node 3 with capacity 4,\nan edge from node 1 to node 2 with capacity 8,\nan edge from node 3 to node 6 with capacity 1,\nan edge from node 5 to node 2 with capacity 1,\nan edge from node 5 to node 0 with capacity 9,\nan edge from node 5 to node 4 with capacity 8,\nan edge from node 6 to node 3 with capacity 9,\nan edge from node 6 to node 0 with capacity 1,\nan edge from node 6 to node 4 with capacity 5.\nQ: What is the maximum flow from node 1 to node 2?\nA:", "answer": "The maximum flow from node 1 to node 2 is 9.", "difficulty": "easy", "doc_id": "291"}