[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of an 8x8 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}

dataset: MATH


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of a 10x10 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of a 12x12 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of a 14x14 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of a 16x16 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of an 18x18 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of a 20x20 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of a 22x22 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of a 24x24 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of a 26x26 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of a 28x28 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have an agent in the bottom left (i=0, j=0) cell of a 30x30 grid, whose objective is to make it to the top right cell. Its valid moves from any cell are to move one cell to the right, or to move one cell upward, and it is further constrained by having to stay strictly below the main diagonal (i = j) at all points in its path except the start and end point. What is the minimum number of cells you can block to make it so that the agent only has one valid path through the grid? (It is not valid for the agent to move into a blocked cell.)

Present the answer in LaTex format: \boxed{Your answer}


[User]
In a 10x10 grid, how many different paths are there from the bottom left (0, 0) to the top right (9, 9), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (2, 0), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (2, 8)?

Present the answer in LaTex format: \boxed{Your answer}


[User]
In a 20x20 grid, how many different paths are there from the bottom left (0, 0) to the top right (19, 19), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (2, 0), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (2, 8), (2, 9), (2, 10), (2, 11), (2, 12), (2, 13), (2, 14), (2, 15), (2, 16), (2, 17), (2, 18)?

Present the answer in LaTex format: \boxed{Your answer}


[User]
In a 10x10 grid, how many different paths are there from the bottom left (0, 0) to the top right (9, 9), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 8)?

Present the answer in LaTex format: \boxed{Your answer}


[User]
In a 20x20 grid, how many different paths are there from the bottom left (0, 0) to the top right (19, 19), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 8), (9, 9), (10, 10), (11, 11), (12, 12), (13, 13), (14, 14), (15, 15), (16, 16), (17, 17), (18, 18)?

Present the answer in LaTex format: \boxed{Your answer}