[ { "id": 1, "question": "A squirrel is following the paths of labyrinth and collecting food for winter. Which stuff it will not be able to take?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image1.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry" ] } }, { "id": 2, "question": "Ingrid has 4 red, 3 blue, 2 green and 1 yellow cube. She uses them to build the following object:\n\nCubes with the same colour don't touch each other. Which colour is the cube with the question mark?\nChoices: A. red\nB. blue\nC. green\nD. Yellow\nE. This cannot be worked out for certain.", "answer": "A", "image_path": "image2.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 3, "question": "Which point in the labyrinth can we get to, starting at point $O$?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "C", "image_path": "image3.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 4, "question": "Lisa has several sheets of construction paper like this\n\nand\n\nShe wants to make 7 identical crowns:\n\nFor that she cuts out the necessary parts.\nWhat is the minimum number of sheets of construction paper that she has to cut up?", "answer": "9", "image_path": "image4.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 5, "question": "Simon has two identical tiles, whose front look like this: The back is white.\n\nWhich pattern can he make with those two tiles?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "A", "image_path": "image5.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 6, "question": "This diagram shows two see-through sheets. You place the sheets on top of each other.Which pattern do you get?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image6.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 7, "question": "Which of the 5 pictures shows a part of this chain?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "C", "image_path": "image7.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Logic" ] } }, { "id": 8, "question": "Five equally big square pieces of card are placed on a table on top of each other. The picture on the side is created this way. The cards are collected up from top to bottom. In which order are they collected?\n\nChoices: A. 5-4-3-2-1\nB. 5-2-3-4-1\nC. 5-4-2-3-1\nD. 5-3-2-1-4\nE. 5-2-3-1-4", "answer": "E", "image_path": "image8.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 9, "question": "A village of 12 houses has four straight streets and four circular streets. The map shows 11 houses. In each straight street there are three houses and in each circular street there are also three houses. Where should the 12th house be placed on this map?\n\nChoices: A. On A\nB. On B\nC. On C\nD. On D\nE. On E", "answer": "D", "image_path": "image9.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 10, "question": "Tom has these nine cards:\n\nHe places these cards on the board next to each other so that each horizontal line and each vertical line has three cards with the three different shapes and the three different amounts of drawings. He has already placed three cards, as shown in the picture. Which card should he place in the colored box?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image10.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 11, "question": "Six different numbers, chosen from integers 1 to 9 , are written on the faces of a cube, one number per face. The sum of the numbers on each pair of opposite faces is always the same. Which of the following numbers could have been written on the opposite side with the number 8 ?\n", "answer": "3", "image_path": "image11.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 12, "question": "Maia the bee can only walk on colorful houses. How many ways can you color exactly three white houses with the same color so that Maia can walk from $A$ to $B$ ?\n", "answer": "16", "image_path": "image12.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 13, "question": "Which figure can be made from the 2 pieces shown on the right?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "A", "image_path": "image13.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 14, "question": "Mara built the square by using 4 of the following 5 shapes. Which shape was not used?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image14.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 15, "question": "Dino walks from the entrance to the exit. He is only allowed to go through each room once. The rooms have numbers (see diagram). Dino adds up all the numbers of the rooms he walks through.\n\nWhat is the biggest result he can get this way?", "answer": "34", "image_path": "image15.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 16, "question": "We consider the perimeter and the area of the region corresponding to the grey squares. How many more squares can we colour grey for the grey area to increase without increasing its perimeter?\n", "answer": "16", "image_path": "image16.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Probability & Statistics", "Number Theory", "Geometry" ] } }, { "id": 17, "question": "The cells of a $4 \\times 4$ table are coloured black and white as shown in the left figure. One move allows us to exchange any two cells positioned in the same row or in the same column. What is the least number of moves necessary to obtain in the right figure?\n", "answer": "4", "image_path": "image17.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 18, "question": "In the box are seven blockss. You want to rearrange the blocks so that another block can placed. What is the minimum number of blocks that have to be moved?\n", "answer": "3", "image_path": "image18.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 19, "question": "Jan cannot draw very accurately but nevertheless he tried to produce a roadmap of his village. The relative position of the houses and the street crossings are all correct but three of the roads are actually straight and only Qurwik street is not. Who lives in Qurwik street?\n\nChoices: A. Amy\nB. Ben\nC. Carol\nD. David\nE. It cannot be determined from the drawing.", "answer": "C", "image_path": "image19.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 20, "question": "Michael wants to write whole numbers into the empty fields of the $3 \\times 3$ table on the right so that the sum of the numbers in each $2 \\times 2$ square equals 10. Four numbers have already been written down. Which of the following values could be the sum of the remaining five numbers?\n\nChoices: A. 9\nB. 10\nC. 12\nD. 13\nE. None of these numbers is possible.", "answer": "E", "image_path": "image20.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 21, "question": "A circular carpet is placed on a floor which is covered by equally big, square tiles. All tiles that have at least one point in common with the carpet are coloured in grey. Which of the following cannot be a result of this?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image21.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 22, "question": "We consider a $5 \\times 5$ square that is split up into 25 fields. Initially all fields are white. In each move it is allowed to change the colour of three fields that are adjacent in a horizontal or vertical line (i.e. white fields turn black and black ones turn white). What is the smallest number of moves needed to obtain the chessboard colouring shown in the diagram?\n\nChoices: A. less than 10\nB. 10\nC. 12\nD. more than 12\nE. This colouring cannot be obtained.", "answer": "A", "image_path": "image22.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic", "Geometry" ] } }, { "id": 23, "question": "The diagram shows the floor plan of Renate's house. Renate enters her house from the terrace (Terrasse) and walks through every door of the house exactly once. Which room does she end up in?\n", "answer": "2", "image_path": "image23.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 24, "question": "In the addition on the right, different letters represent different numbers. Assuming the account is correct, what is the highest possible value for the sum $\\mathrm{C}+\\mathrm{A}+\\mathrm{N}$?\n", "answer": "21", "image_path": "image24.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Number Theory", "Logic" ] } }, { "id": 25, "question": "Which of the rectangles $\\mathbf{A}$ to $\\mathbf{E}$ can be covered by the pattern on the right-hand side in such a way that the result is a totally black rectangle?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image25.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 26, "question": "A kangaroo enters a building. He only passes through triangular rooms. Where does he leave the building?\n\nChoices: A. a\nB. b\nC. c\nD. d\nE. e", "answer": "E", "image_path": "image26.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 27, "question": "Numbers in the picture are ticket prices between neighbouring towns. Peter wants to go from $A$ to $B$ as cheaply as possible. What is the lowest price he has to pay?\n", "answer": "90", "image_path": "image27.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 28, "question": "You can move or rotate each shape as you like, but you are not allowed to flip them over. What shape is not used in the puzzle?\n\n", "answer": "C", "image_path": "image28.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Divide‑and-Conquer", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 29, "question": "Anna made the figure on the right out of five cubes. Which of the following figures (when seen from any direction) cannot she get from the figure on the right side if she is allowed to move exactly one cube?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image29.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 30, "question": "One of the cube faces is cut along its diagonals (see the fig.). Which two of the following nets are impossible?\n\n\nChoices: A. 1 and 3\nB. 1 and 5\nC. 3 and 4\nD. 3 and 5\nE. 2 and 4", "answer": "D", "image_path": "image30.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics" ] } }, { "id": 31, "question": "Which of the following diagrams is impossible to make with the two dominos?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image31.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 32, "question": "In each square of the maze there is a piece of cheese. Ronnie the mouse wants to enter and leave the maze as shown in the picture. He doesn't want to visit a square more than once, but would like to eat as much cheese as possible. What is the maximum number of pieces of cheese that he can eat?\n", "answer": "37", "image_path": "image32.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 33, "question": "You can place together the cards pictured, to make different three digit numbers, for instance 989 or 986. How many different three digit numbers can you make with these cards?\n", "answer": "12", "image_path": "image33.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Probability & Statistics", "Logic", "Number Theory" ] } }, { "id": 34, "question": "Which of the following pieces can be joined to the one pictured so that a rectangle is formed?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "B", "image_path": "image34.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 35, "question": "Leo writes numbers in the multiplication pyramid. Explanation of the multiplication pyramid: By multiplying the numbers which are next to each other, the number directly above (in the middle) is calculated. Which number must Leo write in the grey field?\n", "answer": "8", "image_path": "image35.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 36, "question": "Seven children stand in a circle. Nowhere are two boys found standing next to each other. Nowhere are three girls found standing next to each other. What is possible for the number of girls? The number of girls can...\n\nChoices: A. ... only be 3.\nB. ... be 3 or 4.\nC. ... be 4 or 5.\nD. ... only be 5.\nE. ... only be 4.", "answer": "C", "image_path": "image36.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 37, "question": "Some of the small squares on each of the square transparencies have been coloured black. If you slide the three transparencies on top of each other, without lifting them from the table, a new pattern can be seen. What is the maximum number of black squares which could be seen in the new pattern?\n", "answer": "8", "image_path": "image37.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 38, "question": "The shape in the picture is to be split into three identical pieces. What does one of these pieces look like?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "A", "image_path": "image38.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 39, "question": "A mouse wants to escape a labyrinth. On her way out she is only allowed to go through each opening once at most. How many different ways can the mouse choose to go to get outside?\n", "answer": "4", "image_path": "image39.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 40, "question": "Five sparrows on a rope look in one or the other direction (see diagram). Every sparrow whistles as many times as the number of sparrows he can see in front of him. Azra therefore whistles four times. Then one sparrow turns in the opposite direction and again all sparrows whistle according to the same rule. The second time the sparrows whistle more often in total than the first time. Which sparrow has turned around?\n\nChoices: A. Azra\nB. Bernhard\nC. Christa\nD. David\nE. Elsa", "answer": "B", "image_path": "image40.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 41, "question": "Leonie has hidden a Smiley behind some of the grey boxes. The numbers state how many Smileys there are in the neighbouring boxes. Two boxes are neighbouring if they have one side or one corner in common. How many Smileys has Leonie hidden?\n", "answer": "5", "image_path": "image41.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 42, "question": "Sofie wants to pick 5 different shapes from the boxes. She can only pick 1 shape from each box. Which shape must she pick from box 4?\n\n\nChoices: A. (A)\nB. (B)\nC. (C)\nD. (D)\nE. (E)", "answer": "E", "image_path": "image42.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 43, "question": "A road leads away from each of the six houses (see diagram). A hexagon showing the roads in the middle is however, missing. Which hexagons fit in the middle so\nthat one can travel from $A$ to $B$ and to $E$, but not to $D$?\nChoices: A. 1 and 2\nB. 1 and 4\nC. 1 and 5\nD. 2 and 3\nE. 4 and 5", "answer": "C", "image_path": "image43.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 44, "question": "Using the pieces $A, B, C, D$ and $E$ one can fill this shape completely: Which of the pieces lies on the dot?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image44.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 45, "question": "A building block is made up of five identical rectangles: \nHow many of the patterns shown below can be made with two such building blocks without overlap?\n\n\n\n\n", "answer": "4", "image_path": "image45.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Number Theory", "Logic" ] } }, { "id": 46, "question": "Maria colours exactly 5 cells of this grid in grey. Then she has her 5 friends guess which cells she has coloured in and their answers are the five patterns $A, B, C, D$ and $E$. Maria looks at the patterns and says: \"One of you is right. The others have each guessed exactly four cells correctly.\" Which pattern did Maria paint?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image46.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 47, "question": "You have two identical pieces that you can turn around but not upside down. Which picture can you not make with these two pieces?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image47.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 48, "question": "Each of these two pieces of wire is made of 8 segments of length 1. One of the pieces is placed one above the other so that they coincide partially. What is the largest possible length of their common part?\n", "answer": "5", "image_path": "image48.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 49, "question": "A river goes through a city and there are two islands. There are also six bridges how it is shown in the attached image. How many paths there are going out of a shore of the river (point $A$ ) and come back (to point $B$ ) after having spent one and only one time for each bridge?\n", "answer": "6", "image_path": "image49.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 50, "question": "The multiplication uses each of the digits from 1 to 9 exactly once. What is digit $Y$?", "answer": "5", "image_path": "image50.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 51, "question": "Which of the \"buildings\" A-E, each consisting of 5 cubes, cannot be obtained from the building on the right, if you are allowed to move only one cube?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "C", "image_path": "image51.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 52, "question": "Kangi goes directly from the zoo to school (Schule) and counts the flowers along the way. Which of the following numbers can he not obtain this way?\n", "answer": "11", "image_path": "image52.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 53, "question": "In the box are seven blocks. It is possible to slide the blocks around so that another block can be added to the box. What is the minimum number of blocks that must be moved?\n", "answer": "2", "image_path": "image53.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 54, "question": "In the grid, how many grey squares have to be coloured white, so that in each row and each column there is exactly one grey square?\n", "answer": "6", "image_path": "image54.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 55, "question": "A shape is made by fitting together the four pieces of card with no overlaps. Which of the following shapes is not possible?\n\n\nChoices: A. A)\nB. B)\nC. C)\nD. D)\nE. E)", "answer": "E", "image_path": "image55.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 56, "question": "Fridolin the hamster runs through the maze in the picture. 16 pumpkin seeds are laying on the path. He is only allowed to cross each junction once. What is the maximum number of pumpkin seeds that he can collect?\n", "answer": "13", "image_path": "image56.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 57, "question": "Daniel wants to make a complete square using pieces only like those shown. What is the minimum number of pieces he must use?\n", "answer": "20", "image_path": "image57.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 58, "question": "Nick can turn right but not left on his bicycle. What is the least number of right turns he must make in order to get from $A$ to $B$?\n", "answer": "4", "image_path": "image58.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 59, "question": "Anne has a few grey tiles like the one in the picture.\n\nWhat is the maximum number of these tiles that she can place on the $5 \\times 4$ rectangle without any overlaps?\n", "answer": "4", "image_path": "image59.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Divide‑and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 60, "question": "From an old model train set there are only identical pieces of track to use. Matthias puts 8 such pieces in a circle (picture on the left). Martin begins his track with 2 pieces as shown in the picture on the right. He also wants to build a closed track and use the smallest number of pieces possible. How many pieces will his track use?\n", "answer": "12", "image_path": "image60.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 61, "question": "For the game of Chess a new piece, the Kangaroo, has been invented. With each jump the kangaroo jumps either 3 squares vertically and 1 Horizontally, or 3 horizontally and 1 vertically, as pictured. What is the smallest number of jumps the kangaroo must make to move from its current position to position $\\mathrm{A}$ ?\n", "answer": "3", "image_path": "image61.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 62, "question": "Nina wants to make a cube from the paper net. You can see there are 7 squares Instead of 6. Which square(s) can she remove from the net, so that the other 6 squares remain connected and from the newly formed net a cube can be made?\n\nChoices: A. only 4\nB. only 7\nC. only 3 or 4\nD. only 3 or 7\nE. only 3,4 or 7", "answer": "D", "image_path": "image62.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 63, "question": "In how many ways can the three kangaroos be placed in three different squares so that no kangaroo has an immediate neighbour?\n", "answer": "10", "image_path": "image63.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics" ] } }, { "id": 64, "question": "Lisa has mounted 7 postcards on her fridge door using 8 strong magnets (black dots). What is the maximum amount of magnets she can remove without any postcards falling on the floor?\n", "answer": "4", "image_path": "image64.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Geometry" ] } }, { "id": 65, "question": "Robert has two equally big squares made of paper. He glues them together. Which of the following shapes can he not make?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "A", "image_path": "image65.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 66, "question": "Anna has four identical building blocks that each look like this: Which shape can she not form with them?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image66.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 67, "question": "The 10 islands are connected by 12 bridges (see diagram). All bridges are open for traffic. What is the minimum number of bridges that need to be closed off, so that the traffic between $A$ and $B$ comes to a halt?\n", "answer": "2", "image_path": "image67.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Geometry" ] } }, { "id": 68, "question": "A big cube is made up of 9 identical building blocks. Each building block looks like this: Which big cube is possible?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "A", "image_path": "image68.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 69, "question": "In the diagram the circles represent light bulbs which are connected to some other light bulbs. Initially all light bulbs are switched off. If you touch a light bulb then that light bulb and all directly adjacent light bulbs switch themselves on. What is the minimum number of light bulbs you have to touch in order to switch on all the light bulbs?\n", "answer": "2", "image_path": "image69.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 70, "question": "The four smudges hide four of the numbers $1,2,3,4,5$. The calculations along the two arrows are correct. Which number hides behind the smudge with the star?\n", "answer": "5", "image_path": "image70.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Logic" ] } }, { "id": 71, "question": "Four ladybirds each sit on a different cell of a $4 \\times 4$ grid. One is asleep and does not move. On a whistle the other three each move to an adjacent free cell. They can crawl up, down, to the right or to the left but are not allowed on any account to move back to the cell that they have just come from. Where could the ladybirds be after the fourth whistle?\nInitial position:\n\nAfter the first whistle:\n\nAfter the second whistle:\n\nAfter the third whistle:\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "A", "image_path": "image71.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 72, "question": "Which tile below completes the wall next to it?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image72.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 73, "question": "The figure shows a map with some islands and how they are connected by bridges. A navigator wants to pass through each of the islands exactly once. He started at Cang Island and wants to finish at Uru Island. He has just arrived at the black island in the center of the map. In which direction must he go now to be able to complete his route?\n\nChoices: A. North.\nB. East.\nC. South.\nD. West.\nE. There is more than one possible choice", "answer": "C", "image_path": "image73.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Branch‑and-Bound", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 74, "question": "Mary had a piece of paper. She folded it exactly in half. Then she folded it exactly in half again. She got this shape . Which of the shapes P, Q or R could have been the shape of her original piece of paper?\n\nChoices: A. only P\nB. only Q\nC. only R\nD. only P or Q\nE. any of P, Q or R", "answer": "E", "image_path": "image74.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 75, "question": "Ronja had four white tokens and Wanja had four grey tokens. They played a game in which they took turns to place one of their tokens to create two piles. Ronja placed her first token first. Which pair of piles could they not create?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image75.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Logic" ] } }, { "id": 76, "question": "Five big and four small elephants are marching along a path. Since the path is narrow the elephants cannot change their order. At the fork in the path each elephant either goes to the right or to the left. Which of the following situations cannot happen?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "C", "image_path": "image76.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 77, "question": "Four circles are always connected by a line to form chains of four in a drawing. The numbers 1, 2, 3 and 4 appear in each row, each column and each chain of four.\nWhich number is in the circle with the question mark?\n", "answer": "2", "image_path": "image77.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 78, "question": "Monika wants to find a path through the labyrinth from 'Start' to 'Ziel'. She has to stick to the following rules: She is only allowed to move horizontally and vertically respectively. She has to enter every white circle exactly once but is not allowed to enter a black circle. In which direction does Monika have to move forwards when she reaches the circle marked with $x$ ? \nChoices: A. $\\downarrow$\nB. $\\uparrow$\nC. $\\rightarrow$\nD. $\\leftarrow$\nE. there are several possibilities", "answer": "A", "image_path": "image78.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Branch‑and-Bound", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 79, "question": "Caroline wants to write the numbers $1,2,3,4$ in the square $4 \\times 4$ in such a way that every row and every column has each of the numbers. You see how she started. How many of the 4 numbers can be written in place of $x$?\n", "answer": "2", "image_path": "image79.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 80, "question": "Max and Moritz have drawn a square $5 \\times 5$ and marked the centres of the small squares. Afterwards, they draw obstacles and then find out in how many ways it is possible to go from $A$ to $B$ using the shortest way avoiding the obstacles and going from centre to centre only vertically and horizontally. How many shortest paths are there from $A$ to $B$ under these conditions?\n", "answer": "12", "image_path": "image80.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 81, "question": "How many little squares at least do we have to shade in the picture on the right in order that it have an axis of symmetry?\n", "answer": "3", "image_path": "image81.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 82, "question": "Which of the following objects can be obtained by rotating in space the grey object?\n\n\nChoices: A. W and Y\nB. X and Z\nC. Only Y\nD. None of these\nE. W, X ir Y", "answer": "A", "image_path": "image82.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 83, "question": "We want to paint each square in the grid with the colours P, Q, R and S, so that neighbouring squares always have different colours. (Squares which share the same corner point also count as neighbouring.) Some of the squares are already painted. In which colour(s) could the grey square be painted?\n\nChoices: A. only Q\nB. only R\nC. only S\nD. either R or S\nE. it is not possible.", "answer": "D", "image_path": "image83.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 84, "question": "In the picture on the right we see an L-shaped object which is made up of four squares. We would like to add another equally big square so that the new object has a line of symmetry. How many ways are there to achieve this?\n", "answer": "3", "image_path": "image84.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 85, "question": "Each area in the picture on the right should be coloured using one of the colours, red (R), green (G), blue (B) or orange (O). Areas which touch must be different colours. Which colour is the area marked $X$?\n\nChoices: A. red\nB. blue\nC. green\nD. orange\nE. The colour cannot definitely be determined.", "answer": "A", "image_path": "image85.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 86, "question": "Each of the nine paths in a park are $100 \\mathrm{~m}$ long. Anna wants to walk from $A$ to $B$ without using the same path twice. How long the longest path she can choose?\n\nChoices: A. $900 \\mathrm{~m}$\nB. $800 \\mathrm{~m}$\nC. $700 \\mathrm{~m}$\nD. $500 \\mathrm{~m}$\nE. $400 \\mathrm{~m}$", "answer": "C", "image_path": "image86.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Branch‑and-Bound", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 87, "question": "Werner folds a piece of paper as shown in the diagram. With a pair of scissors he makes two straight cuts into the paper. Then is unfolds it again. Which on the following shapes are not possible for the piece of paper to show afterwards?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image87.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 88, "question": "Gray and white pearls are threaded onto a string. Tony pulls pearls from the ends of the chain. After pulling off the fifth gray pearl he stops. At most, how many white pearls could he have pulled off?\n", "answer": "7", "image_path": "image88.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 89, "question": "On a pond 16 lilly pads are arranged in a $4 \\times 4$ grid as can be seen in the diagram. A frog sits on a lilly pad in one of the corners of the grid (see picture). The frog jumps from one lilly pad to another horizontally or vertically. In doing so he always jumps over at least one lilly pad. He never lands on the same lilly pad twice. What is the maximum number of lilly pads, including the one he is sitting on, on which he can land?\n", "answer": "16", "image_path": "image89.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 90, "question": "Each side of each triangle in the diagram is painted either blue, green or red. Four of the sides are already painted. Which colour can the line marked \"x\" have, if each triangle must have all sides in different colours?\n\nChoices: A. only green\nB. only red\nC. only blue\nD. either red or blue\nE. The question cannot be solved.", "answer": "A", "image_path": "image90.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 91, "question": "Riki wants to write one number in each of the seven sections of the diagram pictured. Two zones are adjacent if they share a part of their outline. The number in each zone should be the sum of all numbers of its adjacent zones. Riki has already placed numbers in two zones. Which number does she need to write in the zone marked \"?\".\n", "answer": "6", "image_path": "image91.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 92, "question": "In a game of luck, A ball rolls downwards towards hammered nails and is diverted either to the right or the left by a nail immediately below it. One possible path is shown in the diagram. How many different ways are there for the ball to reach the second compartment from the left?\n", "answer": "4", "image_path": "image92.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 93, "question": "A $4 \\times 4$ square is made up of the two pieces shown. Which of the following $4 \\times 4$ squares cannot be made this way?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image93.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 94, "question": "Anna has placed matches along the dotted lines to create a path. She has placed the first match as shown in the diagram. The path is in such a way that in the end it leads back to the left end of the first match. The numbers in the small squares state how many sides of the square she has placed matches on. What is the minimum number of matches she has used?\n", "answer": "16", "image_path": "image94.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 95, "question": "Amelia has a paper strip with five equal cells containing different drawings, according to the figure. She folds the strip in such a way that the cells overlap in five layers. Which of the sequences of layers, from top to bottom, is not possible to obtain?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "A", "image_path": "image95.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 96, "question": "The black-white caterpillar shown, rolls up to go to sleep. Which diagram could show the rolled-up caterpillar?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "A", "image_path": "image96.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Branch‑and-Bound", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 97, "question": "What is the minimum number of cells of a $5 \\times 5$ grid that have to be coloured in so that every possible $1 \\times 4$ rectangle and every $4 \\times 1$ rectangle respectively in the grid has at least one cell coloured in?\n", "answer": "6", "image_path": "image97.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 98, "question": "Elisabeth wants to write the numbers 1 to 9 in the fields of the diagram shown so that the product of the numbers of two fields next to each other is no greater than 15. Two fields are called „next to each other“ if they share a common edge. How many ways are there for Elisabeth to label the fields? ", "answer": "16", "image_path": "image98.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 99, "question": "Six integers are marked on the real line (see the fig.). It is known that at least two of them are divisible by 3, and at least two of them are divisible by 5. Which numbers are divisible by 15?\n\nChoices: A. $A$ and $F$\nB. $B$ and $D$\nC. $C$ and $E$\nD. All the six numbers\nE. Only one of them", "answer": "A", "image_path": "image99.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Number Theory" ] } }, { "id": 100, "question": "A barcode as pictured is made up of alternate black and white stripes. The code always starts and ends with a black stripe. Each stripe (black or white) has the width 1 or 2 and the total width of the barcode is 12. How many different barcodes of this kind are there if one reads from left to right?\n", "answer": "114", "image_path": "image100.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Number Theory", "Logic" ] } }, { "id": 101, "question": "How many different ways are there in the diagram shown, to get from point $A$ to point $B$ if you are only allowed to move in the directions indicated?\n", "answer": "12", "image_path": "image101.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 102, "question": "Jilly makes up a multiplication magic square using the numbers $1,2,4,5,10,20,25,50$ and 100. The products of the numbers in each row, column and diagonal should be equal. In the diagram it can be seen how she has started. Which number goes into the cell with the question mark?\n", "answer": "4", "image_path": "image102.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 103, "question": "Paul wants to write a positive whole number onto every tile in the number wall shown, so that every number is equal to the sum of the two numbers on the tiles that are directly below. What is the maximum number of odd numbers he can write on the tiles?\n", "answer": "14", "image_path": "image103.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic", "Combinatorics" ] } }, { "id": 104, "question": "In the diagram shown, you should follow the arrows to get from A to B. How many different ways are there that fulfill this condition?\n", "answer": "16", "image_path": "image104.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 105, "question": "A bee called Maja wants to hike from honeycomb $X$ to honeycomb $Y$. She can only move from one honeycomb to the neighbouring honeycomb if they share an edge. How many, different ways are there for Maja to go from $X$ to $Y$ if she has to step onto every one of the seven honeycombs exactly once?\n", "answer": "5", "image_path": "image105.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 106, "question": "There are three paths running through our park in the city (see diagram). A tree is situated in the centre of the park. What is the minimum number of trees that have to be planted additionally so that there are the same number of trees on either side of each path?\n", "answer": "3", "image_path": "image106.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Logic" ] } }, { "id": 107, "question": "Veronika wears five rings as shown. How many, different ways are there for her to take off the rings one by one?\n", "answer": "20", "image_path": "image107.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 108, "question": "A dark disc with two holes is placed on the dial of a watch as shown in the diagram. The dark disc is now rotated so that the number 10 can be seen through one of the two holes. Which of the numbers could one see through the other hole now? \nChoices: A. 2 and 6\nB. 3 and 7\nC. 3 and 6\nD. 1 and 9\nE. 2 and 7", "answer": "A", "image_path": "image108.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 109, "question": "The numbers from 1 to 9 should be distributed among the 9 squares in the diagram according to the following rules: There should be one number in each square. The sum of three adjacent numbers is always a multiple of 3 . The numbers 3 and 1 are already placed. How many ways are there to place the remaining numbers?", "answer": "24", "image_path": "image109.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Number Theory", "Logic" ] } }, { "id": 110, "question": "Leon has drawn a closed loop on the surface of a cuboid.\nWhich net cannot show his loop? \nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "C", "image_path": "image110.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 111, "question": "Roo has 16 cards: 4 spades ( $(\\boldsymbol{*}), 4$ clubs ( $*$ ), 4 diamonds ( $\\bullet$ ) and 4 hearts $(\\boldsymbol{v})$. He wants to place them in the square shown, so that every row and every column has exactly one card of each suit. The diagram shows how Roo started. How many of the 4 cards can be put in place of the question mark?\n", "answer": "2", "image_path": "image111.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Number Theory", "Logic" ] } }, { "id": 112, "question": "Max and Moritz have drawn out a $5 \\times 5$ grid on the playground, together with three obstacles. They want to walk from $P$ to $Q$ using the shortest route, avoiding the obstacles and always crossing a common edge to go from the centre of one square to the centre of the next. How many such shortest paths are there from $P$ to $Q$ ? ", "answer": "12", "image_path": "image112.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 113, "question": "Dominoes are said to be arranged correctly if, for each pair of adjacent dominoes, the numbers of spots on the adjacent ends are equal. Paul laid six dominoes in a line as shown in the diagram.\n\nHe can make a move either by swapping the position of any two dominoes (without rotating either domino) or by rotating one domino. What is the smallest number of moves he needs to make to arrange all the dominoes correctly?", "answer": "3", "image_path": "image113.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Trial-and-Error" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 114, "question": "Wendy wants to write a number in every cell on the border of a table.\nIn each cell, the number she writes is equal to the sum of the two numbers in the cells with which this cell shares an edge. Two of the numbers are given in the diagram.\nWhat number should she write in the cell marked $x$ ?\n", "answer": "7", "image_path": "image114.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 115, "question": "Which of the following $4 \\times 4$ tiles cannot be formed by combining the two given pieces?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image115.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 116, "question": "In the sum shown, different shapes represent different digits.\n\nWhat digit does the square represent?", "answer": "6", "image_path": "image116.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Algebra", "Probability & Statistics" ] } }, { "id": 117, "question": "Sid is colouring the cells in the grid using the four colours red, blue, yellow and green in such a way that any two cells that share a vertex are coloured differently. He has already coloured some of the cells as shown.\nWhat colour will he use for the cell marked $X$ ?\n\nChoices: A. Red\nB. Blue\nC. Yellow\nD. Green\nE. You can't be certain", "answer": "A", "image_path": "image117.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Branch‑and-Bound", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 118, "question": "Patricia painted some of the cells of a $4 \\times 4$ grid. Carl counted how many red cells there were in each row and in each column and created a table to show his answers.\nWhich of the following tables could Carl have created?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image118.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic", "Probability & Statistics" ] } }, { "id": 119, "question": "Barney has 16 cards: 4 blue $(B), 4$ red $(R), 4$ green $(G)$ and 4 yellow (Y). He wants to place them in the square shown so that every row and every column has exactly one of each card. The diagram shows how he started. How many different ways can he finish?\n", "answer": "4", "image_path": "image119.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Branch‑and-Bound", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 120, "question": "The picture shows seven points and the connections between them. What is the least number of connecting lines that could be added to the picture so that each of the seven points has the same number of connections with other points? (Connecting lines are allowed to cross each other.)\n", "answer": "9", "image_path": "image120.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 121, "question": "A network consists of 16 vertices and 24 edges that connect them, as shown. An ant begins at the vertex labelled Start. Every minute, it walks from one vertex to a neighbouring vertex, crawling along a connecting edge. At which of the vertices labelled $P, Q, R, S, T$ can the ant be after 2019 minutes? \nChoices: A. only $P, R$ or $S$,\nB. not $Q$\nC. only $Q$\nD. only $T$\nE. all of the vertices are possible", "answer": "C", "image_path": "image121.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 122, "question": "The numbers from 1 to 6 are to be placed at the intersections of three circles, one number in each of the six squares. The number 6 is already placed. Which number must replace $x$, so that the sum of the four numbers on each circle is the same? ", "answer": "1", "image_path": "image122.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 123, "question": "On Nadya's smartphone, the diagram shows how much time she spent last week on four of her apps. This week she halved the time spent on two of these apps, but spent the same amount of time as the previous week on the other two apps.\n\nWhich of the following could be the diagram for this week?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image123.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 124, "question": "Cuthbert is going to make a cube with each face divided into four squares. Each square must have one shape drawn on it; either a cross, a triangle or a circle. Squares that share an edge must have different shapes on them. One possible cube is shown in the diagram. Which of the following combinations of crosses and triangles is possible on such a cube (with the other shapes being circles)?\n\nChoices: A. 6 crosses, 8 triangles\nB. 7 crosses, 8 triangles\nC. 5 crosses, 8 triangles\nD. 7 crosses, 7 triangles\nE. none of these are possible", "answer": "E", "image_path": "image124.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 125, "question": "Robin shoots three arrows at a target. He earns points for each shot as shown in the figure. However, if any of his arrows miss the target or if any two of his arrows hit adjacent regions of the target, he scores a total of zero. How many different scores can he obtain?\n", "answer": "13", "image_path": "image125.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 126, "question": "Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct.\nWhat is the answer to 3 ACROSS?\n\n\\section*{ACROSS}\n1. A multiple of 7\n3. The answer to this Question\n5. More than 10\n\\section*{DOWN}\n1. A multiple of a square of an odd prime; neither a square nor a cube\n2. The internal angle of a regular polygon; the exterior angle is between $10^{\\circ}$ and $20^{\\circ}$\n4. A proper factor of $5 \\mathrm{ACROSS}$ but not a proper factor of $1 \\mathrm{DOWN}$", "answer": "961", "image_path": "image126.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic", "Number Theory" ] } }, { "id": 127, "question": "It is possible to place positive integers into the vacant twenty-one squares of the $5 \\times 5$ square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*).\n\n", "answer": "142", "image_path": "image127.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 128, "question": "Hexagon $ABCDEF$ is divided into four rhombuses, $\\mathcal{P, Q, R, S,}$ and $\\mathcal{T,}$ as shown. Rhombuses $\\mathcal{P, Q, R,}$ and $\\mathcal{S}$ are congruent, and each has area $\\sqrt{2006}$. Let $K$ be the area of rhombus $\\mathcal{T}$. Given that $K$ is a positive integer, find the number of possible values for $K$.\n\n", "answer": "89", "image_path": "image128.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 129, "question": "In the $ 6\\times4$ grid shown, $ 12$ of the $ 24$ squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $ N$ be the number of shadings with this property. Find the remainder when $ N$ is divided by $ 1000$.\n", "answer": "860", "image_path": "image129.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Algebra", "Combinatorics", "Logic" ] } }, { "id": 130, "question": "The diagram below shows a $ 4\\times4$ rectangular array of points, each of which is $ 1$ unit away from its nearest neighbors.\nDefine a growing path to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let $ m$ be the maximum possible number of points in a growing path, and let $ r$ be the number of growing paths consisting of exactly $ m$ points. Find $ mr$.", "answer": "240", "image_path": "image130.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 131, "question": "The following analog clock has two hands that can move independently of each other.\n\nInitially, both hands point to the number 12. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move.\n\nLet $N$ be the number of sequences of 144 hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the 144 movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by 1000.", "answer": "608", "image_path": "image131.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic", "Algebra" ] } }, { "id": 132, "question": "Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2\\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3$. One way to do this is shown below. Find the number of positive integer divisors of $N$.\n\n", "answer": "144", "image_path": "image132.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Combinatorics", "Algebra" ] } }, { "id": 133, "question": "A bug travels from $A$ to $B$ along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?\n\n", "answer": "2400", "image_path": "image133.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 134, "question": "In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?\n\n", "answer": "12", "image_path": "image134.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 135, "question": "Suppose that 13 cards numbered $1, 2, 3, \\dots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes?\n\n", "answer": "8178", "image_path": "image135.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 136, "question": "The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\\le|x|\\le7$, $ 3\\le|y|\\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$?\n", "answer": "225", "image_path": "image136.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics" ] } }, { "id": 137, "question": "There are 5 coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?\n\n\nChoices: A. $(C, A, E, D, B)$\nB. $(C, A, D, E, B)$\nC. $(C, D, E, A, B) \\ [1ex]$\nD. $(C, E, A, D, B)$\nE. $(C, E, D, A, B)$", "answer": "E", "image_path": "image137.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Logic" ] } }, { "id": 138, "question": "Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\\widetilde{AB}$, $\\widetilde{AD}$, $\\widetilde{AE}$, $\\widetilde{BC}$, $\\widetilde{BD}$, $\\widetilde{CD}$, $\\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit cities more than once.)\n\n", "answer": "16", "image_path": "image138.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test", "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 139, "question": "The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L,$ without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.) How many different routes can Paula take?\n\n", "answer": "4", "image_path": "image139.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 140, "question": "\n\nUsing only the paths and the directions shown, how many different routes are there from $ M$ to $ N$?", "answer": "6", "image_path": "image140.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 141, "question": "\n\nSuppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box?", "answer": "6", "image_path": "image141.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry" ] } }, { "id": 142, "question": "Six different digits from the set\n\\[\\{ 1,2,3,4,5,6,7,8,9\\}\\]\nare placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12.\nThe sum of the six digits used is\n\n", "answer": "29", "image_path": "image142.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 143, "question": "Points $R, S$ and $T$ are vertices of an equilateral triangle, and points $X, Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices?\n\n", "answer": "4", "image_path": "image143.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Geometry" ] } }, { "id": 144, "question": "How many rectangles are in this figure?\n\n", "answer": "11", "image_path": "image144.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 145, "question": "A $\\triangle$ or $\\bigcirc$ is placed in each of the nine squares in a 3-by-3 grid. Shown below is a sample configuration with three $\\triangle$s in a line.\n\n\nHow many configurations will have three $\\triangle$s in a line and three $\\bigcirc$s in a line?", "answer": "84", "image_path": "image145.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 146, "question": "The digits $2$, $0$, $2$, and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression?\n\n", "answer": "9", "image_path": "image146.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Algebra", "Number Theory", "Logic" ] } }, { "id": 147, "question": "Suppose we have a hexagonal grid in the shape of a hexagon of side length $4$ as shown at left. Define a “chunk” to be four tiles, two of which are adjacent to the other three, and the other two of which are adjacent to just two of the others. The three possible rotations of these are shown at right.\\n\\nIn how many ways can we choose a chunk from the grid?\\n", "answer": "72", "image_path": "image147.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 148, "question": "Suppose that in a group of $6$ people, if $A$ is friends with $B$, then $B$ is friends with $A$. If each of the $6$ people draws a graph of the friendships between the other $5$ people, we get these $6$ graphs, where edges represent\\nfriendships and points represent people.\\n\\nIf Sue drew the first graph, how many friends does she have?\\n", "answer": "4", "image_path": "image148.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 149, "question": "We define the $\\emph{weight}$ of a path to be the sum of the numbers written on each edge of the path. Find the minimum weight among all paths in the graph below that visit each vertex precisely once. \\n", "answer": "65", "image_path": "image149.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 150, "question": "On Misha's new phone, a passlock consists of six circles arranged in a $2\\times 3$ rectangle. The lock is opened by a continuous path connecting the six circles; the path cannot pass through a circle on the way between two others (e.g. the top left and right circles cannot be adjacent). For example, the left path shown below is allowed but the right path is not. (Paths are considered to be oriented, so that a path starting at $A$ and ending at $B$ is different from a path starting at $B$ and ending at $A$. However, in the diagrams below, the paths are valid/invalid regardless of orientation.) How many passlocks are there consisting of all six circles?\\n", "answer": "336", "image_path": "image150.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 151, "question": "Sujay and Rishabh are taking turns marking lattice points within a square board in the Cartesian plane with opposite vertices $(1, 1)$,$(n, n)$ for some constant $n$. Sujay loses when the two-point pattern $P$ below shows up. That is, Sujay loses when there exists a pair of points $(x, y)$ and $(x + 2, y + 1)$. He and Rishabh stop marking points when the pattern $P$ appears on the board. If Rishabh goes first, let $S$ be the set of all integers $3 \\le n \\le 100$ such that Rishabh has a strategy to always trick Sujay into being the one who creates $P$. Find the sum of all elements of $S$.\\n", "answer": "2499", "image_path": "image151.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 152, "question": "In chess, a knight can move by jumping to any square whose center is $\\sqrt{5}$ units away from the center of the square that it is currently on. For example, a knight on the square marked by the horse in the diagram below can move to any of the squares marked with an “X” and to no other squares. How many ways can a knight on the square marked by the horse in the diagram move to the square with a circle in exactly four moves?\\n", "answer": "54", "image_path": "image152.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 153, "question": "Teddy works at Please Forget Meat, a contemporary vegetarian pizza chain in the city of Gridtown, as a deliveryman. Please Forget Meat (PFM) has two convenient locations, marked with “$X$” and “$Y$ ” on the street map of Gridtown shown below. Teddy, who is currently at $X$, needs to deliver an eggplant pizza to $\\nabla$ en route to $Y$ , where he is urgently needed. There is currently construction taking place at $A$, $B$, and $C$, so those three intersections will be completely impassable. How many ways can Teddy get from $X$ to $Y$ while staying on the roads (Traffic tickets are expensive!), not taking paths that are longer than necessary (Gas is expensive!), and that let him pass through $\\nabla$ (Losing a job is expensive!)?\\n", "answer": "1144", "image_path": "image153.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 154, "question": "Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the Primle. For each guess, a digit is highlighted blue if it is in the Primle, but not in the correct place. A digit is highlighted orange if it is in the Primle and is in the correct place. Finally, a digit is left unhighlighted if it is not in the Primle. If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the Primle?\\n", "answer": "79", "image_path": "image154.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 155, "question": "How many lines pass through exactly two points in the following hexagonal grid?\\n", "answer": "60", "image_path": "image155.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 156, "question": "In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right?\\n", "answer": "372", "image_path": "image156.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 157, "question": "A Sudoku matrix is defined as a $ 9\\times9$ array with entries from $ \\{1, 2, \\ldots , 9\\}$ and with the constraint that each row, each column, and each of the nine $ 3 \\times 3$ boxes that tile the array contains each digit from $ 1$ to $ 9$ exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of the squares in this matrix, as shown. What is the probability that the square marked by ? contains the digit $ 3$?\\n", "answer": "$\\frac{2}{21}$", "image_path": "image157.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Probability & Statistics" ] } }, { "id": 158, "question": "Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0), (2,0), (2,1),$ and $(0,1)$. $R$ can be divided into two unit squares, as shown; the resulting figure has seven edges. Compute the number of ways to choose one or more of the seven edges such that the resulting figure is traceable without lifting a pencil. (Rotations and reflections are considered distinct.)\\n", "answer": "61", "image_path": "image158.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Probability & Statistics", "Combinatorics" ] } }, { "id": 159, "question": "What is the shortest distance from node S to node G in this directed graph?", "answer": "29", "image_path": "image159.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 160, "question": "The statements on the right give clues to the identity of a three-digit code. What is the code?", "answer": "275", "image_path": "image160.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 161, "question": "Figure represents the pedigree of a family affected by two types of single-gene inherited diseases. (These two genetic diseases are controlled by gene A/a on an autosome and gene B/b on the X chromosome, respectively.)Please answer the following question:\nWhat is the probability that individual 12 does not carry any disease-causing genes?\nAnswer with results in original form without decimal conversion", "answer": "$\\frac{1}{3}$", "image_path": "image161.png", "annotated": { "difficulty_tier": "Basic", "subject": "Biology", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 162, "question": "As shown in the diagram, use 4 different colors to paint the regions $A$, $B$, $C$, and $D$, requiring that any two adjacent regions cannot use the same color. How many different coloring methods are there?\n A. 24\n B. 48\n C. 72\n D. 96\n Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "D", "image_path": "image162.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 163, "question": "As shown in the diagram, use 5 different colors to color the 6 distinct points $A$, $B$, $C$, $D$, $E$, and $F$ in the graph. Each point must be assigned 1 color, and the two endpoints of every line segment in the graph must be colored differently. How many different coloring methods are there in total?", "answer": "1920", "image_path": "image163.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Combinatorics" ] } }, { "id": 164, "question": "As shown in the diagram, it is known that there are 4 switches in the circuit. Each switch works independently, and the probability of being closed is $\\frac{1}{2}$. What is the probability that the light is on? Provide the raw result of the expression at the end without converting it to the nearest decimal.", "answer": "$\\frac{13}{16}$", "image_path": "image164.png", "annotated": { "difficulty_tier": "Medium", "subject": "Physics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 165, "question": "Use red, yellow, blue, and green to color the six regions shown in the diagram. Adjacent regions cannot have the same color. What is the total number of coloring methods?\n A. 24\n B. 48\n C. 72\n D. 120\n Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "D", "image_path": "image165.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 166, "question": "How many different ways are there to walk from A to F? (Each line segment can only be walked once) Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end.", "answer": "9", "image_path": "image166.png", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 167, "question": "Use 24 matches (id from A to X) and arrange them as shown in the image. Remove 4 matches to make 5 squares of the same size. Please answer the match id.", "answer": "{'id': ['B', 'K', 'N', 'W']}", "image_path": "image167.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 168, "question": "The following figure cannot be drawn with one stroke. Which line should be erased so that the figure can be drawn with one stroke? Answer the id (A-N) of two endpoint of the answer line.", "answer": "{'id': ['I', 'J']}", "image_path": "image168.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Geometry" ] } }, { "id": 169, "question": "Six people numbered A-F are sitting in seats 1-6. A is to the right of B, D is behind B, and E is to the left of C. So which person from A-F is sitting in seat '?' ? Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "A", "image_path": "image169.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 170, "question": "Five origami papers of the same size, a, e, i, o, and u, are placed on top of each other on a flat surface. Which origami paper is on the bottom layer? Please answer the question and provide the correct option letter, e.g., a, b, c, d, at the end.", "answer": "u", "image_path": "image170.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Others", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic" ] } }, { "id": 171, "question": "Eliminate several numbers from the table below so that the sum of each row and column is 15. How many numbers should be eliminated? Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end. ", "answer": "5", "image_path": "image171.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 172, "question": "What number should be in the red box? Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end. ", "answer": "9744", "image_path": "image172.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Algebra", "Combinatorics", "Logic" ] } }, { "id": 173, "question": "There are 5 cards with numbers from 1 to 5. Two cards are taken from the set and given to A and B, one card each. A and B only know the number on their own card. According to the image, what number is on the card given to B? Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end. ", "answer": "3", "image_path": "image173.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic" ] } }, { "id": 174, "question": "There are a total of 20 cards, divided into two types: black and white. Each type has 10 cards, numbered from 0 to 9. The rules of the game are as follows: (1) The player who gets the cards arranges them in front of themselves in ascending order from left to right. The cards must be placed face down. (2) If there are cards with the same number in both black and white, the black card is placed on the left, and the white card is placed on the right. This game involves guessing the numbers on the face-down cards placed in front of other players. As shown in the image, some card numbers are visible. The question is: what are the numbers on the cards marked by star? Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end. ", "answer": "5", "image_path": "image174.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 175, "question": "\"Mate in One \": You are given a chess position where it is **Black's turn to move**. Your task is to make a move that results in **checkmate** in one move. \n\n## Allowed Moves\n- Only legal moves are allowed, meaning moves that do not violate the standard rules of chess.\n- You must only provide a single move that results in checkmate.\n- The move can involve any piece (king, queen, rook, bishop, knight, or pawn).\n\n## Notes\n- The move you provide must **immediately checkmate** the opponent's king.\n- The move should not just check, but must guarantee that the opponent has no legal moves to escape the check.\n\nPlease provide your move in the format of **standard algebraic notation**: \nIn chess, moves are written using standard algebraic notation, which includes the following components: **notation of piece moved – destination square**\n- **Piece**: The piece is represented by a capital letter:\n - **K** for King\n - **Q** for Queen\n - **R** for Rook\n - **B** for Bishop\n - **N** for Knight\n - **P** for Pawn (No letter initial is used for pawns, so e4 means \"pawn moves to e4\".)\n- **Capture**: If the move involves a capture, an **\"x\"** is placed before the destination square. For example, \"Qxh7\" means the queen captures on h7.\n- **Checkmate**: checkmate is indicated by a **\"#\"** (e.g., \"Qh7#\").", "answer": "Qxg2#", "image_path": "image175.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 176, "question": "Place 7 more stars in white squares so that no 2 of the 8 stars are in line horizontally, vertically, or diagonally. Please answer the coordinates (row id:column id) of the stars.", "answer": "[{'row': 'A', 'column': 6}, {'row': 'B', 'column': 3}, {'row': 'C', 'column': 7}, {'row': 'D', 'column': 2}, {'row': 'E', 'column': 8}, {'row': 'F', 'column': 5}, {'row': 'H', 'column': 4}]", "image_path": "image176.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic", "Number Theory" ] } }, { "id": 177, "question": "Remove 2 matches leaving 2 squares of different sizes. Please answer the removed match id and give one possible answer.", "answer": "[['D', 'F'], ['D', 'G'], ['F', 'I'], ['F', 'G']]", "image_path": "image177.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 178, "question": "The figure has 14 squares made out of 8 matches. Remove 2 matches leaving 3 squares. Please answer the removed match id and give one possible answer.", "answer": "[['F', 'C'], ['F', 'D'], ['C', 'G'], ['D', 'G']]", "image_path": "image178.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 179, "question": "This is a multiplication problem. What should A, B, and C be? Please answer in the form of 'id:number'.", "answer": "{'A': 2, 'B': 8, 'C': 6}", "image_path": "image179.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Algebra", "Logic", "Combinatorics" ] } }, { "id": 180, "question": "This is a multiplication problem. What should the two multiplier (A and B) in the red should be? Note that Each '*' in the image represent a number. Answer in 'multiplier id:number' format.", "answer": "{'A': 987, 'B': 121}", "image_path": "image180.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Algebra", "Combinatorics", "Logic" ] } }, { "id": 181, "question": "In the circuit shown in the figure, when the slider $P$ of the variable resistor $R_{3}$ moves towards terminal $b$, ( )\n\nA. The reading of the voltmeter increases, and the reading of the ammeter decreases\nB. The reading of the voltmeter decreases, and the reading of the ammeter increases\nC. The reading of the voltmeter increases, and the reading of the ammeter increases\nD. The reading of the voltmeter decreases, and the reading of the ammeter decreases\nAnswer with the option letters.", "answer": "B", "image_path": "image181.png", "annotated": { "difficulty_tier": "Medium", "subject": "Physics", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic" ] } }, { "id": 182, "question": "(Select one or more answer choices) Outside the black box, there are four terminals numbered $1$, $2$, $3$, and $4$. Between terminals $1$ and $2$, $2$ and $3$, and $3$ and $4$, there is a resistor each. Between the terminals, there is also another resistor $R$ and a DC power source connected. The measured voltages between the terminals are $U_{12} = 3.0$ V, $U_{23} = 2.5$ V, $U_{34} = -1.5$ V. Possible connections that match the above measurement results are ( )\n\nA. The power source is connected between $1$ and $4$, and $R$ is connected between $1$ and $3$\nB. The power source is connected between $1$ and $4$, and $R$ is connected between $2$ and $4$\nC. The power source is connected between $1$ and $3$, and $R$ is connected between $1$ and $4$\nD. The power source is connected between $1$ and $3$, and $R$ is connected between $2$ and $4$\nAnswer with the option letters.", "answer": "C, D", "image_path": "image182.png", "annotated": { "difficulty_tier": "Medium", "subject": "Physics", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Logic" ] } }, { "id": 183, "question": "Fill in the blank squares (id from 1 to 3) with any letter of the alphabet to make valid three-letter words from both top to bottom and left to right. Please answer in the form of 'square id:letter'", "answer": "{'1': 'W', '2': 'A', '3': 'Y'}", "image_path": "image183.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Branch‑and-Bound", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 184, "question": "Fill in the blank squares (id from 1 to 7) with any letter of the alphabet to make valid four-letter words from both top to bottom and left to right. Please answer in the form of 'square id:letter'", "answer": "{'1': 'T', '2': 'H', '3': 'E', '4': 'E', '5': 'S', '6': 'E', '7': 'T'}", "image_path": "image184.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 185, "question": "Select four shapes from A-E to perfectly fill the 4x4 area below. Please answer the question and provide all correct option letter, e.g., A, B, C, D, at the end.", "answer": "['B', 'C', 'D', 'E']", "image_path": "image185.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Others", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Geometry" ] } }, { "id": 186, "question": "The circles below can be filled with the colors red, green, blue, and yellow so that each row, column, and group (linked by lines) contains no duplicate colors. Which color must appear in the circle with the question mark?", "answer": "red", "image_path": "image186.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Others", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 187, "question": "Several identical cubes are placed together, with the front, back, left and right views as shown in the figure. How many cubes are there in this pile at least? Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end.", "answer": "4", "image_path": "image187.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic", "Number Theory" ] } }, { "id": 188, "question": "You have four weights of 1, 2, 3, 4. Place these weights into the pans, one per pan, so that the scales balance. Please answer in the form of 'pan id:weight'", "answer": "{'A': 1, 'B': 3, 'C': 2, 'D': 4}", "image_path": "image188.png", "annotated": { "difficulty_tier": "Medium", "subject": "Physics", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 189, "question": "What number should replace the question mark? Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end.", "answer": "9", "image_path": "image189.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Number Theory" ] } }, { "id": 190, "question": "What number should replace the question mark? Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end.", "answer": "2", "image_path": "image190.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Number Theory" ] } }, { "id": 191, "question": "Four of the five pieces below can be fitted together to form a perfect circle. Which is the odd piece out? Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "D", "image_path": "image191.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics" ] } }, { "id": 192, "question": "What is the minimum number of colours required to fill the spaces in the given diagram without any two adjacent spaces having the same colour?\n A. 6\n B. 5\n C. 4\n D. 3\n Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "D", "image_path": "image192.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic" ] } }, { "id": 193, "question": "Yin-Yang is a logic puzzle with simple rules and challenging solutions.\n\nThe rules are simple.Yin-Yang is played on a rectangular grid with no standard size. Some cells start out filled with black or white. The rest of the cells are empty. Your task is to place black and white stones at the intersections of the grid lines such that:\n1. All black stones must be orthogonally connected to form a single group.\n2. All white stones must be orthogonally connected to form a single group.\n3. No 2x2 region can be monochromatic (i.e., a 2x2 region cannot consist entirely of black or entirely of white stones).\n4. **Do not change the positions of any stones already placed on the grid.**\n\n\nPlease complete the solution for the Yin-Yang puzzle in the diagram. \nAt the end of your response, summary your answer **as a single filled matrix** (list of lists or equivalent). Represent the stones as a 6x6 matrix of \"0\" and \"1\", where \"0\" represents a white stone and \"1\" represents a black stone.", "answer": "VALIDATION RULES:\n1. Provide a 6×6 matrix of 0 / 1 (e.g. [[0,1,0,1,0,1],…]).\n2. Do not change any pre-placed stones in the base grid.\n3. All 0-cells must form one orthogonally connected group, and all 1-cells must form one orthogonally connected group.\n4. No 2×2 square may be entirely 0s or entirely 1s.", "image_path": "image193.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Others", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 194, "question": "Yin-Yang is a logic puzzle with simple rules and challenging solutions.\n\nThe rules are simple.Yin-Yang is played on a rectangular grid with no standard size. Some cells start out filled with black or white. The rest of the cells are empty. Your task is to place black and white stones at the intersections of the grid lines such that:\n1. All black stones must be orthogonally connected to form a single group.\n2. All white stones must be orthogonally connected to form a single group.\n3. No 2x2 region can be monochromatic (i.e., a 2x2 region cannot consist entirely of black or entirely of white stones).\n4. **Do not change the positions of any stones already placed on the grid.**\n\n\nPlease complete the solution for the Yin-Yang puzzle in the diagram. \nAt the end of your response, summary your answer **as a single filled matrix** (list of lists or equivalent). Represent the stones as a 6x6 matrix of \"0\" and \"1\", where \"0\" represents a white stone and \"1\" represents a black stone.", "answer": "VALIDATION RULES:\n1. Provide a 6×6 matrix of 0 / 1 (e.g. [[0,1,0,1,0,1],…]).\n2. Do not change any pre-placed stones in the base grid.\n3. All 0-cells must form one orthogonally connected group, and all 1-cells must form one orthogonally connected group.\n4. No 2×2 square may be entirely 0s or entirely 1s.", "image_path": "image194.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Others", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 195, "question": "As shown in the figure, in rhombus $ABCD$, $AB = 4$, $\\angle ABC = 60^\\circ$, $E$ is the midpoint of $BC$, and $F$ is a moving point on $BC$. The triangle $\\triangle ABF$ is folded along $AF$, and the corresponding point of $B$ after folding is $B'$. Connect $B'E$. What is the minimum value of the length of segment $B'E$?", "answer": "$4-2\\sqrt{3}$", "image_path": "image195.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 196, "question": "In the $4 \\times 4$ grid shown, three coins are randomly placed in different squares. Determine the probability that no two coins lie in the same row or column.", "answer": "$\\\\frac{6}{35}$", "image_path": "image196.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Probability & Statistics", "Geometry" ] } }, { "id": 197, "question": "Tom has these nine cards:\n\nHe places these cards on the board next to each other so that each horizontal line and each vertical line has three cards with the three different shapes and the three different amounts of drawings. He has already placed three cards, as shown in the picture. Which card should he place in the colored box?\n\n Options: A. A, B. B, C. C, D. D, E. E", "answer": "E", "image_path": "image197.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 198, "question": "Adam and Bruna try to find out which is Carla's favorite figure, amongst the figures beside. Adam knows that Carla told Bruna what the shape of the figure was. Bruna knows that Carla told Adam what color the figure was. The following conversation takes place. Adam: \"I don't know what Carla's favorite figure is and I know that Bruna doesn't know either\". Bruna: \"At first I didn't know what Carla's favorite figure was, but now I know\". Adam: \"Now I know too\". What is Carla's favorite figure?\n\n Options: A. A, B. B, C. C, D. D, E. E", "answer": "E", "image_path": "image198.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 199, "question": "There are rectangular cards divided into 4 equal cells with different shapes drawn in each cell. Cards can be placed side by side only if the same shapes appear in adjacent cells on their common side. 9 cards are used to form a rectangle as shown in the figure. Which of the following cards was definitely NOT used to form this rectangle?\n\n Options: A. A, B. B, C. C, D. D, E. E", "answer": "E", "image_path": "image199.jpg", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 200, "question": "Name the most likely mode of inheritance in the following pedigree: Options: A. autosomal recessive (AR), B. autosomal dominant (AD), C. X-linked recessive (XR), D. X-linked dominant (XD)", "answer": "B", "image_path": "image200.png", "annotated": { "difficulty_tier": "Hard", "subject": "Biology", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Probability & Statistics" ] } }, { "id": 201, "question": " In Ayrshire cattle, the mahogany/red locus is an autosomal gene. In both sexes, mahogany homozygotes are mahogany colored, and red homozygotes are red. However, the mahogany/red heterozygote is mahogany if male and red if female. Here is a pedigree for an Ayrshire family that is of great interest to Farmer MacDonald. (In the pedigree, open circles and squares indicate the mahogany phenotype; filled indicate red.) Although Elmer has long ago become a huge pile of Big Macs, Farmer MacDonald has frozen a good supply of Elmer's semen and has used it to artificially inseminate Elmira. What is the probability that Elmira will produce a mahogany female calf? Options: A. 1/2, B. 1/4, C. 3/4, D. 1/8, E. 1/16", "answer": "E", "image_path": "image201.png", "annotated": { "difficulty_tier": "Basic", "subject": "Biology", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Probability & Statistics" ] } }, { "id": 202, "question": "What is the coefficient of relatedness between individuals A and B in ? Options: A. 3/8, B. 5/8, C. 1/4, D. 1/2, E. 3/4", "answer": "A", "image_path": "image202.png", "annotated": { "difficulty_tier": "Hard", "subject": "Biology", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Probability & Statistics" ] } }, { "id": 203, "question": "The pedigree in shows the mode of inheritance of a human disease that is associated with mutations in Gene A. Note: Individuals marrying into this family DO NOT have the disease-associated allele unless shaded and the pedigree is completely penetrant. What is the genotype(s) of Individual 9? Options: A. AA, B. Aa, C. Aa or AA, D. XAXA, E. XAXa, F. XAXA or XAXa, G. XaXa", "answer": "E", "image_path": "image203.png", "annotated": { "difficulty_tier": "Hard", "subject": "Biology", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Probability & Statistics", "Logic" ] } }, { "id": 204, "question": "In an 8×8 grid where at most one chess piece can be placed in each cell, what is the maximum number of pieces that can be placed if the number of pieces in each of the 8 rows, 8 columns, and 30 diagonals (as illustrated below) must all be even?", "answer": "48", "image_path": "image204.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 205, "question": "As shown in the diagram, there is a map with five countries. The task is to color the map using four different colors such that no two adjacent countries share the same color, while non-adjacent countries may have the same color. How many different coloring methods are possible in total?", "answer": "96", "image_path": "image205.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry" ] } }, { "id": 206, "question": "The given picture is a maze, and the black lines represent walls that cannot be walked. Now you want to walk from the blue point to the red point. Is there a feasible path? If so, which of the green marks numbered 1-5 In the picture must be passed in the path?", "answer": "Yes, 3", "image_path": "image206.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Branch‑and-Bound", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 207, "question": "In the given image:\n1. The maze has a clearly marked entry point (black triangle pointing inward) and exit point (black triangle pointing outward);\n2. You are required to enter the maze at the entry point and exit at the exit point;\n3. You must follow the shortest valid path through the maze — that is, a path that does not revisit any part of the passage and does not pass through any walls;\n4. After identifying this path, color all the positions along it (including the start and end points) in black;\n5. The resulting black path will form a pixel-style image.\n\nYour task is:\n\nBased on the resulting pixel art image formed by the path, select the most appropriate description from a list of given options that best describes what the image looks like (e.g., a person, animal, object, etc.).\n\nA. Hamburger B. Ballet dancer C. Fish D. Horse E. Profile of a Viking F. Saxophone player", "answer": "D", "image_path": "image207.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Branch‑and-Bound", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 208, "question": "You are given a puzzle task:\n1. The red shape at the top of the image is the target shape that you need to assemble;\n2. Below it are several blue puzzle pieces, each labeled with a unique number;\n3. You may select any number of these pieces to complete the puzzle, but the following rules must be followed:\n\t- You are not allowed to rotate or flip any of the blue pieces;\n\t- Each piece can be used at most once;\n\t- You may move the pieces around, but they must not overlap;\n\t- The selected pieces must fit together exactly to form the red target shape;\n4. It is known that there is exactly one valid solution.\n\nYour task is:\n\nIdentify which puzzle pieces are used in the solution, and output their numbers in ascending order.", "answer": "[2, 3, 4, 5, 6]", "image_path": "image208.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry" ] } }, { "id": 209, "question": "In the diagram, there are three equilateral triangles. Fill the numbers 1 through 9, without repetition, into the nine circles located at the triangle vertices so that the sum of the three numbers at the vertices of each triangle is equal, and the sum of the four numbers on each straight line passing through four circles is also equal. How many valid arrangements are there in total?", "answer": "2", "image_path": "image209.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 210, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "A", "image_path": "image210.png", "annotated": { "difficulty_tier": "Medium", "subject": "Physics", "answer_type": "Multiple-choice questions", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Combinatorics" ] } }, { "id": 211, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image211.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 212, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "A", "image_path": "image212.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 213, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image213.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 214, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "C", "image_path": "image214.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 215, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "A", "image_path": "image215.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 216, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "C", "image_path": "image216.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 217, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image217.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 218, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "D", "image_path": "image218.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 219, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image219.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 220, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "G", "image_path": "image220.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 221, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "D", "image_path": "image221.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 222, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "A", "image_path": "image222.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 223, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image223.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 224, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "A", "image_path": "image224.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 225, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "A", "image_path": "image225.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 226, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "G", "image_path": "image226.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 227, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "B", "image_path": "image227.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry" ] } }, { "id": 228, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "C", "image_path": "image228.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 229, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "J", "image_path": "image229.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 230, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "F", "image_path": "image230.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 231, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "B", "image_path": "image231.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 232, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "B", "image_path": "image232.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 233, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "B", "image_path": "image233.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 234, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "C", "image_path": "image234.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Geometry" ] } }, { "id": 235, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "C", "image_path": "image235.png", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 236, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "J", "image_path": "image236.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 237, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "C", "image_path": "image237.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 238, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "A", "image_path": "image238.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 239, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "H", "image_path": "image239.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 240, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "C", "image_path": "image240.png", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 241, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "C", "image_path": "image241.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 242, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "C", "image_path": "image242.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 243, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "C", "image_path": "image243.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 244, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image244.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 245, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "I", "image_path": "image245.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 246, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "F", "image_path": "image246.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 247, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "A", "image_path": "image247.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 248, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "B", "image_path": "image248.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 249, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "I", "image_path": "image249.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 250, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "C", "image_path": "image250.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 251, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "D", "image_path": "image251.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 252, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image252.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 253, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "A", "image_path": "image253.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 254, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image254.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 255, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "I", "image_path": "image255.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 256, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "E", "image_path": "image256.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 257, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image257.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 258, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "D", "image_path": "image258.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry" ] } }, { "id": 259, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "H", "image_path": "image259.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 260, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "B", "image_path": "image260.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 261, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "G", "image_path": "image261.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 262, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "C", "image_path": "image262.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 263, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image263.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 264, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "J", "image_path": "image264.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 265, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "G", "image_path": "image265.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 266, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image266.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 267, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "A", "image_path": "image267.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 268, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image268.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 269, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "I", "image_path": "image269.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 270, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image270.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 271, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "D", "image_path": "image271.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 272, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "C", "image_path": "image272.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 273, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "I", "image_path": "image273.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 274, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D\nE. E\nF. F\nG. G\nH. H\nI. I\nJ. J", "answer": "C", "image_path": "image274.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 275, "question": "Sophie is at (0, 0) on a coordinate grid and would like to get to (3, 3). If Sophie is at (x, y), in a single step she can move to one of (x + 1, y), (x, y + 1), (x - 1, y + 1), or (x + 1, y - 1). She cannot revisit any points along her path, and neither her x-coordinate nor her y-coordinate can ever be less than 0 or greater than 3. Compute the number of ways for Sophie to reach (3, 3).", "answer": "2304", "image_path": "image275.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Geometry" ] } }, { "id": 276, "question": "Jerry places at most one rook in each cell of a 2025 \\times 2025 grid of cells. A rook attacks another rook if the two rooks are in the same row or column and there are no other rooks between them. Determine, with proof, the maximum number of rooks Jerry can place on the grid such that no rook attacks 4 other rooks.", "answer": "8096", "image_path": "image276.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 277, "question": "Compute the number of ways to divide a 20 \\times 24 rectangle into 4 \\times 5 rectangles. (Rotations and reflections are considered distinct.)", "answer": "6", "image_path": "image277.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 278, "question": "Sally the snail sits on the 3 \\times 24 lattice of points (i, j) for all 1 ≤ i ≤ 3 and 1 ≤ j ≤ 24. She wants to visit every point in the lattice exactly once. In a move, Sally can move to a point in the lattice exactly one unit away. Given that Sally starts at (2, 1), compute the number of possible paths Sally can take.", "answer": "4096", "image_path": "image278.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 279, "question": "A peacock is a ten-digit positive integer that uses each digit exactly once. Compute the number of peacocks that are exactly twice another peacock.", "answer": "184320", "image_path": "image279.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic", "Number Theory" ] } }, { "id": 280, "question": "Compute the number of ways to select 99 cells of a 19 \\times 19 square grid such that no two selected cells share an edge or vertex.", "answer": "1000", "image_path": "image280.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 281, "question": "Farmer James wishes to cover a circle with circumference 10π with six different types of colored arcs. Each type of arc has radius 5, has length either π or 2π, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to completely cover his circle without overlap, subject to the following conditions: • Any two adjacent arcs are of different colors. • Any three adjacent arcs where the middle arc has length π are of three different colors. Find the number of distinct ways Farmer James can cover his circle. Here, two coverings are equivalent if and only if they are rotations of one another. In particular, two colorings are considered distinct if they are reflections of one another, but not rotations of one another.", "answer": "93", "image_path": "image281.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 282, "question": "How many ways can one fill a 3 \\\times 3 square grid with nonnegative integers such that no nonzero integer appears more than once in the same row or column and the sum of the numbers in every row and column equals 7?", "answer": "216", "image_path": "image282.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 283, "question": "Fred the Four-Dimensional Fluffy Sheep is walking in 4-dimensional space. He starts at the origin. Each minute, he walks from his current position (a1, a2, a3, a4) to some position (x1, x2, x3, x4) with integer coordinates satisfying $(x_1-a_1)^2+(x_2-a_2)^2+(x_3-a_3)^2+(x_4-a_4)^2=4\\mathrm{~and~}\\quad|(x_1+x_2+x_3+x_4)-(a_1+a_2+a_3+a_4)|=2.$. In how many ways can Fred reach (10, 10, 10, 10) after exactly 40 minutes, if he is allowed to pass through this point during his walk?", "answer": "$\\binom{40}{10}\\binom{40}{20}^3$", "image_path": "image283.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 284, "question": "Sarah stands at (0, 0) and Rachel stands at (6, 8) in the Euclidean plane. Sarah can only move 1 unit in the positive x or y direction, and Rachel can only move 1 unit in the negative x or y direction. Each second, Sarah and Rachel see each other, independently pick a direction to move at the same time, and move to their new position. Sarah catches Rachel if Sarah and Rachel are ever at the same point. Rachel wins if she is able to get to (0, 0) without being caught; otherwise, Sarah wins. Given that both of them play optimally to maximize their probability of winning, what is the probability that Rachel wins?", "answer": "\\frac{63}{64}", "image_path": "image284.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Probability & Statistics" ] } }, { "id": 285, "question": "Lily has a 300 \\\times 300 grid of squares. She now removes 100 \\\times 100 squares from each of the four corners and colors each of the remaining 50000 squares black and white. Given that no 2 \\\times 2 square is colored in a checkerboard pattern, find the maximum possible number of (unordered) pairs of squares such that one is black, one is white and the squares share an edge.", "answer": "49998", "image_path": "image285.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 286, "question": "There are 2017 jars in a row on a table, initially empty. Each day, a nice man picks ten consecutivejars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that N of the jars all contain the same positive integer number of coins (i.e. there is an integer d > 0 such that N of the jars have exactly d coins). What is the maximum possible value of N ?", "answer": "2014", "image_path": "image286.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Probability & Statistics", "Logic", "Combinatorics" ] } }, { "id": 287, "question": "Kristoff is planning to transport a number of indivisible ice blocks with positive integer weights from the north mountain to Arendelle. He knows that when he reaches Arendelle, Princess Anna and Queen Elsa will name an ordered pair (p, q) of nonnegative integers satisfying p + q ≤ 2016. Kristoff must then give Princess Anna exactly p kilograms of ice. Afterward, he must give Queen Elsa exactly q kilograms of ice. What is the minimum number of blocks of ice Kristoff must carry to guarantee that he can always meet Anna and Elsa's demands, regardless of which p and q are chosen?", "answer": "18", "image_path": "image287.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 288, "question": "Let n be an odd positive integer, and suppose that n people sit on a committee that is in the process of electing a president. The members sit in a circle, and every member votes for the person either to his/her immediate left, or to his/her immediate right. If one member wins more votes than all the other members do, he/she will be declared to be the president; otherwise, one of the the members who won at least as many votes as all the other members did will be randomly selected to be the president. If Hermia and Lysander are two members of the committee, with Hermia sitting to Lysander's left and Lysander planning to vote for Hermia, determine the probability that Hermia is elected president, assuming that the other n - 1 members vote randomly.", "answer": "$\\frac{2^n-1}{n2^{n-1}}$", "image_path": "image288.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Probability & Statistics", "Combinatorics", "Geometry" ] } }, { "id": 289, "question": "Compute the number of ways to pick two rectangles in a 5 \\times 5 grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not matter.", "answer": "56", "image_path": "image289.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 290, "question": "In an 11 \\times 11 grid of cells, each pair of edge-adjacent cells is connected by a door. Karthik wants towalk a path in this grid. He can start in any cell, but he must end in the same cell he started in, and he cannot go through any door more than once (not even in opposite directions). Compute the maximum number of doors he can go through in such a path.", "answer": "200", "image_path": "image290.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 291, "question": "Compute the number of ways to arrange 3 copies of each of the 26 lowercase letters of the English alphabet such that for any two distinct letters x_1 and x_2, the number of x_2's between the first and second occurrences of x_1 equals the number of x_2's between the second and third occurrences of x_1.", "answer": "2^{25} \\cdot 26!", "image_path": "image291.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Probability & Statistics", "Combinatorics" ] } }, { "id": 292, "question": "The circumference of a circle is divided into 45 arcs, each of length 1. Initially, there are 15 snakes, each of length 1, occupying every third arc. Every second, each snake independently moves either one arc left or one arc right, each with probability \\frac{1}{2} . If two snakes ever touch, they merge to form a single snake occupying the arcs of both of the previous snakes, and the merged snake moves as one snake. Compute the expected number of seconds until there is only one snake left.", "answer": "\\frac{448}{3}", "image_path": "image292.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Probability & Statistics" ] } }, { "id": 293, "question": "There is a grid of height 2 stretching infinitely in one direction. Between any two edge-adjacent cells of the grid, there is a door that is locked with probability \\frac{1}{2} independent of all other doors. Philip starts in a corner of the grid (in the starred cell). Compute the expected number of cells that Philip can reach, assuming he can only travel between cells if the door between them is unlocked.", "answer": "\\frac{32}{7}", "image_path": "image293.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics", "Probability & Statistics", "Logic" ] } }, { "id": 294, "question": "Elbert and Yaiza each draw 10 cards from a 20-card deck with cards numbered 1, 2, 3, . . . , 20. Then, starting with the player with the card numbered 1, the players take turns placing down the lowestnumbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends. Given that Yaiza lost and 5 cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player's hand does not matter.)", "answer": "324", "image_path": "image294.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 295, "question": "Teresa the bunny has a fair 8-sided die. Seven of its sides have fixed labels 1, 2, ... , 7, and the label on the eighth side can be changed and begins as 1. She rolls it several times, until each of 1, 2, ... , 7 appears at least once. After each roll, if k is the smallest positive integer that she has not rolled so far, she relabels the eighth side with k. The probability that 7 is the last number she rolls is \\frac{a}{b} , where a and b are relatively prime positive integers. Compute 100a+b", "answer": "104", "image_path": "image295.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Probability & Statistics", "Logic" ] } }, { "id": 296, "question": "The integers 1, 2, . . . , 64 are written in the squares of a 8 \\times 8 chess board, such that for each 1 ≤ i < 64, the numbers i and i + 1 are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals?", "answer": "432", "image_path": "image296.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Probability & Statistics", "Combinatorics", "Logic" ] } }, { "id": 297, "question": "In the figure below, how many ways are there to select 5 bricks, one in each row, such that any two bricks in adjacent rows are adjacent?", "answer": "61", "image_path": "image297.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 298, "question": "Dizzy Daisy is standing on the point (0, 0) on the xy-plane and is trying to get to the point (6, 6). She starts facing rightward and takes a step 1 unit forward. On each subsequent second, she either takes a step 1 unit forward or turns 90 degrees counterclockwise then takes a step 1 unit forward. She may never go on a point outside the square defined by |x| ≤ 6, |y| ≤ 6, nor may she ever go on the same point twice. How many different paths may Daisy take?", "answer": "131922", "image_path": "image298.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 299, "question": "A parking lot consists of 2012 parking spots equally spaced in a line, numbered 1 through 2012. Oneby one, 2012 cars park in these spots under the following procedure: the first car picks from the 2012 spots uniformly randomly, and each following car picks uniformly randomly among all possible choices which maximize the minimal distance from an already parked car. What is the probability that the last car to park must choose spot 1?", "answer": "\\frac{1}{2062300}", "image_path": "image299.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Divide‑and-Conquer" ], "sub_categories": [ "Probability & Statistics", "Combinatorics" ] } }, { "id": 300, "question": "Alice and Bob play a game in which two thousand and eleven 2011 \\times 2011 grids are distributed between the two of them, 1 to Bob, and the other 2010 to Alice. They go behind closed doors and fill their grid(s) with the numbers 1, 2, . . . , 20112 so that the numbers across rows (left-to-right) and down columns (top-to-bottom) are strictly increasing. No two of Alice's grids may be filled identically. After the grids are filled, Bob is allowed to look at Alice's grids and then swap numbers on his own grid, two at a time, as long as the numbering remains legal (i.e. increasing across rows and down columns) after each swap. When he is done swapping, a grid of Alice's is selected at random. If there exist two integers in the same column of this grid that occur in the same row of Bob's grid, Bob wins. Otherwise, Alice wins. If Bob selects his initial grid optimally, what is the maximum number of swaps that Bob may need in order to guarantee victory?", "answer": "1", "image_path": "image300.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch‑and-Bound", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Probability & Statistics", "Combinatorics" ] } }, { "id": 301, "question": "Manya has a stack of 85 = 1 + 4 + 16 + 64 blocks comprised of 4 layers (the kth layer from the top has 4^{k-1} blocks; see the diagram below). Each block rests on 4 smaller blocks, each with dimensions half those of the larger block. Laura removes blocks one at a time from this stack, removing only blocks that currently have no blocks on top of them. Find the number of ways Laura can remove precisely 5 blocks from Manya's stack (the order in which they are removed matters).", "answer": "3384", "image_path": "image301.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics", "Probability & Statistics", "Logic" ] } }, { "id": 302, "question": "The squares of a 3 \\times 3 grid are filled with positive integers such that 1 is the label of the upperleftmost square, 2009 is the label of the lower-rightmost square, and the label of each square divides the one directly to the right of it and the one directly below it. How many such labelings are possible?", "answer": "2448", "image_path": "image302.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 303, "question": "A 3 \\times 3 \\times 3 cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a 3 \\times 3 \\times 1 block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a 45 degree angle with the horizontal plane.", "answer": "60", "image_path": "image303.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics" ] } }, { "id": 304, "question": "On the Cartesian grid, Johnny wants to travel from (0, 0) to (5, 1), and he wants to pass through all twelve points in the set S = {(i, j) | 0 ≤ i ≤ 1, 0 ≤ j ≤ 5, i, j ∈ Z}. Each step, Johnny may go from one point in S to another point in S by a line segment connecting the two points. How many ways are there for Johnny to start at (0, 0) and end at (5, 1) so that he never crosses his own path?", "answer": "252", "image_path": "image304.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 305, "question": "In how many ways can we enter numbers from the set {1, 2, 3, 4} into a 4 \\times 4 array so that all of the following conditions hold? (a) Each row contains all four numbers. (b) Each column contains all four numbers. (c) Each “quadrant” contains all four numbers. (The quadrants are the four corner \\times 2 squares.)", "answer": "288", "image_path": "image305.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 306, "question": "Eight coins are arranged in a circle heads up. A move consists of flipping over two adjacent coins. How many different sequences of six moves leave the coins alternating heads up and tails up?", "answer": "7680", "image_path": "image306.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 307, "question": "The numbers 1, 2, ..., 8 are placed in the 3 × 3 grid below, leaving exactly one blank square. Such a placement is called okay if in every pair of adjacent squares, either one square is blank or the difference between the two numbers is at most 2 (two squares are considered adjacent if they share a common side). If reflections, rotations, etc. of placements are considered distinct, compute the number of distinct okay placements.", "answer": "32", "image_path": "image307.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch‑and-Bound", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 308, "question": "Derek starts at the point (0, 0), facing the point (0, 1), and he wants to get to the point (1, 1). He takes unit steps parallel to the coordinate axes. A move consists of either a step forward, or a 90 degree right (clockwise) turn followed by a step forward, so that his path does not contain any left turns. His path is restricted to the square region defined by 0 ≤ x ≤ 17 and 0 ≤ y ≤ 17. Compute the number of ways he can get to (1, 1) without returning to any previously visited point.", "answer": "529", "image_path": "image308.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Number Theory", "Geometry" ] } }, { "id": 309, "question": "Let ABC be a triangle with m∠B = m∠C = 80◦. Compute the number of points P in the plane such that triangles PAB, PBC, and PCA are all isosceles and non-degenerate.", "answer": "6", "image_path": "image309.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 310, "question": "T-10. Let $S = \\{1, 2, \\dots, 20\\}$, and let $f$ be the function from $S$ to $S$ as the example shown in the figure below. Define the sequence $s_1, s_2, s_3, \\dots$ by setting $$s_n = \\sum_{k=1}^{20} \\underbrace{(f \\circ \\cdots \\circ f)}_{n \\text{ times}}(k)$$ That is, $s_1 = f(1) + \\dots + f(20)$, $s_2 = f(f(1)) + \\dots + f(f(20))$, $s_3 = f(f(f(1))) + f(f(f(2))) + \\dots + f(f(f(20)))$, etc. Compute the smallest integer $p$ such that the following statement is true: The sequence $s_1, s_2, s_3, \\dots$ must be periodic after a certain point, and its period is at most $p$. (If the sequence is never periodic, then write $\\infty$ as your answer.)", "answer": "140", "image_path": "image310.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Algebra", "Combinatorics" ] } }, { "id": 311, "question": "Let W = (0, 0), A = (7, 0), S = (7, 1), and H = (0, 1). Compute the number of ways to tile rectangle WASH with triangles of area 1/2 and vertices at lattice points on the boundary of WASH.", "answer": "3432", "image_path": "image311.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 312, "question": "To solve a KenKen puzzle, you fill in an n × n grid with the digits 1, . . . , n according to the following two rules: \n1. Each row and column contains exactly one of each digit. \n2. Each bold-outlined group of cells is a cage containing digits which achieve the specified result using the specified mathematical operation: addition (+), subtraction (-), multiplication (×), and division (÷) on the digits in some order. Digits may repeat inside a cage. A solved 3×3 KenKen appears to the left. Now there is a 5×5 KenKen puzzle. On your answer sheet, enter the digits in the starred squares in the correctly solved KenKen puzzle, in order from left to right. In the example, you would enter 1, 2, 3. Digits may appear more than once in the starred squares.", "answer": "2,1,4,2,2", "image_path": "image312.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Number Theory", "Logic" ] } }, { "id": 313, "question": "A n-sided die has the integers between 1 and n (inclusive) on its faces. A roll refers to the value that shows on the topmost face of the die after it is thrown. All values on the faces of the die are equally likely to be rolled. For the following question, we use the notation: for a positive integer n, let $R_{1},R_{2},\\dots,R_{2n}$ denote $2n$ rolls of a six-sided die. We define $P_{1},P_{2},\\dots,P_{n}$ by $P_{i}=R_{2i-1}\\times R_{2i}$, the product of the $i^{\\text{th}}$ pair of rolls. A six-sided die is thrown 12 times. For $3\\le k\\le6$, $P_{k}$ satisfies the Fibonacci recurrence $P_{k}=P_{k-1}+P_{k-2}$. Compute the number of possible values of $P_{6}$.", "answer": "3", "image_path": "image313.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Algebra", "Number Theory", "Logic" ] } }, { "id": 314, "question": "Compute the number of distinct ways to color the nine triangles in the figure below either red, white, or blue such that no two triangles that share a side are the same color.", "answer": "528", "image_path": "image314.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 315, "question": "In chess, a knight can move either two squares horizontally and one square vertically, or two squares vertically and one square horizontally. Unlike all other standard chess pieces, the knight can ‘jump over’ all other pieces (of either color) to its destination square. Compute the minimum number of moves to exchange the positions of the two white and black knights as shown in the graphic below. Two knights may not occupy the same square at the same time. Alternating black and white knight moves is not required.", "answer": "16", "image_path": "image315.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 316, "question": "In chess, a knight can move either two squares horizontally and one square vertically, or two squares vertically and one square horizontally. Unlike all other standard chess pieces, the knight can ‘jump over’ all other pieces (of either color) to its destination square. Compute the minimum number of moves to exchange the positions of the two white and black knights as shown in the graphic below. Two knights may not occupy the same square at the same time. Alternating black and white knight moves is not required.", "answer": "14", "image_path": "image316.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 317, "question": "A hexaknight can move either two squares horizontally, or two squares vertically and one square horizontally. Consider the path formed by joining in order the centers of the squares visited by a knight. The path is simple if none of the edges cross in their interiors and is closed if the knight returns to the square where it began. In the board below, the white knight’s path is simple but not closed, while the black knight’s path is closed but not simple. If the centers of adjacent squares are one unit apart, compute the maximum area enclosed by a knight’s simple closed path on an 8 × 8 board.", "answer": "37", "image_path": "image317.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 318, "question": "In chess, a knight can move either two squares horizontally and one square vertically, or two squares vertically and one square horizontally. The graphic below shows the eight possible locations to which the knight in the center of the 5 5 board can move. Unlike all other standard chess pieces, the knight can ‘jump over’ all other pieces (of either color) to its destination square. This question is an estimation problem. If the answer given is within 10% of the correct answer, your team will receive credit. A knight is on a square on an infinite chess board. Compute the number of distinct squares where the knight can end up after exactly 10 moves.", "answer": "741, thus any answer between 666.9 and 815.1 is considered correct.", "image_path": "image318.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Others", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 319, "question": "In this game, there is an area ten squares wide and a number of squares tall. Pieces chosen randomly from among the seven “tetrominoes” made up of four squares glued together, as shown below, fall from the top of the screen. As the pieces fall, the player may rotate them or slide them left or right, but once they touch a piece below them they stick in place. If the player is able to fit the pieces together so as to leave no gaps in a row, that row disappears and all the blocks above fall to leave more room for new blocks. Otherwise the screen fills up with blocks and the game ends. If the puzzle is ten squares wide, in the pattern with only two squares remaining at the bottom line, how many types can be achieved by eliminating three lines? The picture at the bottom shows one of the possible solutions.", "answer": "11", "image_path": "image319.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 320, "question": "In this game, there is an area ten squares wide and a number of squares tall. Pieces chosen randomly from among the seven “tetrominoes” made up of four squares glued together, as shown below, fall from the top of the screen. As the pieces fall, the player may rotate them or slide them left or right, but once they touch a piece below them they stick in place. If the player is able to fit the pieces together so as to leave no gaps in a row, that row disappears and all the blocks above fall to leave more room for new blocks. Otherwise the screen fills up with blocks and the game ends. If the puzzle is ten squares wide, in the pattern with only two squares remaining at the bottom line. How many possible numbers of rows can be eliminated? ", "answer": "2", "image_path": "image320.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 321, "question": "In this game, there is an area ten squares wide and a number of squares tall. Pieces chosen randomly from among the seven “tetrominoes” made up of four squares glued together, as shown below, fall from the top of the screen. As the pieces fall, the player may rotate them or slide them left or right, but once they touch a piece below them they stick in place. If the player is able to fit the pieces together so as to leave no gaps in a row, that row disappears and all the blocks above fall to leave more room for new blocks. Otherwise the screen fills up with blocks and the game ends. If the puzzle is ten squares wide, in the pattern with only two squares remaining at the bottom line, how many types can be achieved simply by eliminating one line? The picture at the bottom shows one of the possible solutions.", "answer": "34", "image_path": "image321.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 322, "question": "In this game, there is an area ten squares wide and a number of squares tall. Pieces chosen randomly from among the seven “tetrominoes” made up of four squares glued together, as shown below, fall from the top of the screen. As the pieces fall, the player may rotate them or slide them left or right, but once they touch a piece below them they stick in place. If the player is able to fit the pieces together so as to leave no gaps in a row, that row disappears and all the blocks above fall to leave more room for new blocks. Otherwise the screen fills up with blocks and the game ends. If the puzzle is ten squares wide, in the pattern with only two squares left at the bottom line, There are 11 types solusion can be achieved by eliminating three line. If you try to achive this situation but without using the “T”-shaped block, how many are solusions can be achieved?", "answer": "6", "image_path": "image322.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 323, "question": "In this game, there is an area ten squares wide and a number of squares tall. Pieces chosen randomly from among the seven “tetrominoes” made up of four squares glued together, as shown below, fall from the top of the screen. As the pieces fall, the player may rotate them or slide them left or right, but once they touch a piece below them they stick in place. If the player is able to fit the pieces together so as to leave no gaps in a row, that row disappears and all the blocks above fall to leave more room for new blocks. Otherwise the screen fills up with blocks and the game ends. If the puzzle is ten squares wide, in the pattern with only two squares left at the bottom line, There are 34 types solusion can be achieved simply by eliminating one line. If you try to achive this situation but without using the “T”-shaped block, how many are solusions can be achieved?", "answer": "19", "image_path": "image323.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 324, "question": "In this game, there is an area ten squares wide and a number of squares tall. Pieces chosen randomly from among the seven “tetrominoes” made up of four squares glued together, as shown below, fall from the top of the screen. As the pieces fall, the player may rotate them or slide them left or right, but once they touch a piece below them they stick in place. If the player is able to fit the pieces together so as to leave no gaps in a row, that row disappears and all the blocks above fall to leave more room for new blocks. Otherwise the screen fills up with blocks and the game ends. If the puzzle is ten squares wide, in the pattern with only two squares left at the bottom line, there are 11 types of solutions that can be achieved simply by eliminating three lines. If you try to achive this situation but without using the “T”-shaped block, how many are solusions can be achieved? Among the three situations shown in the following figure, how many can be confirmed to have been successfully eliminated?", "answer": "3", "image_path": "image324.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 325, "question": "Identify the right image for the empty slot '?'", "answer": "B", "image_path": "image325.png", "annotated": { "difficulty_tier": "Basic", "subject": "Physics", "answer_type": "Multiple-choice questions", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic" ] } }, { "id": 326, "question": "The Planar National Park is a subset of the Euclidean plane consisting of several trails which meet at junctions. Every trail has its two endpoints at two different junctions whereas each junction is the endpoint of exactly three trails. Trails only intersect at junctions (in particular, trails only meet at endpoints). Finally, no trails begin and end at the same two junctions. (An example of one possible layout of the park is shown to the left below, in which there are six junctions and nine trails.)A visitor walks through the park as follows: she begins at a junction and starts walking along a trail. At the end of that first trail, she enters a junction and turns left. On the next junction she turns right, and so on, alternating left and right turns at each junction. She does this until she gets back to the junction where she started. What is the largest possible number of times she could have entered any junction during her walk, over all possible layouts of the park?", "answer": "3", "image_path": "image326.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Geometry", "Combinatorics" ] } }, { "id": 327, "question": "Let $n \\geq 2$ be an integer. An $n \times n$ board is initially empty. Each minute, you may perform one of three moves:\nIf there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells.\nIf all cells in a column have a stone, you may remove all stones from that column.\nIf all cells in a row have a stone, you may remove all stones from that row. For which $n$ is it possible that, after some non-zero number of moves, the board has no stones?", "answer": "\\boxed{n \\bmod 3 = 0}", "image_path": "image327.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic", "Number Theory" ] } }, { "id": 328, "question": "Let $$n$$ be a positive integer. A Japanese triangle consists of $$1 + 2 + \\dots + n$$ circles arranged in an equilateral triangular shape such that for each $i = 1$, $2$, $\\dots$, $n$, the $$i^{th}$$ row contains exactly $$i$$ circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of $$n$$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with $n = 6$, along with a ninja path in that triangle containing two red circles.In terms of $n$, find the greatest $$k$$ such that in each Japanese triangle there is a ninja path containing at least $$k$$ red circles.", "answer": "k = \\lfloor \\log_2 n \\rfloor + 1", "image_path": "image328.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 329, "question": "Each square of a $33\times 33$ square grid is colored in one of the three colors: red, yellow or blue, such that the numbers of squares in each color are the same. If two squares sharing a common edge are in different colors, call that common edge a separating edge. Find the minimal number of separating edges in the grid.", "answer": "56", "image_path": "image329.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 330, "question": "From the four given options, select the most suitable one to fill in the question mark to present a certain regularity.\nChoices: A. A\nB. B\nC. C\nD. D", "answer": "B", "image_path": "image330.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 331, "question": "Let $n$ be a positive integer. A Nordic square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:\n(i) the first cell in the sequence is a valley,\n(ii) each subsequent cell in the sequence is adjacent to the previous cell, and \n(iii) the numbers written in the cells in the sequence are in increasing order. Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square.", "answer": "2n(n-1)+1", "image_path": "image331.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Probability & Statistics", "Combinatorics", "Logic" ] } }, { "id": 332, "question": "Let n be a positive integer satisfying the following property: If n dominoes are placed on a 6 x 6 chessboard with each domino covering exactly two unit squares, then one can always place one more domino on the board without moving any other dominoes. Determine the maximum value of n.", "answer": "11", "image_path": "image332.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 333, "question": "Let $n$ be a positive integer. We are given an $n \\times n$ chessboard and $n$ rooks. Let A and B be two points on the edges of the squares determining the chessboard. A path connecting A and B is a continuous curve along the edges of the squares of the chessboard with A and B as endpoints. The length of the curve is the length of the path. Claudia is asked to complete the following task. First, she places all of the rooks on the chessboard so that the rooks cannot attack each other. Next, she draws a path of length $2n$ connecting the top left corner to the bottom right corner of the chessboard so that all of the rooks are on the same side of the path. In how many different ways can Claudia complete this task? (Two different paths for the same placement of rooks are considered as different.) ", "answer": "2 x (2n- 1)!!", "image_path": "image333.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 334, "question": "Mr. Fat is baking $m$ different cakes with different kinds of cake mix. Some of the kinds of cake mix are sweetened. Each cake is made from five different kinds of mix with at least one kind of sweetened mix among them. It is known that for every three kinds of mix there is exactly one cake containing them. If there exists at least one very sweet cake: a cake made from at least four kinds of sweetened mix, compute the minimum value of $m$.", "answer": "68", "image_path": "image334.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 335, "question": "The bottom right unit square is removed from a 2 x 2 unit grid to form a tromino. Find the number of ways that k trominoes can be placed, without being rotated and without overlapping, on a 3 x n rectangular unit grid. (Each tromino covers exactly three unit squares on the grid.)", "answer": "$\\binom{2n-2k}{k}$", "image_path": "image335.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 336, "question": "We say a polygon is orthogonal if all its angles are of 90 or 270 degrees. Firstly, we give it a chess- board coloring and let $b$ and $w$ as the number of black and white squares, respectively. Then let $B$ and $W$ be the length of the boundary that ends up painted black and the length of the boundary that ends up painted white, respectively. \n(1) Could an orthogonal polygon whose sides are all of odd integer length be tiled with 2 × 1 domino tiles? Just answer yes or no. \n(2) Determine the relation of $b$, $w$, $B$, $W$.", "answer": "(1) no (2) 4(b - w) = B - W", "image_path": "image336.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic", "Geometry", "Combinatorics" ] } }, { "id": 337, "question": "Find all pairs $(m,n)$ such that an $m\\times n$ board can be tiled with the following base tile: (Note: The tile can be rotated and flipped upside down.)", "answer": "$m\\times n$ is a multiple of 12, at least one of them is divisible by 4 and neither of them is 1, 2 or 5", "image_path": "image337.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Others", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Logic", "Number Theory" ] } }, { "id": 338, "question": "Question: (1) If a $5 \\times n$ board can be completely tiled with the given L-shaped tiles, then $n$ must satisfy that ___ (2) Determine the smallest number of ways to tile a $5 \\times 2k$ board with these tiles", "answer": "(1) n is even; (2) $2\\times 3^{k-1}$", "image_path": "image338.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 339, "question": "In a bridge tournament, $110$ teams play $6$ rounds. In each round, the teams are split into $55$ pairs, with each pair playing one match. No two teams play more than once. (1) What is the number of teams you can find such that no two of them have ever played each other? (2) If team number can be denoted as $6k + 2$ (or more), what the answer?", "answer": "(1) 19; (2) $k+1$", "image_path": "image339.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Number Theory", "Combinatorics" ] } }, { "id": 340, "question": "In a country some cities are connected by roads. We know that, using these roads, we can get from any city to any other city, although it may not be directly. Denote by $t$ the smallest possible integer for which there is a city from which it is possible to get to any other city using at most $t$ roads. Give the necessary and sufficient condition (about $i$ and $j$) for the existence of cities $A_1, A_2, \\dots, A_{2t-1}$ such that for any $1 \\leq i < j \\leq 2t-1$, there is a road between $A_i$ and $A_j$.", "answer": "$i+1=j$", "image_path": "image340.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Number Theory", "Geometry" ] } }, { "id": 341, "question": "A tower path in a rectangular board of unit squares is a path made by a sequence of movements parallel to the sides of the board from one unit square to one of its neighbors, in which every movement begins where the last one ended and so that no movement crosses a square that was previously visited by a movement of the path. That is, a tower path does not intersect itself. Let $R(m,n)$ be the number of tower paths in an $m\\times n$ board (m rows, n columns) that start in the bottom left corner and end in the top left corner. Find a formula of $R(3,n)$ for every positive integer $n$.", "answer": "$R(3,n)=\\frac{(1+\\sqrt{2})^{n+1}-(1-\\sqrt{2})^{n+1}}{2\\sqrt{2}}-1$", "image_path": "image341.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 342, "question": "A finite number of coins are arranged in some points with integer coordinates $(x, y)$ on the Cartesian plane. Only one coin can be placed at one point. Coins may be moved and removed according to the following rules. Let us consider three points $A$, $B$, and $C$ with integer coordinates such that the following conditions are satisfied: \n(C1) The points $A$, $B$,and $C$ belong to the same horizontal line or to the same vertical line; the point B is the midpoint of the segment $AC$; the distance between the points $A$ and $C$ is equal to 2.\n (C2) There is a coin on points $A$ and $B$, and there is no coin on point $C$. Then, it is allowed to move the coin from point $A$ to point $C$, and remove the coin from point $B$ at the same time. If a coin is placed on the point whose coordinates are $(x,y)$, where $y=k$, we say that this coin is at the level $k$. Suppose that a finite number of coins are arranged on some points with integer coordinates, and that all these points are on the $x$-axis or below the $x$-axis. The player chooses how many coins will be used, and the points where these coins will be placed at the beginning of the game. The goal of the game is to put a coin at the highest possible level. Question: \n(1) How many coins and steps are needed to reache level 2?\n(2) How many coins and steps are needed to reache level 3?\n(3) How many coins and steps are needed to reache level 4?\n(4) How many coins and steps are needed to reache level 5? NOTE: If level $k$ can be reached, answer with numbers. If not, just answer 'that is impossible'", "answer": "(1) 4 coins, 3 steps; (2) 8 coins, 7 steps; (3) 20 coins, 19 steps; (4) that is impossible.", "image_path": "image342.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 343, "question": "Every city in a state is connected to exactly three other cities by direct air flights. One can fly from each city to any other city with at least one stop. Determine the maximal number of cities in the state.", "answer": "10", "image_path": "image343.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 344, "question": "A chessboard consists of 64 squares (fields) arranged in eight rows and eight columns. The squares are arranged in two alternating colors (light and dark). The columns of the chessboard are denoted by $a$, $b$, $c$, $d$, $e$, $f$, $g$, and $h$, and the rows by 1, 2, ..., 8. The chess pieces are: king, queen, rooks, bishops, knights, and pawns. Twenty-one rectangles $3 \\times 1$ are placed on a chessboard $8 \\times 8$ such that only one field of the chessboard is not covered. Determine all the fields of the chessboard that can appear uncovered this way.", "answer": "$c3$, $f3$, $c6$, and $f6$", "image_path": "image344.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 345, "question": "A chessboard consists of 64 squares (fields) arranged in eight rows and eight columns. The squares are arranged in two alternating colors (light and dark). The columns of the chessboard are denoted by $a$, $b$, $c$, $d$, $e$, $f$, $g$, and $h$, and the rows by 1, 2, ..., 8. The chess pieces are: king, queen, rooks, bishops, knights, and pawns. In following problems, we assume that the chess pieces can move and attack each other according to chess rules. Question: \n (1) How many ways can a black and a white knight be placed on a chessboard, such that they do not attack each other? \n(2) How many ways can twenty chips be placed on the fields of a square table $8\\times 8$, such that the arrangement of the chips remains the same after rotating the table, if the angle of rotation is $\\alpha \\in {90◦,180◦,270◦}$?", "answer": "(1) 3796; (2) C(16,5)=4368", "image_path": "image345.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 346, "question": "On an infinite chessboard a game is played as follows. At the start, $n^2$ pieces are arranged on the chessboard in an $n \\times n$ block of adjoining squares, one piece in each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece which has been jumped over is removed. Find those values of $n$ for which the game can end with only one piece remaining on the board.", "answer": "n is not divisible by 3 ($3 \\nmid n$)", "image_path": "image346.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Others", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 347, "question": "Consider an $n \\times n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^2$ unit squares. We say that two different squares on the board are adjacent if they have a common side. $N$ unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square. Determine the smallest possible value of $N$.", "answer": "$\\frac{n(n+2)}{4}$", "image_path": "image347.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 348, "question": "Let $n \\geq 2$ be an integer. Consider an $n \\times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for each peaceful configuration of $n$ rooks there is a $k \\times k$ square which does not contain a rook on any of its $k^2$ unit squares.", "answer": "$k = \\left\\lfloor \\sqrt{n}-1 \\right\\rfloor$", "image_path": "image348.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 349, "question": "On a $5 \\times 5$ board, two players alternately mark numbers on empty cells. The first player always marks 1's, the second 0's. One number is marked per turn, until the board is filled. For each of the nine $3 \\times 3$ squares the sum of the nine numbers on its cells is computed. Denote by A the maximum of these sums. How large can the first player make A, regardless of the responses of the second player?", "answer": "6", "image_path": "image349.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 350, "question": "Let $n$ be an even positive integer. We say that two different cells of an $n \\times n$ board are neighboring if they have a common side. Find the minimal number of cells on the $n \\times n$ board that must be marked so that every cell (marked or not marked) has a marked neighboring cell.", "answer": "$n(n+2)/4$", "image_path": "image350.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Number Theory" ] } }, { "id": 351, "question": "Let $n$ and $k$ be positive integers such that $n/2 < k \\le 2n/3$. Find the least number $m$ for which it is possible to place $m$ pawns on $m$ squares of an $n \\times n$ chessboard so that no column or row contains a block of $k$ adjacent unoccupied squares.", "answer": "$4(n-k)$", "image_path": "image351.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 352, "question": "A box is a rectangle in the plane whose sides are parallel to the coordinate axes and have positive lengths. Two boxes intersect if they have a common point in their interior or on their boundary. Find the largest $n$ for which there exist $n$ boxes $B_1, \\dots, B_n$ such that $B_i$ and $B_j$ intersect if and only if $i \\ne j \\pm 1 \\pmod{n}$.", "answer": "6", "image_path": "image352.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 353, "question": "On a $999 \\times 999$ board a limp rook can move in the following way: From any square it can move to any of its adjacent squares, i.e., a square having a common side with it, and every move must be a turn: i.e., the directions of any two consecutive moves must be perpendicular. A nonintersecting route of the limp rook consists of a sequence of distinct squares that the limp rook can visit in that order by an admissible sequence of moves. Such a nonintersecting route is called cyclic if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, nonintersecting route of a limp rook visit?", "answer": "996000", "image_path": "image353.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 354, "question": "n($n \\geq 5$) football teams participate in a single round-robin tournament. Each two teams play once, and the winner gets three points, the loser gets zero points, and both get one point in case of a draw. After the tournament, the third last team has a score strictly lower than any team before it, and strictly higher than the two teams after it; it also has strictly more wins than the teams before it, and strictly less wins than the two teams after it. Find the minimum value of $n$.", "answer": "13", "image_path": "image354.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 355, "question": "Let P be a point in a regular tetrahedron T with volume 1 (including the boundary). Draw four planes passing through P, such that they are parallel to the four faces of T respectively, and divide T into 14 regions. Let f(P) be the sum of the volumes of the regions that are neither tetrahedra or parallelepipeds. Find the range of f(P).", "answer": "The range of $f(P)$ is $0 \\le f(P) \\le \\frac{3}{4}$.", "image_path": "image355.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 356, "question": "For a positive integer M, if there are integers a, b, c, d such that M ≤ a < b < c < d ≤ M + 49, and ad = bc, we say that M is a good number; otherwise M is called a bad number. Find the biggest good number and the smallest bad number.", "answer": "The biggest good number is 576, and the smallest bad number is 443.", "image_path": "image356.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Probability & Statistics" ] } }, { "id": 357, "question": "Assume there are 1988 unit cubes. Put them together to construct three regular square prisms A, B, C with height 1 and bottom side lengths a, b, c respectively. Put A, B, and C in the first quadrant, so that the bottom sides are parallel to the axes. Assume that one vertex of C is the origin. B is placed on top of C, and each unit cube of B lies on top of exactly one unit cube of C, but the boundary of B does not touch the boundary of C. Similarly, A is placed on top of B, each unit cube of A lies on top of exactly one unit cube of B, but the boundary of A does not touch the boundary of B. A three-storey building is built like this. What is the value of a, b, and c, such that the number of different buildings is maximized?", "answer": "When (a, b, c) = (1, 17, 41), the maximum value of P is 345².", "image_path": "image357.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Combinatorics", "Geometry" ] } }, { "id": 358, "question": "Consider a quadrilateral ABCD inscribed in a circle, whose four sides have lengths being positive integers. We know DA = 2005, ∠ABC = ∠ADC = 90°, and max{AB, BC, CD} < 2005. Find the maximum and minimum value of the perimeter of quadrilateral ABCD.", "answer": "7772, 4160", "image_path": "image358.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry" ] } }, { "id": 359, "question": "Let a point $P$ start from $A(1, 1)$, move along lattice paths, and arrive at $B(m, n)$ ($m, n \\in \\mathbb{N}^*$). At each step, $P$ moves to an adjacent lattice point, so that either the $x$-coordinate or the $y$-coordinate increases by 1. Find the maximum value of the sum $S$ of the products of $x$- and $y$-coordinates of all lattice points $P$ passes through.", "answer": "$S_{max} = \\frac{1}{6} n(3m^2 + n^2 + 3m - 1)$, when $m < n$, and similarly for when $m > n$.", "image_path": "image359.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Branch‑and-Bound", "Divide‑and-Conquer" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 360, "question": "The MO space station consists of 99 space stations, where any two stations are connected by a tubular channel. Set 99 of the channels to be two-way channels, and the rest are strictly one-way. For a group of four stations, if starting from any station one can reach any other station through the channels, the group of four stations is called a connected four-station group. Find the maximum number of connected four-station groups, and justify your answer.", "answer": "2052072", "image_path": "image360.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 361, "question": "Let $x_1, x_2, \\dots, x_n$ be in an interval of length 1. Define $x = \\frac{1}{n} \\sum_{j=1}^n x_j$, $y = \\frac{1}{n} \\sum_{j=1}^n x_j$. Find the maximum value of $f = y - x^2$. ", "answer": "The maximum value of $f$ is $1/4$ when $n$ is even and $\\frac{n^2 - 1}{4n^2}$ when $n$ is odd.", "image_path": "image361.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Formula", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Algebra", "Combinatorics", "Logic" ] } }, { "id": 362, "question": "Let $n$ be a fixed even positive integer. Consider an $n \times n$ square chessboard. Two grids are called 'neighboring' if they share a common edge. Now, mark $N$ grids in the chessboard, such that any grid in the chessboard (marked or unmarked) is the neighbor of a marked grid. Find the minimum value of $N$. ", "answer": "\\frac{1}{4}(n+2)", "image_path": "image362.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry", "Number Theory", "Combinatorics" ] } }, { "id": 363, "question": "There are 2000 points in the $xOy$ plane, forming a point set $S$. We know that the line connecting any two points is not parallel to the axes. For any two points $P, Q$ in $S$, consider the rectangle $MPQ$ whose diagonal is $PQ$ and whose edges are parallel to the axes. We use $W_{PQ}$ to represent the number of points in rectangle $MPQ$. If the statement 'no matter how the points in $S$ distribute in the plane, there are at least a pair of points $P, Q$, which makes $W_{PQ} \\geq N$' is true, find the maximum value of $N$. ", "answer": "400", "image_path": "image363.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 364, "question": "In one competition there are 20 gymnasts and 9 referees. Each referee ranks the gymnasts from 1 to 20 after their performance. It is known that the differences between the nine positions given by the referees for each gymnast are at most 3. Now for each gymnast, consider the sum of the nine ranks he/she gets, and rearrange these sums as $c_1 \\leq c_2 \\leq \\cdots \\leq c_{20}$. Find the maximum value of $c_1$.", "answer": "24", "image_path": "image364.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 365, "question": "There are $r$ people participating in a chess tournament, where any two people play against each other once. The winner of each game gains two points, the loser gains zero points, and each gains one point in the case of a draw. After the tournament, there is only one player who has the least number of wins and the highest score. Find the minimum value of $r$.", "answer": "6", "image_path": "image365.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 366, "question": "There are $n$ people greeting each other by phone on a holiday. It is known that each one called at most three others, that each two people talked on the phone at most once, and that for any three people, at least one of them called one of the others. Find the maximum value of $n$.", "answer": "14", "image_path": "image366.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 367, "question": "In an $n \times n$ chessboard $C$, each cell is filled with a number, in some particular order. First the cells on the boundary of the chessboard are filled with $-1$. Then the rest of the cells are filled in some order, such that each cell is filled with the number which is the product of the two numbers which are in either the same row or the same column of the cell, and closest to the cell. Find the maximum number $f(n)$ of 1s and the minimum number $g(n)$ of 1s. (The 20th USSR Mathematical Olympiad)", "answer": "The maximum number of 1s is $f(n) = (n-2)^2 - 1$, and the minimum number of -1s is $g(n) = n-2$.", "image_path": "image367.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 368, "question": "There are h 8 × 8 chessboards, and each board is filled with 1, 2, 3, ..., 64 in each cell respectively, such that when any two of the boards overlap in any way, the numbers at the same position are different. Find the maximum value of h.", "answer": "16", "image_path": "image368.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 369, "question": "There are 50 blanks in a lottery ticket. Each participant writes the numbers from 1 to 50 in the blanks (each number appears only once in the same ticket) of a lottery ticket. The tickets are then compared with the official ticket (the winning ticket). If a ticket has at least one position which is the same with the official ticket, then this ticket wins the lottery. How many lotteries should a participant write, such that one of them is guaranteed to win?", "answer": "26", "image_path": "image369.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Probability & Statistics", "Combinatorics" ] } }, { "id": 370, "question": "If a positive integer n satisfies the following condition: there exists a sequence of n real numbers, where the sum of any 17 consecutive terms is positive, and the sum of any 10 consecutive terms is negative, find the maximum value of n.", "answer": "25", "image_path": "image370.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 371, "question": "There are n middle schools in a city. The ith middle school sends ci students (1 ≤ ci ≤ 39) to watch a football game in a stadium, where \\( \\sum_{i=1}^{n} c_i = 1990 \\). There are 199 seats in each row of the stand. It is required that the students in the same school sit in the same row. At least how many rows should there be, so that this is always possible?", "answer": "12", "image_path": "image371.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 372, "question": "How many stones can be put on a 19 × 89 chessboard, with any 2 × 2 square in the chessboard containing no more than two stones?", "answer": "890", "image_path": "image372.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Number Theory", "Combinatorics" ] } }, { "id": 373, "question": "There is a 9 × 9 chessboard with grids colored black and white. The grids next to each white grid include more black grids than white grids, and the grids next to each white grid include more white grids than black grids (a grid is next to another when these two grids share an edge). Find the maximum value of the difference between the numbers of black and white grids for all possible painting patterns.", "answer": "3", "image_path": "image373.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 374, "question": "A diagonal of a 2006-sided regular polygon is called 'good' if the two end-points of it divide the boundary of P into two parts, and that each part contains an odd number of sides. In particular, the sides of P are also called good. Assume that P is divided into triangles with 2003 diagonals which do not intersect each other, find the maximum number of isosceles triangles, two of whose edges are good.", "answer": "1003", "image_path": "image374.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 375, "question": "Given 21 points on a circle, how many pairs of points subtend an angle less than or equal to 120° at the center?", "answer": "100", "image_path": "image375.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Number Theory", "Geometry" ] } }, { "id": 376, "question": "Let A_1, A_2, ..., A_101 be different subsets of the set {1, 2, ..., n}. Suppose that the union of any 50 subsets has more than n/51 elements. How many subsets among these 50 must have common elements?", "answer": "3", "image_path": "image376.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 377, "question": "Let G be a simple graph. To every vertex of G one assigns a nonnegative real number such that the sum of the numbers assigned to all vertices is 1. For any two vertices connected by an edge, compute the product of the numbers associated to these vertices. What is the maximal value of the sum of these products?", "answer": "\\frac{1}{2}\\left(1-\\frac{1}{k}\\right)", "image_path": "image377.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 378, "question": "What is the maximal number of complete maximal subgraphs that a graph on \\(n\\) vertices can have if \\(n-1\\) is not a multiple of 3?", "answer": "2 * 3^((n-2)/3)", "image_path": "image378.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 379, "question": "Find the greatest N for which there are N consecutive positive integers such that the sum of digits of the k-th number is divisible by k, for k = 1, 2, ..., N.", "answer": "21", "image_path": "image379.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Number Theory" ] } }, { "id": 380, "question": "How many positive integers $n\\leq10^{2005}$ can be written as the sum of two positive integers with the same sum of digits?", "answer": "10^{2005} - 9023", "image_path": "image380.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 381, "question": "There are 50 boxes, each containing a collection of balls. The balls come in $n$ different types. The only thing we know is that for any given box, the number of different types of balls inside it is more than half of the total number of types, $n$. We want to create a 'test kit' by selecting a small number of ball types. This test kit is considered successful if it contains at least one ball type that is also present in **each** of the 50 boxes. What is the minimum number of ball types we need in our test kit to **guarantee** success?", "answer": "5", "image_path": "image381.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Number Theory" ] } }, { "id": 382, "question": "Let $A$ be a set with 100 elements, and let $A_1, A_2, \\dots, A_m$ be a collection of subsets of $A$. These subsets satisfy the following conditions: \\begin{itemize} \\item Each subset has exactly 4 elements. \\item The intersection of any two distinct subsets contains at most 2 elements. \\end{itemize} We are told that we can guarantee the existence of a minimal cover of $A$ formed by exactly 49 of these subsets. A minimal cover is defined as a collection of subsets whose union is equal to $A$, but the union of any smaller sub-collection is not equal to $A$. What is the minimum integer value of $m$ for which this guarantee holds?", "answer": "40425", "image_path": "image382.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 383, "question": "On an $n^2 \times n^2$ chessboard, a positive integer is written in each square. The difference between the numbers in any two adjacent squares (sharing an edge) is at most $n$. What is the minimum number of squares that are guaranteed to contain the same number? Express your answer in terms of $n$.", "answer": "1+\\left\\lfloor\\frac{n}{2}\\right\\rfloor", "image_path": "image383.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Number Theory" ] } }, { "id": 384, "question": "50 students compete in a contest consisting of 8 problems. Across all students, a total of 171 correct solutions were received. What is the minimum number of problems that are guaranteed to have been solved by at least 3 students?", "answer": "3", "image_path": "image384.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Number Theory", "Logic" ] } }, { "id": 385, "question": "Let E be a set of 2n —1 points on a circle, with n > 2. Suppose that precisely k points of E are colored black. We say that this coloring is admissible if there is at least one pair of black points such that the interior of one of the arcs they determine contains exactly n points of E. What is the smallest k such that any coloring of k points of E is admissible?", "answer": "k = \\begin{cases} \\lfloor \\frac{2n-1}{2} \\rfloor + 1 & \\text{if } 3 \\nmid (2n-1) \\\\ 3\\lfloor \\frac{2n-1}{6} \\rfloor + 1 & \\text{if } 3 \\mid (2n-1) \\end{cases}", "image_path": "image385.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Mathematics", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 386, "question": "Draw horizontal or vertical bridges to link up all the islands. Bridges may be single, double OR more; they may not cross; the islands must all end up connected to each other; the number in each island must match the number of bridges that end at that island (counting double bridges as two). Note that loops of bridges are permitted. Provide the final answer in the following format: AB1, EF3, which mean node A and B connected with single bridge and node E and F connected with 3 bridges.", "answer": "AB1, AF1, BC2, BD2, CJ2, EL1, FG1, FK1, GH2, IJ2, KL1", "image_path": "image386.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 387, "question": "Draw horizontal or vertical bridges to link up all the islands. Bridges may be single, double OR more; they may not cross; the islands must all end up connected to each other; the number in each island must match the number of bridges that end at that island (counting double bridges as two). Note that loops of bridges are permitted. Provide the final answer in the following format: AB1, EF3, which mean node A and B connected with single bridge and node E and F connected with 3 bridges.", "answer": "AF3, BC1, CD2, CG1, EJ2, FG1, FH2, HI3, HT1, IJ3, IZ1, JS2, KL1, KN2, MN2, MP2, ZO1, PQ1, PR2, RS2, TU2", "image_path": "image387.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 388, "question": "Draw horizontal or vertical bridges to link up all the islands. Bridges may be single, double OR more; they may not cross; the islands must all end up connected to each other; the number in each island must match the number of bridges that end at that island (counting double bridges as two). Note that loops of bridges are permitted. Provide the final answer in the following format: AB1, EF3, which mean node A and B connected with single bridge and node E and F connected with 3 bridges.", "answer": "AB1, AG2, BC1, BL1, DO3, EF2, EH3, FJ1, GK2, HM1, IJ3, LM3, MN3, NO3", "image_path": "image388.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 389, "question": "Tile the rectangle with dominoes (1×2 rectangles) so that every possible domino appears exactly once (that is, every possible pair of numbers, including doubles). Output a matrix of the same size, where each position is 0, 1, 2, or 3, which mean a cell with its up, down, left, or right neighbor get pair are paired as the 1×2 rectangle.", "answer": "[\\n[3 2 1 1 3 2 1 1]\\n[1 1 0 0 1 1 0 0]\\n[0 0 3 2 0 0 3 2]\\n[1 3 2 1 1 3 2 1]\\n[0 3 2 0 0 1 1 0]\\n[1 3 2 3 2 0 0 1]\\n[0 3 2 3 2 3 2 0]\\n]", "image_path": "image389.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 390, "question": "Tile the rectangle with dominoes (1×2 rectangles) so that every possible domino appears exactly once (that is, every possible pair of numbers, including doubles). Output a matrix of the same size, where each position is 0, 1, 2, or 3, which mean a cell with its up, down, left, or right neighbor get pair are paired as the 1×2 rectangle.", "answer": "[\\n[1 1 3 2 1 3 2 1 1]\\n[0 0 1 1 0 3 2 0 0]\\n[1 1 0 0 1 3 2 3 2]\\n[0 0 3 2 0 3 2 3 2]\\n[1 3 2 1 3 2 3 2 1]\\n[0 1 1 0 1 3 2 1 0]\\n[1 0 0 1 0 3 2 0 1]\\n[0 3 2 0 3 2 3 2 0]]", "image_path": "image390.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Number Theory", "Combinatorics" ] } }, { "id": 391, "question": "Tile the rectangle with dominoes (1×2 rectangles) so that every possible domino appears exactly once (that is, every possible pair of numbers, including doubles). Output a matrix of the same size, where each position is 0, 1, 2, or 3, which mean a cell with its up, down, left, or right neighbor get pair are paired as the 1×2 rectangle.", "answer": "[\\n[1 3 2 3 2 3 2 1 1 1]\\n[0 1 3 2 3 2 1 0 0 0]\\n[1 0 3 2 3 2 0 1 1 1]\\n[0 1 3 2 1 3 2 0 0 0]\\n[1 0 3 2 0 3 2 3 2 1]\\n[0 1 3 2 1 3 2 3 2 0]\\n[1 0 3 2 0 1 1 3 2 1]\\n[0 3 2 3 2 0 0 3 2 0]\\n[3 2 3 2 3 2 3 2 3 2]]", "image_path": "image391.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic", "Combinatorics" ] } }, { "id": 392, "question": "Write a number in every blank square of the grid. When the grid is full, every orthogonally connected group of identical numbers should have an area equal to that number: so 1s always appear alone, 2s in pairs, and so on. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times M$. Determine the number in the position (7,9), (4,1), (13,1), (13,8).", "answer": "(7,9): 5, (4,1): 7, (13,1): 9, (13,8): 7", "image_path": "image392.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic", "Combinatorics" ] } }, { "id": 393, "question": "Write a number in every blank square of the grid. When the grid is full, every orthogonally connected group of identical numbers should have an area equal to that number: so 1s always appear alone, 2s in pairs, and so on. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times M$. Determine the number in the position (1,1), (5,8), (7,9), (13,3).", "answer": "(1,1): 4, (5,8): 6, (7,9): 8, (13,3): 4", "image_path": "image393.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 394, "question": "Write a number in every blank square of the grid. When the grid is full, every orthogonally connected group of identical numbers should have an area equal to that number: so 1s always appear alone, 2s in pairs, and so on. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times M$. Determine the number in the position (1,7), (4,4), (11,1), (12,8).", "answer": "(1,7): 6, (4,4): 8, (11,1): 4, (12,8): 5", "image_path": "image394.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 395, "question": "Try to light up all the squares in the grid by flipping combinations of them. Flipping a square will also flip some of its neighbours. The diagram in each square indicates which other squares will flip. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times M$. Determine which square to be flipped.", "answer": "(2,2), (2,3), (2,5), (3,3), (4,3), (4,4), (4,5), (5,3), (5,5)", "image_path": "image395.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 396, "question": "Try to light up all the squares in the grid by flipping combinations of them. Flipping a square will also flip some of its neighbours. The diagram in each square indicates which other squares will flip. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times M$. Determine which square to be flipped.", "answer": "(1,1), (1,2), (1,5), (2,1), (2,4), (2,5), (3,1), (3,3), (4,3), (4,4), (5,2)", "image_path": "image396.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 397, "question": "Try to light up all the squares in the grid by flipping combinations of them. Flipping a square will also flip some of its neighbours. The diagram in each square indicates which other squares will flip. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times M$. Determine which square to be flipped.", "answer": "(1,2), (1,3), (2,4), (3,1), (4,3), (4,5)", "image_path": "image397.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 398, "question": "Draw lines along grid edges so as to divide the grid up into connected regions of squares. Every region should have two-way rotational symmetry, should contain exactly one dot which is in its centre, and should contain no lines separating two of its own squares from each other. A region satisfying all of these requirements will be automatically highlighted. Let the area of a basic spuare is 1, determines the area of the region with dot A,B,C,D,E,F？", "answer": "A1, B11, C20, D10, E8, F12", "image_path": "image398.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 399, "question": "Draw lines along grid edges so as to divide the grid up into connected regions of squares. Every region should have two-way rotational symmetry, should contain exactly one dot which is in its centre, and should contain no lines separating two of its own squares from each other. A region satisfying all of these requirements will be automatically highlighted. Let the area of a basic spuare is 1, determines the area of the region with dot A,B,C,D,E,F？", "answer": "A4, B17, C1, D11, E5", "image_path": "image399.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 400, "question": "Draw lines along grid edges so as to divide the grid up into connected regions of squares. Every region should have two-way rotational symmetry, should contain exactly one dot which is in its centre, and should contain no lines separating two of its own squares from each other. A region satisfying all of these requirements will be automatically highlighted. Let the area of a basic spuare is 1, determines the area of the region with dot A,B,C,D,E,F？", "answer": "A1, B7, C20, D6, E8", "image_path": "image400.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 401, "question": "Fill in the grid with digits from 1 to the grid size N, so that every digit appears exactly once in each row and column, and so that all the arithmetic clues are satisfied (i.e. the clue number in each thick box should be possible to construct from the digits in the box using the specified arithmetic operation). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Determine all numbers and output answer in the format of matrix.", "answer": "[[2 3 5 4 6 1]\\n[3 1 6 5 4 2]\\n[4 5 2 6 1 3]\\n[1 2 4 3 5 6]\\n[6 4 3 1 2 5]\\n[5 6 1 2 3 4]]", "image_path": "image401.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 402, "question": "Fill in the grid with digits from 1 to the grid size N, so that every digit appears exactly once in each row and column, and so that all the arithmetic clues are satisfied (i.e. the clue number in each thick box should be possible to construct from the digits in the box using the specified arithmetic operation). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Determine all numbers and output final answer as the $N\\times N$ matrix.", "answer": "[\\n[1 2 5 3 4 6]\\n[2 5 6 1 3 4]\\n[4 6 2 5 1 3]\\n[3 1 4 6 2 5]\\n[6 4 3 2 5 1]\\n[5 3 1 4 6 2]\\n]", "image_path": "image402.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 403, "question": "Fill in the grid with digits from 1 to the grid size N, so that every digit appears exactly once in each row and column, and so that all the arithmetic clues are satisfied (i.e. the clue number in each thick box should be possible to construct from the digits in the box using the specified arithmetic operation). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Determine all numbers and output final answer as the $N\\times N$ matrix.", "answer": "[\\n[2 3 4 5 6 1]\\n[6 1 2 3 4 5]\\n[3 6 5 4 1 2]\\n[5 2 6 1 3 4]\\n[4 5 1 6 2 3]\\n[1 4 3 2 5 6]\\n]", "image_path": "image403.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 404, "question": "Place light bulbs in the grid so as to light up all the blank squares. A light illuminates its own square and all the squares in the same row or column unless blocked by walls (black squares). Lights may not illuminate each other. Each numbered square must be orthogonally adjacent to exactly the given number of lights. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Determine all the positions of lights.", "answer": "(1,2), (1,6), (1,9), (2,8), (3,4), (4,3), (4,10), (5,1), (6,7), (7,8), (8,3), (8,6), (9,5), (10,1), (10,10)", "image_path": "image404.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Branch‑and-Bound", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 405, "question": "Place light bulbs in the grid so as to light up all the blank squares. A light illuminates its own square and all the squares in the same row or column unless blocked by walls (black squares). Lights may not illuminate each other. Each numbered square must be orthogonally adjacent to exactly the given number of lights. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Determine all the positions of lights.", "answer": "(1,5), (1,9), (2,3), (2,8), (3,2), (3,6), (3,10), (4,5), (5,3), (5,7), (5,10), (6,1), (6,4), (6,8), (7,10), (8,2), (8,6), (9,1), (9,5), (9,8), (10,3), (10,9)", "image_path": "image405.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Branch‑and-Bound", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 406, "question": "Place light bulbs in the grid so as to light up all the blank squares. A light illuminates its own square and all the squares in the same row or column unless blocked by walls (black squares). Lights may not illuminate each other. Each numbered square must be orthogonally adjacent to exactly the given number of lights. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Determine all the positions of lights.", "answer": "(1,1), (1,3), (1,10), (2,7), (3,4), (3,10), (4,8), (5,5), (6,2), (6,8), (7,6), (8,1), (8,7), (8,9), (9,8), (10,1), (10,4), (10,10)", "image_path": "image406.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Branch‑and-Bound", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 407, "question": "Form a single closed loop out of the grid edges, in such a way that every numbered square has exactly that many of its edges included in the loop.Label some region (the smallest triangle or quadrilateral) in the alphabetical order. Determine whether the labeled polygon unit is included in the closed loop. Give final answer in the format: A0, B1, which mean the polygon unit region A is in and region B is not in the closed loop.", "answer": "A1, B0, C0, D1, E0", "image_path": "image407.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 408, "question": "Form a single closed loop out of the grid edges, in such a way that every numbered square has exactly that many of its edges included in the loop.Label some region (the smallest triangle or quadrilateral) in the alphabetical order. Determine whether the labeled polygon unit is included in the closed loop. Give final answer in the format: A0, B1, which mean the polygon unit region A is in and region B is not in the closed loop.", "answer": "A1, B1, C0, D0, E1", "image_path": "image408.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Branch‑and-Bound", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 409, "question": "Form a single closed loop out of the grid edges, in such a way that every numbered square has exactly that many of its edges included in the loop.Label some region (the smallest triangle or quadrilateral) in the alphabetical order. Determine whether the labeled polygon unit is included in the closed loop. Give final answer in the format: A0, B1, which mean the polygon unit region A is in and region B is not in the closed loop.", "answer": "A0, B1, C0, D1, E0", "image_path": "image409.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Branch‑and-Bound", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 410, "question": "Fill each domino shape with either a magnet (consisting of a + and − pole) or a neutral domino (green). The number of + poles that in each row and column must match the numbers along the top and left; the number of − poles must match the numbers along the bottom and right. Two + poles may not be orthogonally adjacent to each other, and similarly two − poles. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: A+, B-, Cx, which means the unit region A is + poles, B is - poles, and C is neither.", "answer": "A+, B-, C+, D-, E+", "image_path": "image410.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 411, "question": "Fill each domino shape with either a magnet (consisting of a + and − pole) or a neutral domino (green). The number of + poles that in each row and column must match the numbers along the top and left; the number of − poles must match the numbers along the bottom and right. Two + poles may not be orthogonally adjacent to each other, and similarly two − poles. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: A+, B-, Cx, which means the unit region A is + poles, B is - poles, and C is neither.", "answer": "A-, B-, C+, D+, Ex", "image_path": "image411.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 412, "question": "Fill each domino shape with either a magnet (consisting of a + and − pole) or a neutral domino (green). The number of + poles that in each row and column must match the numbers along the top and left; the number of − poles must match the numbers along the bottom and right. Two + poles may not be orthogonally adjacent to each other, and similarly two − poles. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: A+, B-, Cx, which means the unit region A is + poles, B is - poles, and C is neither.", "answer": "A-, Bx, C+, D+, Ex", "image_path": "image412.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 413, "question": "Colour the map with four colours, so that no two adjacent regions have the same colour. (Regions touching at only one corner do not count as adjacent.) There is a unique colouring consistent with the coloured regions you are already given. Label some region in the alphabetical order. Give final answer in following format: [RegionIndex][Color]. NOTE: [Color] can be g(green), r(red), b(brown) and y(yellow). The example answer: Ag means the region A is in green.", "answer": "Ag, Bg, Cg, Db, Er", "image_path": "image413.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 414, "question": "Colour the map with four colours, so that no two adjacent regions have the same colour. (Regions touching at only one corner do not count as adjacent.) There is a unique colouring consistent with the coloured regions you are already given. Label some region in the alphabetical order. Give final answer in following format: [RegionIndex][Color]. NOTE: [Color] can be g(green), r(red), b(brown) and y(yellow). The example answer: Ag means the region A is in green.", "answer": "Ab, Br, Cb, Dg, Ey", "image_path": "image414.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 415, "question": "Colour the map with four colours, so that no two adjacent regions have the same colour. (Regions touching at only one corner do not count as adjacent.) There is a unique colouring consistent with the coloured regions you are already given. Label some region in the alphabetical order. Give final answer in following format: [RegionIndex][Color]. NOTE: [Color] can be g(green), r(red), b(brown) and y(yellow). The example answer: Ag means the region A is in green.", "answer": "Ag, Bb, Cb, Dy, Eg", "image_path": "image415.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 416, "question": "Colour every square either black or white. Each number indicates how many black squares are in the 3×3 square surrounding the number – including the clue square itself. Label some $1\\time 1$ square region (the smallest square unit) in the alphabetical order. Give final answer in following format: [RegionIndex][Color]. NOTE: [Color] can be g(green), r(red), b(brown) and y(yellow). The example answer: Ag means the region A is in green.", "answer": "A1, B1, C1, D1, E0", "image_path": "image416.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 417, "question": "Colour every square either black or white. Each number indicates how many black squares are in the 3×3 square surrounding the number – including the clue square itself. Label some $1\\time 1$ square region (the smallest square unit) in the alphabetical order. Give final answer in following format: [RegionIndex][Color]. NOTE: [Color] can be g(green), r(red), b(brown) and y(yellow). The example answer: Ag means the region A is in green.", "answer": "A0, B1, C1, D1, E1", "image_path": "image417.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 418, "question": "Colour every square either black or white. Each number indicates how many black squares are in the 3×3 square surrounding the number – including the clue square itself. Label some $1\\time 1$ square region (the smallest square unit) in the alphabetical order. Give final answer in following format: [RegionIndex][Color]. NOTE: [Color] can be g(green), r(red), b(brown) and y(yellow). The example answer: Ag means the region A is in green.", "answer": "A1, B1, C1, D1, E0", "image_path": "image418.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 419, "question": "Rotate the grid squares so that they all join up into a single connected network with no loops. You can rotate a square anticlockwise or clockwise. Middle-click. Squares connected to the middle square are lit up. Aim to light up every square in the grid (not just the endpoint blobs). We enable grid lines to run off one edge of the playing area and come back on the opposite edge! Label some $1\\time 1$ square region (the smallest square unit) in the alphabetical order. Give final answer in following format: [RegionIndex][Roation Direction][Roation Angle]. NOTE: [Roation Direction] can be +(clockwise), or -(anticlockwise). And Roation Angle can be 0°,90°, or 180°. The example answer: A+90 means that region A need be rotated 90° clockwise to get final result. Due to the fact that +0°=-0°, +180°=-180°, you can just write [RegionIndex][Roation Angle] when the angle is 0°, 180°, as B0 or C180", "answer": "A-90, B+180, C-90, D-90, E-90", "image_path": "image419.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 420, "question": "Rotate the grid squares so that they all join up into a single connected network with no loops. You can rotate a square anticlockwise or clockwise. Middle-click. Squares connected to the middle square are lit up. Aim to light up every square in the grid (not just the endpoint blobs). We enable grid lines to run off one edge of the playing area and come back on the opposite edge! Label some $1\\time 1$ square region (the smallest square unit) in the alphabetical order. Give final answer in following format: [RegionIndex][Roation Direction][Roation Angle]. NOTE: [Roation Direction] can be +(clockwise), or -(anticlockwise). And Roation Angle can be 0°,90°, or 180°. The example answer: A+90 means that region A need be rotated 90° clockwise to get final result. Due to the fact that +0°=-0°, +180°=-180°, you can just write [RegionIndex][Roation Angle] when the angle is 0°, 180°, as B0 or C180", "answer": "A0, B180, C+90, D-90, E-90", "image_path": "image420.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 421, "question": "Rotate the grid squares so that they all join up into a single connected network with no loops. You can rotate a square anticlockwise or clockwise. Middle-click. Squares connected to the middle square are lit up. Aim to light up every square in the grid (not just the endpoint blobs). We enable grid lines to run off one edge of the playing area and come back on the opposite edge! Label some $1\\time 1$ square region (the smallest square unit) in the alphabetical order. Give final answer in following format: [RegionIndex][Roation Direction][Roation Angle]. NOTE: [Roation Direction] can be +(clockwise), or -(anticlockwise). And Roation Angle can be 0°,90°, or 180°. The example answer: A+90 means that region A need be rotated 90° clockwise to get final result. Due to the fact that +0°=-0°, +180°=-180°, you can just write [RegionIndex][Roation Angle] when the angle is 0°, 180°, as B0 or C180", "answer": "A+90, B+90, C180, D+90, E-90", "image_path": "image421.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 422, "question": "Draw lines along the grid edges, in such a way that the grid is divided into connected regions, all of the size shown in the status line. Also, each square containing a number should have that many of its edges drawn in. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Judge the region it belongs to, and determine the sum of all numbers contained within that region. Give the final answer in format of A10, which means the sum of all numbers of the region which square marked A also belongs to is 10.", "answer": "A4, B3, C4, D6, E9", "image_path": "image422.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 423, "question": "Draw lines along the grid edges, in such a way that the grid is divided into connected regions, all of the size shown in the status line. Also, each square containing a number should have that many of its edges drawn in. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Judge the region it belongs to, and determine the sum of all numbers contained within that region. Give the final answer in format of A10, which means the sum of all numbers of the region which square marked A also belongs to is 10.", "answer": "A4, B5, C2, D2, E2", "image_path": "image423.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 424, "question": "Draw lines along the grid edges, in such a way that the grid is divided into connected regions, all of the size shown in the status line. Also, each square containing a number should have that many of its edges drawn in. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Judge the region it belongs to, and determine the sum of all numbers contained within that region. Give the final answer in format of A10, which means the sum of all numbers of the region which square marked A also belongs to is 10.", "answer": "A3, B4, C4, D3, E1", "image_path": "image424.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Divide‑and-Conquer" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 425, "question": "Fill in the grid with a pattern of black and white squares, so that the numbers in each row and column match the lengths of consecutive runs of black squares. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C0, D1, E0", "image_path": "image425.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics", "Number Theory" ] } }, { "id": 426, "question": "Fill in the grid with a pattern of black and white squares, so that the numbers in each row and column match the lengths of consecutive runs of black squares. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B1, C1, D0, E0", "image_path": "image426.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics", "Number Theory" ] } }, { "id": 427, "question": "Fill in the grid with a pattern of black and white squares, so that the numbers in each row and column match the lengths of consecutive runs of black squares. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B0, C1, D1, E0", "image_path": "image427.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics", "Number Theory" ] } }, { "id": 428, "question": "Draw a single closed loop by connecting together the centres of adjacent grid squares, so that some squares end up as corners, some as straights (horizontal or vertical), and some may be empty. Every square containing a black circle must be a corner not connected directly to another corner; every square containing a white circle must be a straight which is connected to at least one corner. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. For each labeled square, identify the edge crossed by the final single closed loop: top (0), bottom (1), left (2), or right (3). Answer in format: A01, which means the top and bottom edge are crossed by the final single closed loop line. NOTE: The first number need to smaller than the second number. If no edges are crossed, just answer the letter without digits.", "answer": "A03, B12, C12, D13, E02", "image_path": "image428.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic", "Combinatorics" ] } }, { "id": 429, "question": "Draw a single closed loop by connecting together the centres of adjacent grid squares, so that some squares end up as corners, some as straights (horizontal or vertical), and some may be empty. Every square containing a black circle must be a corner not connected directly to another corner; every square containing a white circle must be a straight which is connected to at least one corner. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. For each labeled square, identify the edge crossed by the final single closed loop: top (0), bottom (1), left (2), or right (3). Answer in format: A01, which means the top and bottom edge are crossed by the final single closed loop line. NOTE: The first number need to smaller than the second number. If no edges are crossed, just answer the letter without digits.", "answer": "A13, B12, C13, D12, E02", "image_path": "image429.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic", "Combinatorics" ] } }, { "id": 430, "question": "Draw a single closed loop by connecting together the centres of adjacent grid squares, so that some squares end up as corners, some as straights (horizontal or vertical), and some may be empty. Every square containing a black circle must be a corner not connected directly to another corner; every square containing a white circle must be a straight which is connected to at least one corner. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. For each labeled square, identify the edge crossed by the final single closed loop: top (0), bottom (1), left (2), or right (3). Answer in format: A01, which means the top and bottom edge are crossed by the final single closed loop line. NOTE: The first number need to smaller than the second number. If no edges are crossed, just answer the letter without digits.", "answer": "A, B01, C13, D12, E02", "image_path": "image430.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic", "Combinatorics" ] } }, { "id": 431, "question": "Colour some squares black, so as to meet the following conditions: No two black squares are orthogonally adjacent. No group of white squares is separated from the rest of the grid by black squares. Each numbered cell can see precisely that many white squares in total by looking in all four orthogonal directions, counting itself. (Black squares block the view. So, for example, a 2 clue must be adjacent to three black squares or grid edges, and in the fourth direction there must be one white square and then a black one beyond it.) Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: [RegionIndex][Color]. NOTE: [Color] can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B0, C0, D1, E1", "image_path": "image431.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 432, "question": "Colour some squares black, so as to meet the following conditions: No two black squares are orthogonally adjacent. No group of white squares is separated from the rest of the grid by black squares. Each numbered cell can see precisely that many white squares in total by looking in all four orthogonal directions, counting itself. (Black squares block the view. So, for example, a 2 clue must be adjacent to three black squares or grid edges, and in the fourth direction there must be one white square and then a black one beyond it.) Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: [RegionIndex][Color]. NOTE: [Color] can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B0, C0, D1, E1", "image_path": "image432.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 433, "question": "Colour some squares black, so as to meet the following conditions: No two black squares are orthogonally adjacent. No group of white squares is separated from the rest of the grid by black squares. Each numbered cell can see precisely that many white squares in total by looking in all four orthogonal directions, counting itself. (Black squares block the view. So, for example, a 2 clue must be adjacent to three black squares or grid edges, and in the fourth direction there must be one white square and then a black one beyond it.) Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: [RegionIndex][Color]. NOTE: [Color] can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B0, C1, D0, E0", "image_path": "image433.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 434, "question": "Draw lines along the grid edges to divide the grid into rectangles, so that each rectangle contains exactly one numbered square and its area is equal to the number written in that square. Judge which edge is selected. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. For each labeled square, identify the edge selected as the edge in one rectangle: top (0), bottom (1), left (2), or right (3). Answer in format: A01, which means the top and bottom edge are selected as the edge in one rectangle. The digits should be in ascending order from left to right. If no edges are selected, just answer the letter without digits.", "answer": "A2, B123, C13, D03, E013", "image_path": "image434.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics", "Number Theory" ] } }, { "id": 435, "question": "Draw lines along the grid edges to divide the grid into rectangles, so that each rectangle contains exactly one numbered square and its area is equal to the number written in that square. Judge which edge is selected. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. For each labeled square, identify the edge selected as the edge in one rectangle: top (0), bottom (1), left (2), or right (3). Answer in format: A01, which means the top and bottom edge are selected as the edge in one rectangle. The digits should be in ascending order from left to right. If no edges are selected, just answer the letter without digits.", "answer": "A123, B13, C023, D23, E123", "image_path": "image435.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics", "Number Theory" ] } }, { "id": 436, "question": "Draw lines along the grid edges to divide the grid into rectangles, so that each rectangle contains exactly one numbered square and its area is equal to the number written in that square. Judge which edge is selected. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. For each labeled square, identify the edge selected as the edge in one rectangle: top (0), bottom (1), left (2), or right (3). Answer in format: A01, which means the top and bottom edge are selected as the edge in one rectangle. The digits should be in ascending order from left to right. If no edges are selected, just answer the letter without digits.", "answer": "A012, B012, C23, D13, E01", "image_path": "image436.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics", "Number Theory" ] } }, { "id": 437, "question": "Connect all the squares together into a sequence, so that every square's arrow points towards the square that follows it (though the next square can be any distance away in that direction). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Give final answer in the format of $N\\times N$ matrix with corresponding index number (from 1 to $N^2$).", "answer": "[\\n[6 12 14 13 7]\\n[10 19 18 17 16]\\n[1 11 20 21 2]\\n[5 8 15 24 4]\\n[9 22 25 23 3]]", "image_path": "image437.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 438, "question": "Connect all the squares together into a sequence, so that every square's arrow points towards the square that follows it (though the next square can be any distance away in that direction). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Give final answer in the format of $N\\times N$ matrix with corresponding index number (from 1 to $N^2$).", "answer": "[\\n[19 18 10 1 9]\\n[16 24 17 4 23]\\n[11 12 20 5 6]\\n[2 8 22 25 7]\\n[3 13 21 15 14]]", "image_path": "image438.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Geometry", "Logic" ] } }, { "id": 439, "question": "Connect all the squares together into a sequence, so that every square's arrow points towards the square that follows it (though the next square can be any distance away in that direction). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Give final answer in the format of $N\\times N$ matrix with corresponding index number (from 1 to $N^2$).", "answer": "[\\n[8 19 9 25 1]\\n[3 20 4 12 2]\\n[10 18 5 24 11]\\n[16 17 22 6 23]\\n[15 21 14 13 7]]", "image_path": "image439.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Number Theory", "Logic" ] } }, { "id": 440, "question": "Black out some of the squares, in such a way that: no number appears twice in any row or column, no two black squares are adjacent, the white squares form a single connected group (connections along diagonals do not count). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Output a $2\\times 8$ matrix, whose first and second row contains the number of black square of each row(from top to bottom), and each colomn (from left to right), respectively.", "answer": "[[3 2 3 2 2 2 3 1]\\n[2 2 2 3 2 2 3 2]]", "image_path": "image440.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 441, "question": "Black out some of the squares, in such a way that: no number appears twice in any row or column, no two black squares are adjacent, the white squares form a single connected group (connections along diagonals do not count). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Output a $2\\times 8$ matrix, whose first and second row contains the number of black square of each row(from top to bottom), and each colomn (from left to right), respectively.", "answer": "[[1 3 3 1 3 2 3 2]\\n[3 2 2 3 1 3 1 3]]", "image_path": "image441.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 442, "question": "Black out some of the squares, in such a way that: no number appears twice in any row or column, no two black squares are adjacent, the white squares form a single connected group (connections along diagonals do not count). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Output a $2\\times 8$ matrix, whose first and second row contains the number of black square of each row(from top to bottom), and each colomn (from left to right), respectively.", "answer": "[[2 3 1 3 3 2 2 2]\\n[3 1 3 2 3 1 2 3]]", "image_path": "image442.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 443, "question": "Fill in a diagonal line in every grid square so that there are no loops in the grid, and so that every numbered point has that many lines meeting at it. Label a square to with a \\(Sign 0), /(Sign 1) or just empty(Sign 2). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the sign in labeled squares and output final answer in format: [SquareID][SignID]. [SignID] can be 0 (means \\) and 1(means /). For example, A0 means fill '\\' in the square A. ", "answer": "A0, B0, C1, D0, E1", "image_path": "image443.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 444, "question": "Fill in a diagonal line in every grid square so that there are no loops in the grid, and so that every numbered point has that many lines meeting at it. Label a square to with a \\(Sign 0), /(Sign 1) or just empty(Sign 2). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the sign in labeled squares and output final answer in format: [SquareID][SignID]. [SignID] can be 0 (means \\) and 1(means /). For example, A0 means fill '\\' in the square A. ", "answer": "A1, B0, C0, D1, E1", "image_path": "image444.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 445, "question": "Fill in a diagonal line in every grid square so that there are no loops in the grid, and so that every numbered point has that many lines meeting at it. Label a square to with a \\(Sign 0), /(Sign 1) or just empty(Sign 2). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the sign in labeled squares and output final answer in format: [SquareID][SignID]. [SignID] can be 0 (means \\) and 1(means /). For example, A0 means fill '\\' in the square A. ", "answer": "A1, B0, C1, D0, E0", "image_path": "image445.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 446, "question": "Fill in a number in every square so that every number appears exactly once in each row, each column and each block marked by thick lines.Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][Digit].", "answer": "A9, B2, C1, D1, E5", "image_path": "image446.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 447, "question": "Fill in a number in every square so that every number appears exactly once in each row, each column and each block marked by thick lines.Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][Digit].", "answer": "A3, B5, C8, D6, E8", "image_path": "image447.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 448, "question": "Fill in a number in every square so that every number appears exactly once in each row, each column and each block marked by thick lines.Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][Digit].", "answer": "A2, B2, C7, D5, E2", "image_path": "image448.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 449, "question": "Place tents in the empty squares in such a way that: no two tents are adjacent, even diagonally, the number of tents in each row and column matches the numbers around the edge of the grid, it is possible to match tents to trees so that each tree is orthogonally adjacent to its own tent (but may also be adjacent to other tents). Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the sign in labeled squares and output final answer in format: [SquareID][SignID]. [SignID] can be 0 (means empty) and 1(means tents). For example, A0 means the square A is empty. ", "answer": "A0, B0, C1, D0, E1", "image_path": "image449.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 450, "question": "Place tents in the empty squares in such a way that: no two tents are adjacent, even diagonally, the number of tents in each row and column matches the numbers around the edge of the grid, it is possible to match tents to trees so that each tree is orthogonally adjacent to its own tent (but may also be adjacent to other tents). Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the sign in labeled squares and output final answer in format: [SquareID][SignID]. [SignID] can be 0 (means empty) and 1(means tents). For example, A0 means the square A is empty. ", "answer": "A1, B0, C1, D0, E0", "image_path": "image450.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 451, "question": "Place tents in the empty squares in such a way that: no two tents are adjacent, even diagonally, the number of tents in each row and column matches the numbers around the edge of the grid, it is possible to match tents to trees so that each tree is orthogonally adjacent to its own tent (but may also be adjacent to other tents). Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the sign in labeled squares and output final answer in format: [SquareID][SignID]. [SignID] can be 0 (means empty) and 1(means tents). For example, A0 means the square A is empty. ", "answer": "A0, B0, C1, D0, E1", "image_path": "image451.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 452, "question": "Fill in the grid with towers whose heights range from 1 to the grid size, so that every possible height appears exactly once in each row and column, and so that each clue around the edge counts the number of towers that are visible when looking into the grid from that direction. (Taller towers hide shorter ones behind them. So the sequence 2,1,4,3,5 would match a clue of 3 on the left, because the 1 is hidden behind the 2 and the 3 is hidden behind the 4. On the right, it would match a clue of 1 because the 5 hides everything else.) Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][Digit] (e.g. A4, B2, C3, D2, E4).", "answer": "A6, B6, C1, D6, E4", "image_path": "image452.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic", "Number Theory" ] } }, { "id": 453, "question": "Fill in the grid with towers whose heights range from 1 to the grid size, so that every possible height appears exactly once in each row and column, and so that each clue around the edge counts the number of towers that are visible when looking into the grid from that direction. (Taller towers hide shorter ones behind them. So the sequence 2,1,4,3,5 would match a clue of 3 on the left, because the 1 is hidden behind the 2 and the 3 is hidden behind the 4. On the right, it would match a clue of 1 because the 5 hides everything else.) Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][Digit] (e.g. A4, B2, C3, D2, E4).", "answer": "A5, B6, C5, D1, E2", "image_path": "image453.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 454, "question": "Fill in the grid with towers whose heights range from 1 to the grid size, so that every possible height appears exactly once in each row and column, and so that each clue around the edge counts the number of towers that are visible when looking into the grid from that direction. (Taller towers hide shorter ones behind them. So the sequence 2,1,4,3,5 would match a clue of 3 on the left, because the 1 is hidden behind the 2 and the 3 is hidden behind the 4. On the right, it would match a clue of 1 because the 5 hides everything else.) Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][Digit] (e.g. A4, B2, C3, D2, E4).", "answer": "A3, B2, C3, D4, E1", "image_path": "image454.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 455, "question": "Complete the track from A to B so that the rows and columns contain the same number of track segments as are indicated in the clues to the top and right of the grid. There are only straight and 90-degree curved rail sections, and the track may not cross itself. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. For each labeled square, identify the edge crossed by the final track: top (0), bottom (1), left (2), or right (3). Answer in format: A01, which means the top and bottom edge are crossed by the final track. NOTE: The first number need to smaller than the second number. If no edges are crossed, just answer the letter without digits.", "answer": "A, B13, C02, D, E01", "image_path": "image455.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 456, "question": "Complete the track from A to B so that the rows and columns contain the same number of track segments as are indicated in the clues to the top and right of the grid. There are only straight and 90-degree curved rail sections, and the track may not cross itself. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. For each labeled square, identify the edge crossed by the final track: top (0), bottom (1), left (2), or right (3). Answer in format: A01, which means the top and bottom edge are crossed by the final track. NOTE: The first number need to smaller than the second number. If no edges are crossed, just answer the letter without digits.", "answer": "A01, B, C13, D, E02", "image_path": "image456.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 457, "question": "Complete the track from A to B so that the rows and columns contain the same number of track segments as are indicated in the clues to the top and right of the grid. There are only straight and 90-degree curved rail sections, and the track may not cross itself. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. For each labeled square, identify the edge crossed by the final track: top (0), bottom (1), left (2), or right (3). Answer in format: A01, which means the top and bottom edge are crossed by the final track. NOTE: The first number need to smaller than the second number. If no edges are crossed, just answer the letter without digits.", "answer": "A, B23, C03, D, E", "image_path": "image457.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Divide‑and-Conquer", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 458, "question": "Fill in every grid square which doesn't contain a mirror with either a ghost, a vampire, or a zombie. The numbers round the grid edges show how many monsters must be visible along your line of sight if you look directly into the grid from that position, along a row or column. Zombies are always visible; ghosts are only visible when reflected in at least one mirror; vampires are only visible when not reflected in any mirror. Select a square to mark it with the sign letter: G for a ghost, V for a vampire or Z for a zombie. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the sign in labeled squares and output final answer in format: [SquareID][TypeID]. [TypeID] can be g (means ghost), v(means vampire ) and z(means zombie). For example, Ag means the square A is with a ghost.", "answer": "Av, Bv, Cz, Dg, Ev", "image_path": "image458.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 459, "question": "Fill in every grid square which doesn't contain a mirror with either a ghost, a vampire, or a zombie. The numbers round the grid edges show how many monsters must be visible along your line of sight if you look directly into the grid from that position, along a row or column. Zombies are always visible; ghosts are only visible when reflected in at least one mirror; vampires are only visible when not reflected in any mirror. Select a square to mark it with the sign letter: G for a ghost, V for a vampire or Z for a zombie. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the sign in labeled squares and output final answer in format: [SquareID][TypeID]. [TypeID] can be g (means ghost), v(means vampire ) and z(means zombie). For example, Ag means the square A is with a ghost.", "answer": "Ag, Bv, Cz, Dg, Ev", "image_path": "image459.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 460, "question": "Fill in every grid square which doesn't contain a mirror with either a ghost, a vampire, or a zombie. The numbers round the grid edges show how many monsters must be visible along your line of sight if you look directly into the grid from that position, along a row or column. Zombies are always visible; ghosts are only visible when reflected in at least one mirror; vampires are only visible when not reflected in any mirror. Select a square to mark it with the sign letter: G for a ghost, V for a vampire or Z for a zombie. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the sign in labeled squares and output final answer in format: [SquareID][TypeID]. [TypeID] can be g (means ghost), v(means vampire ) and z(means zombie). For example, Ag means the square A is with a ghost.", "answer": "Az, Bv, Cz, Dv, Eg", "image_path": "image460.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 461, "question": "Fill in the grid with numbers from 1 to the grid size, so that every number appears exactly once in each row and column, and so that all the < signs represent true inequalities (i.e. the number at the pointed end is smaller than the number at the open end). Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][Digit]. For example, A4 means that digit 4 should be filled in square A.", "answer": "A3, B6, C3, D3, E6", "image_path": "image461.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 462, "question": "Fill in the grid with numbers from 1 to the grid size, so that every number appears exactly once in each row and column, and so that all the < signs represent true inequalities (i.e. the number at the pointed end is smaller than the number at the open end). Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][Digit]. For example, A4 means that digit 4 should be filled in square A.", "answer": "A1, B7, C7, D7, E2", "image_path": "image462.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 463, "question": "Fill in the grid with numbers from 1 to the grid size, so that every number appears exactly once in each row and column, and so that all the < signs represent true inequalities (i.e. the number at the pointed end is smaller than the number at the open end). Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][Digit]. For example, A4 means that digit 4 should be filled in square A.", "answer": "A7, B7, C7, D4, E6", "image_path": "image463.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 464, "question": "Colour every square either black or white, in such a way that: no three consecutive squares, horizontally or vertically, are the same colour, each row and column contains the same number of black and white squares. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][TypeID]. [TypeID] can be 0(black) or 1(white). For example, A0 means that the square A is in black.", "answer": "A1, B1, C0, D1, E1", "image_path": "image464.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 465, "question": "Colour every square either black or white, in such a way that: no three consecutive squares, horizontally or vertically, are the same colour, each row and column contains the same number of black and white squares. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][TypeID]. [TypeID] can be 0(black) or 1(white). For example, A0 means that the square A is in black.", "answer": "A1, B0, C1, D0, E1", "image_path": "image465.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 466, "question": "Colour every square either black or white, in such a way that: no three consecutive squares, horizontally or vertically, are the same colour, each row and column contains the same number of black and white squares. Label some $1\\time 1$ squares (the smallest square unit) in the alphabetical order. Determine the digit in labeled squares and output final answer in format: [SquareID][TypeID]. [TypeID] can be 0(black) or 1(white). For example, A0 means that the square A is in black.", "answer": "A0, B1, C0, D1, E0", "image_path": "image466.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 467, "question": "The goal is to shade some cells in the grid to reveal a single white path from S (Start) to G (Goal) according to a series of rules. Rules: Lettered, Circled, or Triangular Cells are Always White: Any cell containing one of these symbols cannot be shaded black. White Path from S to G: The resulting white cells must form a single, continuous path from S to G. This path must always be exactly one cell wide, meaning it cannot branch off or become wider. Role of Circles and Triangles in the Path: All cells containing a circle must be part of the S-to-G path. All cells containing a triangle must not be part of the S-to-G path. 2x2 Area Color Restriction: No 2x2 area of cells within the grid can be all the same color (i.e., you cannot have a 2x2 area of all-white cells or all-black cells). Orthogonal Connectivity of White Cells: All white cells must be connected to each other by sharing sides (orthogonally). They cannot be connected only by their corners (diagonally). White Cells Cannot Form Loops: The S-to-G path must be unique, and no part of the white path can form a closed loop. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: [SquareID][Condition]. NOTE: [Condition] only can be 0(not on the path), or 1(on the path). The example answer: A1 means the square A is on the path.", "answer": "A0, B0, C1, D0, E1", "image_path": "image467.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 468, "question": "The goal is to shade some cells in the grid to reveal a single white path from S (Start) to G (Goal) according to a series of rules. Rules: Lettered, Circled, or Triangular Cells are Always White: Any cell containing one of these symbols cannot be shaded black. White Path from S to G: The resulting white cells must form a single, continuous path from S to G. This path must always be exactly one cell wide, meaning it cannot branch off or become wider. Role of Circles and Triangles in the Path: All cells containing a circle must be part of the S-to-G path. All cells containing a triangle must not be part of the S-to-G path. 2x2 Area Color Restriction: No 2x2 area of cells within the grid can be all the same color (i.e., you cannot have a 2x2 area of all-white cells or all-black cells). Orthogonal Connectivity of White Cells: All white cells must be connected to each other by sharing sides (orthogonally). They cannot be connected only by their corners (diagonally). White Cells Cannot Form Loops: The S-to-G path must be unique, and no part of the white path can form a closed loop. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: [SquareID][Condition]. NOTE: [Condition] only can be 0(not on the path), or 1(on the path). The example answer: A1 means the square A is on the path.", "answer": "A0, B0, C1, D1, E0", "image_path": "image468.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 469, "question": "The goal is to shade some cells in the grid to reveal a single white path from S (Start) to G (Goal) according to a series of rules. Rules: Lettered, Circled, or Triangular Cells are Always White: Any cell containing one of these symbols cannot be shaded black. White Path from S to G: The resulting white cells must form a single, continuous path from S to G. This path must always be exactly one cell wide, meaning it cannot branch off or become wider. Role of Circles and Triangles in the Path: All cells containing a circle must be part of the S-to-G path. All cells containing a triangle must not be part of the S-to-G path. 2x2 Area Color Restriction: No 2x2 area of cells within the grid can be all the same color (i.e., you cannot have a 2x2 area of all-white cells or all-black cells). Orthogonal Connectivity of White Cells: All white cells must be connected to each other by sharing sides (orthogonally). They cannot be connected only by their corners (diagonally). White Cells Cannot Form Loops: The S-to-G path must be unique, and no part of the white path can form a closed loop. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ square (the smallest square unit) in the alphabetical order. Give final answer in following format: [SquareID][Condition]. NOTE: [Condition] only can be 0(not on the path), or 1(on the path). The example answer: A1 means the square A is on the path.", "answer": "A0, B1, C0, D0, E1", "image_path": "image469.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic", "Combinatorics" ] } }, { "id": 470, "question": "This puzzle consists of a rectangular grid of an arbitrary size, which is divided into multiple regions. In each region, four cells must be shaded to form one of the following tetrominoes: L(1), I(2), T(3), or S(4). The tetrominoes can be rotated or reflected. Rules: Same Shape Restriction: When two tetrominoes in adjacent regions share an edge, they cannot be the same type (e.g., an L-shape cannot share an edge with another L-shape). Connectivity: All of the shaded tetrominoes must form a single, orthogonally-connected area (meaning all shaded cells must connect through their sides, not just at the corners). 2x2 Area Rule: The shaded cells cannot form any 2x2 squares. Now, please determine which two tetrominoes are contained in each of the regions A, B, C, D, and E. The number codes for the tetrominoes are as follows: 1 = L, 2 = I, 3 = T, 4 = S Give the final answer in the following format: [RegionIndex][tetrominoe1][tetrominoe2]. NOTE: For each region, the two tetrominoe numbers must be listed in ascending order (from 1 to 4). For example, an answer of A13 means that region A contains an L(1) and a T(3). The answers should be given in alphabetical order and the tetrominoe code (e.g. A13, B24, C11, D13, E14).", "answer": "A12, B44, C13, D34, E12", "image_path": "image470.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 471, "question": "This puzzle consists of a rectangular grid of an arbitrary size, which is divided into multiple regions. In each region, four cells must be shaded to form one of the following tetrominoes: L(1), I(2), T(3), or S(4). The tetrominoes can be rotated or reflected. Rules: Same Shape Restriction: When two tetrominoes in adjacent regions share an edge, they cannot be the same type (e.g., an L-shape cannot share an edge with another L-shape). Connectivity: All of the shaded tetrominoes must form a single, orthogonally-connected area (meaning all shaded cells must connect through their sides, not just at the corners). 2x2 Area Rule: The shaded cells cannot form any 2x2 squares. Now, please determine which two tetrominoes are contained in each of the regions A, B, C, D, and E. The number codes for the tetrominoes are as follows: 1 = L, 2 = I, 3 = T, 4 = S Give the final answer in the following format: [RegionIndex][tetrominoe1][tetrominoe2]. NOTE: For each region, the two tetrominoe numbers must be listed in ascending order (from 1 to 4). For example, an answer of A13 means that region A contains an L(1) and a T(3). The answers should be given in alphabetical order and the tetrominoe code (e.g. A13, B24, C11, D13, E14).", "answer": "A12, B14, C44, D34, E23", "image_path": "image471.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 472, "question": "This puzzle consists of a rectangular grid of an arbitrary size, which is divided into multiple regions. In each region, four cells must be shaded to form one of the following tetrominoes: L(1), I(2), T(3), or S(4). The tetrominoes can be rotated or reflected. Rules: Same Shape Restriction: When two tetrominoes in adjacent regions share an edge, they cannot be the same type (e.g., an L-shape cannot share an edge with another L-shape). Connectivity: All of the shaded tetrominoes must form a single, orthogonally-connected area (meaning all shaded cells must connect through their sides, not just at the corners). 2x2 Area Rule: The shaded cells cannot form any 2x2 squares. Now, please determine which two tetrominoes are contained in each of the regions A, B, C, D, and E. The number codes for the tetrominoes are as follows: 1 = L, 2 = I, 3 = T, 4 = S Give the final answer in the following format: [RegionIndex][tetrominoe1][tetrominoe2]. NOTE: For each region, the two tetrominoe numbers must be listed in ascending order (from 1 to 4). For example, an answer of A13 means that region A contains an L(1) and a T(3). The answers should be given in alphabetical order and the tetrominoe code .", "answer": "A13, B44, C11, D13, E14", "image_path": "image472.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 473, "question": "In the puzzle grid, you will see several thermometers, which may be unfilled, partially filled, or completely filled. The numbers outside the grid indicate the number of filled cells in that row or column. Puzzle Rules: The filling of each thermometer must always start from the bottom (the circular bulb) and fill upwards continuously. There can be no empty cells between the bulb and a filled cell, or between two filled cells within the same thermometer. The actual orientation of a thermometer (vertical, horizontal, or diagonal) does not change its filling rule; it always fills from the bottom(the bulb) to the top(the end). The goal of this puzzle is to use logical deduction, based on the numbers outside the grid, to determine the filled state of every thermometer. In the alphabetical order of the thermometer labels, please output the number of mercury-filled cells for each thermometer. Give the final answer in the following format: [ThermometerIndex]-[Length]-[Mercury-filledLength]. The example answer: A-10-5 means the total length of thermometer A is 10, and the length of Mercury-filled cells is 5", "answer": "A-7-4, B-5-2, C-7-4, D-4-3, E-7-3", "image_path": "image473.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 474, "question": "In the puzzle grid, you will see several thermometers, which may be unfilled, partially filled, or completely filled. The numbers outside the grid indicate the number of filled cells in that row or column. Puzzle Rules: The filling of each thermometer must always start from the bottom (the circular bulb) and fill upwards continuously. There can be no empty cells between the bulb and a filled cell, or between two filled cells within the same thermometer. The actual orientation of a thermometer (vertical, horizontal, or diagonal) does not change its filling rule; it always fills from the bottom(the bulb) to the top(the end). The goal of this puzzle is to use logical deduction, based on the numbers outside the grid, to determine the filled state of every thermometer. In the alphabetical order of the thermometer labels, please output the number of mercury-filled cells for each thermometer. Give the final answer in the following format: [ThermometerIndex]-[Length]-[Mercury-filledLength]. The example answer: A-10-5 means the total length of thermometer A is 10, and the length of Mercury-filled cells is 5", "answer": "A-8-6, B-8-2, C-9-2, D-8-7, E-8-5", "image_path": "image474.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 475, "question": "In the puzzle grid, you will see several thermometers, which may be unfilled, partially filled, or completely filled. The numbers outside the grid indicate the number of filled cells in that row or column. Puzzle Rules: The filling of each thermometer must always start from the bottom (the circular bulb) and fill upwards continuously. There can be no empty cells between the bulb and a filled cell, or between two filled cells within the same thermometer. The actual orientation of a thermometer (vertical, horizontal, or diagonal) does not change its filling rule; it always fills from the bottom(the bulb) to the top(the end). The goal of this puzzle is to use logical deduction, based on the numbers outside the grid, to determine the filled state of every thermometer. In the alphabetical order of the thermometer labels, please output the number of mercury-filled cells for each thermometer. Give the final answer in the following format: [ThermometerIndex]-[Length]-[Mercury-filledLength]. The example answer: A-10-5 means the total length of thermometer A is 10, and the length of Mercury-filled cells is 5", "answer": "A-12-2, B-11-0, C-16-5, D-12-9, E-10-5", "image_path": "image475.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Branch‑and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 476, "question": "Some cells in the grid contain lines (bridges). The goal is to shade some cells black to create pairs of identical shapes (twins or lenses). Each pair of lenses must be symmetrical with respect to one of the bridges. Rules: The two resulting shaded regions (the lenses) cannot share an edge. Cells that contain a bridge cannot be shaded black. The numbers outside the grid indicate the number of black cells in the corresponding row or column. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C1, D0, E1", "image_path": "image476.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 477, "question": "Some cells in the grid contain lines (bridges). The goal is to shade some cells black to create pairs of identical shapes (twins or lenses). Each pair of lenses must be symmetrical with respect to one of the bridges. Rules: The two resulting shaded regions (the lenses) cannot share an edge. Cells that contain a bridge cannot be shaded black. The numbers outside the grid indicate the number of black cells in the corresponding row or column. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B0, C1, D0, E1", "image_path": "image477.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 478, "question": "Some cells in the grid contain lines (bridges). The goal is to shade some cells black to create pairs of identical shapes (twins or lenses). Each pair of lenses must be symmetrical with respect to one of the bridges. Rules: The two resulting shaded regions (the lenses) cannot share an edge. Cells that contain a bridge cannot be shaded black. The numbers outside the grid indicate the number of black cells in the corresponding row or column. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C1, D1, E0", "image_path": "image478.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Divide‑and-Conquer", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 479, "question": "Trilogy consists of a grid where some cells contain a shape: a square, a circle, or a triangle. The goal of the puzzle is to fill every cell with a shape, following two key rules for any three consecutive cells. In any row, column, or diagonal: Three consecutive shapes cannot be all the same. Three consecutive shapes cannot be all different. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][ShapeType]. NOTE: [ShapeType] only can be 1(square), 2(circle), or 3(triangle). The example answer: A2 means the cell A is filled by a shape of circle.", "answer": "A3, B3, C1, D1, E2", "image_path": "image479.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 480, "question": "Trilogy consists of a grid where some cells contain a shape: a square, a circle, or a triangle. The goal of the puzzle is to fill every cell with a shape, following two key rules for any three consecutive cells. In any row, column, or diagonal: Three consecutive shapes cannot be all the same. Three consecutive shapes cannot be all different. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][ShapeType]. NOTE: [ShapeType] only can be 1(square), 2(circle), or 3(triangle). The example answer: A2 means the cell A is filled by a shape of circle.", "answer": "A2, B2, C1, D3, E2", "image_path": "image480.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 481, "question": "Trilogy consists of a grid where some cells contain a shape: a square, a circle, or a triangle. The goal of the puzzle is to fill every cell with a shape, following two key rules for any three consecutive cells. In any row, column, or diagonal: Three consecutive shapes cannot be all the same. Three consecutive shapes cannot be all different. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][ShapeType]. NOTE: [ShapeType] only can be 1(square), 2(circle), or 3(triangle). The example answer: A2 means the cell A is filled by a shape of circle.", "answer": "A1, B1, C2, D1, E3", "image_path": "image481.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 482, "question": "The grid is divided into several regions. The goal is to place exactly one triangle, one square, and one circle in each region. The rules are: Identical shapes cannot be placed in adjacent cells, not even diagonally. All cells containing shapes must form a single, orthogonally connected group. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][ShapeType]. NOTE: [ShapeType] only can be 0(None), 1(square), 2(circle), or 3(triangle). The example answer: A2 means the cell A is filled by a shape of circle, A0 means the cell A should not be filled by any shape above.", "answer": "A0, B2, C0, D1, E3", "image_path": "image482.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 483, "question": "The grid is divided into several regions. The goal is to place exactly one triangle, one square, and one circle in each region. The rules are: Identical shapes cannot be placed in adjacent cells, not even diagonally. All cells containing shapes must form a single, orthogonally connected group. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][ShapeType]. NOTE: [ShapeType] only can be 0(None), 1(square), 2(circle), or 3(triangle). The example answer: A2 means the cell A is filled by a shape of circle, A0 means the cell A should not be filled by any shape above.", "answer": "A1, B0, C3, D2, E1", "image_path": "image483.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 484, "question": "The grid is divided into several regions. The goal is to place exactly one triangle, one square, and one circle in each region. The rules are: Identical shapes cannot be placed in adjacent cells, not even diagonally. All cells containing shapes must form a single, orthogonally connected group. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][ShapeType]. NOTE: [ShapeType] only can be 0(None), 1(square), 2(circle), or 3(triangle). The example answer: A2 means the cell A is filled by a shape of circle, A0 means the cell A should not be filled by any shape above.", "answer": "A1, B1, C3, D0, E0", "image_path": "image484.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 485, "question": "Makaro is a logic puzzle. The grid is divided into several regions. Rules: Each region must be filled with the numbers from 1 to the number of cells in that region. The grid may contain black cells with arrows. An arrow points to the largest number among the four orthogonally adjacent cells (up, down, left, or right). When two numbers are orthogonally adjacent across a region boundary, they must be different. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Number]. NOTE: [Number] means the number filled in the specific cell. The example answer: A4 means the cell A is filled in number 4.", "answer": "A2, B3, C2, D2, E1", "image_path": "image485.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 486, "question": "Makaro is a logic puzzle. The grid is divided into several regions. Rules: Each region must be filled with the numbers from 1 to the number of cells in that region. The grid may contain black cells with arrows. An arrow points to the largest number among the four orthogonally adjacent cells (up, down, left, or right). When two numbers are orthogonally adjacent across a region boundary, they must be different. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Number]. NOTE: [Number] means the number filled in the specific cell. The example answer: A4 means the cell A is filled in number 4.", "answer": "A2, B1, C2, D3, E1", "image_path": "image486.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 487, "question": "Makaro is a logic puzzle. The grid is divided into several regions. Rules: Each region must be filled with the numbers from 1 to the number of cells in that region. The grid may contain black cells with arrows. An arrow points to the largest number among the four orthogonally adjacent cells (up, down, left, or right). When two numbers are orthogonally adjacent across a region boundary, they must be different. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Number]. NOTE: [Number] means the number filled in the specific cell. The example answer: A4 means the cell A is filled in number 4.", "answer": "A3, B2, C1, D3, E4", "image_path": "image487.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 488, "question": "The grid is divided into several regions. The goal of the puzzle is to place one number in each region. The value of the number must be equal to the size of its region. The distance between two horizontally or vertically adjacent numbers must be equal to the absolute difference between those two numbers. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID]-[CellNumber]-[RawID]-[ColumnID]. NOTE: [CellNumber] means the number of cells in the specific region. The RawID and ColumnID are the specific position of the cell placed a number. The example answer: A-4-1-2 means the region A has 4 cells totally, and the number 4 will be placed at (1, 2)", "answer": "A-8-6-2, B-6-7-8, C-8-6-1, D-8-3-4, E-6-1-7", "image_path": "image488.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 489, "question": "The grid is divided into several regions. The goal of the puzzle is to place one number in each region. The value of the number must be equal to the size of its region. The distance between two horizontally or vertically adjacent numbers must be equal to the absolute difference between those two numbers. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID]-[CellNumber]-[RawID]-[ColumnID]. NOTE: [CellNumber] means the number of cells in the specific region. The RawID and ColumnID are the specific position of the cell placed a number. The example answer: A-4-1-2 means the region A has 4 cells totally, and the number 4 will be placed at (1, 2)", "answer": "A-5-9-3, B-8-6-7, C-5-5-2, D-10-3-4, E-6-1-7", "image_path": "image489.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 490, "question": "The grid is divided into several regions. The goal of the puzzle is to place one number in each region. The value of the number must be equal to the size of its region. The distance between two horizontally or vertically adjacent numbers must be equal to the absolute difference between those two numbers. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID]-[CellNumber]-[RawID]-[ColumnID]. NOTE: [CellNumber] means the number of cells in the specific region. The RawID and ColumnID are the specific position of the cell placed a number. The example answer: A-4-1-2 means the region A has 4 cells totally, and the number 4 will be placed at (1, 2)", "answer": "A-5-8-3, B-11-7-12, C-13-10-12, D-12-7-7, E-13-2-9", "image_path": "image490.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 491, "question": "Akiperago is a logic puzzle. The grid contains some cells with numbers. The goal is to shade some cells black according to the following rules: Islands: The black cells form islands. An island is a group of orthogonally connected black cells. Any two islands cannot share an edge; they can only connect via their corners. If an island contains a numbered cell, that number represents the total count of black cells in that island. An island can contain several numbered cells (if so, all numbers within that island will be the same). Archipelagos: All islands are part of archipelagos. An archipelago is a group of two or more islands that are connected via their corners. If an archipelago is composed of N islands, then the sizes of those islands must be the numbers 1 through N (in no particular order). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$.Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C0, D1, E1", "image_path": "image491.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 492, "question": "Akiperago is a logic puzzle. The grid contains some cells with numbers. The goal is to shade some cells black according to the following rules: Islands: The black cells form islands. An island is a group of orthogonally connected black cells. Any two islands cannot share an edge; they can only connect via their corners. If an island contains a numbered cell, that number represents the total count of black cells in that island. An island can contain several numbered cells (if so, all numbers within that island will be the same). Archipelagos: All islands are part of archipelagos. An archipelago is a group of two or more islands that are connected via their corners. If an archipelago is composed of N islands, then the sizes of those islands must be the numbers 1 through N (in no particular order). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$.Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B1, C1, D0, E0", "image_path": "image492.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 493, "question": "Akiperago is a logic puzzle. The grid contains some cells with numbers. The goal is to shade some cells black according to the following rules: Islands: The black cells form islands. An island is a group of orthogonally connected black cells. Any two islands cannot share an edge; they can only connect via their corners. If an island contains a numbered cell, that number represents the total count of black cells in that island. An island can contain several numbered cells (if so, all numbers within that island will be the same). Archipelagos: All islands are part of archipelagos. An archipelago is a group of two or more islands that are connected via their corners. If an archipelago is composed of N islands, then the sizes of those islands must be the numbers 1 through N (in no particular order). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$.Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B0, C1, D0, E0", "image_path": "image493.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 494, "question": "Nurimisaki (from Japanese, literally meaning to paint a cape) is a logic puzzle. Some cells in the grid contain circles, which may or may not include a number. The task is to shade some cells in the grid black according to the following rules: Path of White Cells: All white cells must form a single, orthogonally connected path that is exactly one cell wide. Cells containing circles are always part of this white path. Rules for Circles (Capes): A cell with a circle is a cape and must be a dead end of the path. This means it must have exactly one orthogonally adjacent white cell. A number inside a circle indicates the total number of white cells visible horizontally and vertically from that cell, including the cell itself. Rules for Other Cells: Any white cell that does not contain a circle must have at least two orthogonally adjacent white cells (to continue the path). Any 2x2 area of cells in the grid cannot be entirely the same color (i.e., no 2x2 blocks of all-white or all-black cells). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B0, C1, D1, E1", "image_path": "image494.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 495, "question": "Nurimisaki (from Japanese, literally meaning to paint a cape) is a logic puzzle. Some cells in the grid contain circles, which may or may not include a number. The task is to shade some cells in the grid black according to the following rules: Path of White Cells: All white cells must form a single, orthogonally connected path that is exactly one cell wide. Cells containing circles are always part of this white path. Rules for Circles (Capes): A cell with a circle is a cape and must be a dead end of the path. This means it must have exactly one orthogonally adjacent white cell. A number inside a circle indicates the total number of white cells visible horizontally and vertically from that cell, including the cell itself. Rules for Other Cells: Any white cell that does not contain a circle must have at least two orthogonally adjacent white cells (to continue the path). Any 2x2 area of cells in the grid cannot be entirely the same color (i.e., no 2x2 blocks of all-white or all-black cells). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C0, D0, E1", "image_path": "image495.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 496, "question": "Nurimisaki (from Japanese, literally meaning to paint a cape) is a logic puzzle. Some cells in the grid contain circles, which may or may not include a number. The task is to shade some cells in the grid black according to the following rules: Path of White Cells: All white cells must form a single, orthogonally connected path that is exactly one cell wide. Cells containing circles are always part of this white path. Rules for Circles (Capes): A cell with a circle is a cape and must be a dead end of the path. This means it must have exactly one orthogonally adjacent white cell. A number inside a circle indicates the total number of white cells visible horizontally and vertically from that cell, including the cell itself. Rules for Other Cells: Any white cell that does not contain a circle must have at least two orthogonally adjacent white cells (to continue the path). Any 2x2 area of cells in the grid cannot be entirely the same color (i.e., no 2x2 blocks of all-white or all-black cells). Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B0, C0, D1, E1", "image_path": "image496.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 497, "question": "Regional Yajilin is a puzzle solved on a grid that is divided into different regions. You need to accomplish two goals: Shade some cells black: A number given in a region indicates the exact number of cells that must be shaded black within that region. Regions without a number can have any quantity of black cells. Draw a single, non-intersecting loop: You must draw a single, continuous, non-crossing loop that passes through all the white (unshaded) cells. Additionally, two important rules must be followed: No adjacent black cells: Any two shaded cells cannot be orthogonally adjacent (they cannot share a side). The loop must pass through all cells that are not shaded black. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white and and is definitely on the line of that loop.", "answer": "A0, B1, C1, D0, E1", "image_path": "image497.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Divide‑and-Conquer", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 498, "question": "Regional Yajilin is a puzzle solved on a grid that is divided into different regions. You need to accomplish two goals: Shade some cells black: A number given in a region indicates the exact number of cells that must be shaded black within that region. Regions without a number can have any quantity of black cells. Draw a single, non-intersecting loop: You must draw a single, continuous, non-crossing loop that passes through all the white (unshaded) cells. Additionally, two important rules must be followed: No adjacent black cells: Any two shaded cells cannot be orthogonally adjacent (they cannot share a side). The loop must pass through all cells that are not shaded black. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white and and is definitely on the line of that loop.", "answer": "A1, B1, C0, D0, E0", "image_path": "image498.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 499, "question": "Regional Yajilin is a puzzle solved on a grid that is divided into different regions. You need to accomplish two goals: Shade some cells black: A number given in a region indicates the exact number of cells that must be shaded black within that region. Regions without a number can have any quantity of black cells. Draw a single, non-intersecting loop: You must draw a single, continuous, non-crossing loop that passes through all the white (unshaded) cells. Additionally, two important rules must be followed: No adjacent black cells: Any two shaded cells cannot be orthogonally adjacent (they cannot share a side). The loop must pass through all cells that are not shaded black. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white and and is definitely on the line of that loop.", "answer": "A1, B1, C0, D1, E0", "image_path": "image499.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend", "Divide‑and-Conquer" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 500, "question": "Sujiken, derived from the Japanese word sujikai (meaning diagonal), is a variation of Sudoku invented by American George Heineman. The puzzle consists of a triangular grid. The goal is to fill the grid with digits from 1 to 9 such that: No digit is repeated in any row, column, or diagonal. No digit is repeated within each of the three large 3x3 square regions. No digit is repeated within each of the three large triangular regions marked by bold borders. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Fill some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E) with number 1 to 9. Give final answer in following format: [SquareID][Number]. The example answer: A1 means the square A will be filled with number 1.", "answer": "A9, B8, C5, D1, E7", "image_path": "image500.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic", "Number Theory" ] } }, { "id": 501, "question": "Sujiken, derived from the Japanese word sujikai (meaning diagonal), is a variation of Sudoku invented by American George Heineman. The puzzle consists of a triangular grid. The goal is to fill the grid with digits from 1 to 9 such that: No digit is repeated in any row, column, or diagonal. No digit is repeated within each of the three large 3x3 square regions. No digit is repeated within each of the three large triangular regions marked by bold borders. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Fill some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E) with number 1 to 9. Give final answer in following format: [SquareID][Number]. The example answer: A1 means the square A will be filled with number 1.", "answer": "A2, B1, C5, D3, E5", "image_path": "image501.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 502, "question": "Sujiken, derived from the Japanese word sujikai (meaning diagonal), is a variation of Sudoku invented by American George Heineman. The puzzle consists of a triangular grid. The goal is to fill the grid with digits from 1 to 9 such that: No digit is repeated in any row, column, or diagonal. No digit is repeated within each of the three large 3x3 square regions. No digit is repeated within each of the three large triangular regions marked by bold borders. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Fill some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E) with number 1 to 9. Give final answer in following format: [SquareID][Number]. The example answer: A1 means the square A will be filled with number 1.", "answer": "A2, B6, C7, D7, E1", "image_path": "image502.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 503, "question": "Some cells in the grid contain circles with numbers. The goal of Mobiriti is to shade some of the cells black. The rules are as follows: Cells with circles cannot be shaded black. The number in a circle indicates how many white cells can be reached by moving horizontally or vertically from that cell. This is the total size of the contiguous white region that the circle belongs to.Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B0, C0, D1, E1", "image_path": "image503.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 504, "question": "Some cells in the grid contain circles with numbers. The goal of Mobiriti is to shade some of the cells black. The rules are as follows: Cells with circles cannot be shaded black. The number in a circle indicates how many white cells can be reached by moving horizontally or vertically from that cell. This is the total size of the contiguous white region that the circle belongs to.Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C1, D0, E0", "image_path": "image504.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 505, "question": "Some cells in the grid contain circles with numbers. The goal of Mobiriti is to shade some of the cells black. The rules are as follows: Cells with circles cannot be shaded black. The number in a circle indicates how many white cells can be reached by moving horizontally or vertically from that cell. This is the total size of the contiguous white region that the circle belongs to.Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C1, D1, E0", "image_path": "image505.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 506, "question": "Oases is a logic puzzle. Some cells in the rectangular or square grid contain circles with numbers. The goal of the game is to shade some cells black (leaving the others white) so that the following rules are met: All unshaded (white) cells must be orthogonally connected, forming a single area. Shaded cells cannot share an edge (they cannot be orthogonally adjacent). Cells containing circles cannot be shaded black. The unshaded cells must not form any 2x2 areas. The number in each circled cell represents the number of other circles that can be reached from it by traveling only through empty (unshaded) cells. (A circle is only counted once, even if it can be reached via multiple paths.) Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B0, C1, D1, E1", "image_path": "image506.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 507, "question": "Oases is a logic puzzle. Some cells in the rectangular or square grid contain circles with numbers. The goal of the game is to shade some cells black (leaving the others white) so that the following rules are met: All unshaded (white) cells must be orthogonally connected, forming a single area. Shaded cells cannot share an edge (they cannot be orthogonally adjacent). Cells containing circles cannot be shaded black. The unshaded cells must not form any 2x2 areas. The number in each circled cell represents the number of other circles that can be reached from it by traveling only through empty (unshaded) cells. (A circle is only counted once, even if it can be reached via multiple paths.) Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B0, C0, D1, E1", "image_path": "image507.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 508, "question": "Oases is a logic puzzle. Some cells in the rectangular or square grid contain circles with numbers. The goal of the game is to shade some cells black (leaving the others white) so that the following rules are met: All unshaded (white) cells must be orthogonally connected, forming a single area. Shaded cells cannot share an edge (they cannot be orthogonally adjacent). Cells containing circles cannot be shaded black. The unshaded cells must not form any 2x2 areas. The number in each circled cell represents the number of other circles that can be reached from it by traveling only through empty (unshaded) cells. (A circle is only counted once, even if it can be reached via multiple paths.) Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B1, C0, D1, E1", "image_path": "image508.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 509, "question": "Some cells in the Stars and Arrows grid contain arrows. The goal of the puzzle is to place stars into some of the empty cells. The rules are as follows: Each arrow must point to exactly one star. Each star must be pointed to by exactly one arrow. The numbers outside the grid indicate the number of stars in the corresponding row or column. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][PlaceStar]. NOTE: [PlaceStar] only can be 0(not place star), or 1(place star). The example answer: A1 means the square A is placed a star.", "answer": "A1, B1, C0, D0, E1", "image_path": "image509.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 510, "question": "Some cells in the Stars and Arrows grid contain arrows. The goal of the puzzle is to place stars into some of the empty cells. The rules are as follows: Each arrow must point to exactly one star. Each star must be pointed to by exactly one arrow. The numbers outside the grid indicate the number of stars in the corresponding row or column. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][PlaceStar]. NOTE: [PlaceStar] only can be 0(not place star), or 1(place star). The example answer: A1 means the square A is placed a star.", "answer": "A1, B0, C0, D0, E0", "image_path": "image510.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 511, "question": "Some cells in the Stars and Arrows grid contain arrows. The goal of the puzzle is to place stars into some of the empty cells. The rules are as follows: Each arrow must point to exactly one star. Each star must be pointed to by exactly one arrow. The numbers outside the grid indicate the number of stars in the corresponding row or column. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][PlaceStar]. NOTE: [PlaceStar] only can be 0(not place star), or 1(place star). The example answer: A1 means the square A is placed a star.", "answer": "A0, B1, C0, D1, E0", "image_path": "image511.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive‑and‑Comprehend" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 512, "question": "Hebi-Ichigo (from Japanese, literally Snake-Strawberry) is a logic puzzle. The goal is to draw one or more snakes each five cells long, on the grid according to the following rules: The Snakes: Each snake is a continuous chain of five cells, connected horizontally or vertically, and numbered sequentially from 1 (the head) to 5 (the tail). Snakes cannot touch each other horizontally or vertically, but they can touch diagonally. Line of Sight: A snake cannot be in front of another snake's head. The eyes of a snake are on its head (the '1' cell), looking in the direction opposite to its body (the '2' cell). This line of sight continues until it reaches a black cell or the edge of the grid, and it cannot pass through any part of another snake. Black Cells (Strawberries): The black cells are the clues. A number inside a black cell indicates the sum of the first digits it sees when looking in all four orthogonal directions (up, down, left, and right). If a line of sight from a black cell hits the grid edge or another black cell before seeing a snake, that direction contributes 0 to the sum. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Number]. NOTE: [Number] only can be 0(Nothing to place), or 1,2, ..., 5(number from 1 to 5 to place). The example answer: A1 means the cell A will be placed a number 1.", "answer": "A2, B0, C0, D3, E3", "image_path": "image512.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 513, "question": "Hebi-Ichigo (from Japanese, literally Snake-Strawberry) is a logic puzzle. The goal is to draw one or more snakes each five cells long, on the grid according to the following rules: The Snakes: Each snake is a continuous chain of five cells, connected horizontally or vertically, and numbered sequentially from 1 (the head) to 5 (the tail). Snakes cannot touch each other horizontally or vertically, but they can touch diagonally. Line of Sight: A snake cannot be in front of another snake's head. The eyes of a snake are on its head (the '1' cell), looking in the direction opposite to its body (the '2' cell). This line of sight continues until it reaches a black cell or the edge of the grid, and it cannot pass through any part of another snake. Black Cells (Strawberries): The black cells are the clues. A number inside a black cell indicates the sum of the first digits it sees when looking in all four orthogonal directions (up, down, left, and right). If a line of sight from a black cell hits the grid edge or another black cell before seeing a snake, that direction contributes 0 to the sum. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Number]. NOTE: [Number] only can be 0(Nothing to place), or 1,2, ..., 5(number from 1 to 5 to place). The example answer: A1 means the cell A will be placed a number 1.", "answer": "A3, B0, C2, D0, E0", "image_path": "image513.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 514, "question": "Hebi-Ichigo (from Japanese, literally Snake-Strawberry) is a logic puzzle. The goal is to draw one or more snakes each five cells long, on the grid according to the following rules: The Snakes: Each snake is a continuous chain of five cells, connected horizontally or vertically, and numbered sequentially from 1 (the head) to 5 (the tail). Snakes cannot touch each other horizontally or vertically, but they can touch diagonally. Line of Sight: A snake cannot be in front of another snake's head. The eyes of a snake are on its head (the '1' cell), looking in the direction opposite to its body (the '2' cell). This line of sight continues until it reaches a black cell or the edge of the grid, and it cannot pass through any part of another snake. Black Cells (Strawberries): The black cells are the clues. A number inside a black cell indicates the sum of the first digits it sees when looking in all four orthogonal directions (up, down, left, and right). If a line of sight from a black cell hits the grid edge or another black cell before seeing a snake, that direction contributes 0 to the sum. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Number]. NOTE: [Number] only can be 0(Nothing to place), or 1,2, ..., 5(number from 1 to 5 to place). The example answer: A1 means the cell A will be placed a number 1.", "answer": "A1, B5, C0, D0, E0", "image_path": "image514.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Branch‑and-Bound" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 515, "question": "Gaidoaro is a logic puzzle where the grid contains some arrows and a single star. The goal is to shade some cells black, such that all white cells form a single, orthogonally connected area. The following rules must be observed: Shaded cells cannot share any edges (i.e., they cannot be horizontally or vertically adjacent). Cells containing an arrow or the star cannot be shaded black. From every white cell, there must be a single, unique path to the star, moving only horizontally and vertically through other white cells. The arrows are clues that indicate the direction of the first step along the path from that cell towards the star. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B0, C1, D0, E0", "image_path": "image515.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 516, "question": "Gaidoaro is a logic puzzle where the grid contains some arrows and a single star. The goal is to shade some cells black, such that all white cells form a single, orthogonally connected area. The following rules must be observed: Shaded cells cannot share any edges (i.e., they cannot be horizontally or vertically adjacent). Cells containing an arrow or the star cannot be shaded black. From every white cell, there must be a single, unique path to the star, moving only horizontally and vertically through other white cells. The arrows are clues that indicate the direction of the first step along the path from that cell towards the star. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B0, C0, D1, E0", "image_path": "image516.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 517, "question": "Gaidoaro is a logic puzzle where the grid contains some arrows and a single star. The goal is to shade some cells black, such that all white cells form a single, orthogonally connected area. The following rules must be observed: Shaded cells cannot share any edges (i.e., they cannot be horizontally or vertically adjacent). Cells containing an arrow or the star cannot be shaded black. From every white cell, there must be a single, unique path to the star, moving only horizontally and vertically through other white cells. The arrows are clues that indicate the direction of the first step along the path from that cell towards the star. Let the coordinates of left-bottom and right-top cell be $1\\time 1$ and $N\\times N$. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B0, C0, D1, E0", "image_path": "image517.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 518, "question": "Nuraf (also known as Araf Nurikabe) is a logic puzzle that combines the rules of Nurikabe and Araf. Some cells in the grid contain numbers. The goal of the puzzle is to shade some cells black according to the following rules: The black cells divide the grid into regions of white cells (islands). Cells with numbers are always white. The black cells form a single, continuous sea that separates the islands. (This covers the rules Two islands cannot be connected and All black cells must be connected). The black cells cannot form any 2x2 squares. Each island must contain exactly two numbers. The area of an island (its cell count) must be strictly between the two numbers it contains. For example, if an island contains the numbers 1 and 4, its area must be either 2 or 3. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C0, D0, E1", "image_path": "image518.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 519, "question": "Nuraf (also known as Araf Nurikabe) is a logic puzzle that combines the rules of Nurikabe and Araf. Some cells in the grid contain numbers. The goal of the puzzle is to shade some cells black according to the following rules: The black cells divide the grid into regions of white cells (islands). Cells with numbers are always white. The black cells form a single, continuous sea that separates the islands. (This covers the rules Two islands cannot be connected and All black cells must be connected). The black cells cannot form any 2x2 squares. Each island must contain exactly two numbers. The area of an island (its cell count) must be strictly between the two numbers it contains. For example, if an island contains the numbers 1 and 4, its area must be either 2 or 3. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C0, D0, E1", "image_path": "image519.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics", "Number Theory" ] } }, { "id": 520, "question": "Nuraf (also known as Araf Nurikabe) is a logic puzzle that combines the rules of Nurikabe and Araf. Some cells in the grid contain numbers. The goal of the puzzle is to shade some cells black according to the following rules: The black cells divide the grid into regions of white cells (islands). Cells with numbers are always white. The black cells form a single, continuous sea that separates the islands. (This covers the rules Two islands cannot be connected and All black cells must be connected). The black cells cannot form any 2x2 squares. Each island must contain exactly two numbers. The area of an island (its cell count) must be strictly between the two numbers it contains. For example, if an island contains the numbers 1 and 4, its area must be either 2 or 3. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B1, C0, D0, E0", "image_path": "image520.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics", "Number Theory" ] } }, { "id": 521, "question": "Stostone (also written as Sto-Stone) is a logic puzzle invented by Nikoli. The grid is divided into several regions. The goal is to shade some cells (the stones) black according to the following rules: All shaded cells within a single region must be orthogonally connected. A number in a region indicates the exact number of cells that must be shaded in that region. In a region without a number, any quantity of cells (at least one) may be shaded. When two cells are orthogonally adjacent across a region border, at least one of them must be white. (In other words, shaded cells cannot touch across borders). If all the stones were to fall to the bottom of the grid within their respective columns, they must perfectly cover the bottom half of the grid. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A1, B0, C0, D1, E1", "image_path": "image521.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Geometry", "Combinatorics" ] } }, { "id": 522, "question": "Stostone (also written as Sto-Stone) is a logic puzzle invented by Nikoli. The grid is divided into several regions. The goal is to shade some cells (the stones) black according to the following rules: All shaded cells within a single region must be orthogonally connected. A number in a region indicates the exact number of cells that must be shaded in that region. In a region without a number, any quantity of cells (at least one) may be shaded. When two cells are orthogonally adjacent across a region border, at least one of them must be white. (In other words, shaded cells cannot touch across borders). If all the stones were to fall to the bottom of the grid within their respective columns, they must perfectly cover the bottom half of the grid. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C1, D0, E0", "image_path": "image522.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 523, "question": "Stostone (also written as Sto-Stone) is a logic puzzle invented by Nikoli. The grid is divided into several regions. The goal is to shade some cells (the stones) black according to the following rules: All shaded cells within a single region must be orthogonally connected. A number in a region indicates the exact number of cells that must be shaded in that region. In a region without a number, any quantity of cells (at least one) may be shaded. When two cells are orthogonally adjacent across a region border, at least one of them must be white. (In other words, shaded cells cannot touch across borders). If all the stones were to fall to the bottom of the grid within their respective columns, they must perfectly cover the bottom half of the grid. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Color]. NOTE: [Color] only can be 0(black), or 1(white). The example answer: A1 means the square A is in white.", "answer": "A0, B1, C1, D1, E1", "image_path": "image523.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Perceive‑and‑Comprehend", "Hypothesize-and-Test", "Divide‑and-Conquer" ], "sub_categories": [ "Logic", "Combinatorics", "Geometry" ] } }, { "id": 524, "question": "The puzzle is played on a grid where every cell contains a number. The top-left cell is always 1, and the bottom-right cell is N (the highest number). The other cells contain various numbers from 1 to N. The goal is to find a path from the top-left corner to the bottom-right corner that passes through a sequence of exactly N cells, visiting the numbers from 1 to N in order. The path connects horizontally or vertically adjacent cells and cannot visit the same cell twice. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Direction]. NOTE: [Direction] only can be 1(up), 2(down), 3(left), 4(right), or 0(Not on the path). The example answer: A1 means the cell A is on the path, and the next cell in the direction toward the destination is located up to cell A.", "answer": "A3, B4, C0, D0, E0", "image_path": "image524.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 525, "question": "The puzzle is played on a grid where every cell contains a number. The top-left cell is always 1, and the bottom-right cell is N (the highest number). The other cells contain various numbers from 1 to N. The goal is to find a path from the top-left corner to the bottom-right corner that passes through a sequence of exactly N cells, visiting the numbers from 1 to N in order. The path connects horizontally or vertically adjacent cells and cannot visit the same cell twice. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Direction]. NOTE: [Direction] only can be 1(up), 2(down), 3(left), 4(right), or 0(Not on the path). The example answer: A1 means the cell A is on the path, and the next cell in the direction toward the destination is located up to cell A.", "answer": "A2, B1, C3, D0, E4", "image_path": "image525.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 526, "question": "The puzzle is played on a grid where every cell contains a number. The top-left cell is always 1, and the bottom-right cell is N (the highest number). The other cells contain various numbers from 1 to N. The goal is to find a path from the top-left corner to the bottom-right corner that passes through a sequence of exactly N cells, visiting the numbers from 1 to N in order. The path connects horizontally or vertically adjacent cells and cannot visit the same cell twice. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Direction]. NOTE: [Direction] only can be 1(up), 2(down), 3(left), 4(right), or 0(Not on the path). The example answer: A1 means the cell A is on the path, and the next cell in the direction toward the destination is located up to cell A.", "answer": "A3, B1, C4, D2, E4", "image_path": "image526.png", "annotated": { "difficulty_tier": "Olympiad", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 527, "question": "The goal is to fill a number into every empty cell of the grid. Rules: Orthogonally adjacent cells with the same number are considered part of the same region.The number of cells in every region must be equal to the number written in those cells. (For example, a region made of 4 cells must be filled with the number 4). As a consequence, two regions with the same number cannot be orthogonally adjacent. There is an additional constraint: No region is allowed to be a rectangle or a square. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Number]. NOTE: [Number] means that the cell is filled with the [Number], which also indicates that the cell belongs to a region with a total of [Number] cells. The example answer: A5 means the cell A is a part of 5 region which have 5 cells totally.", "answer": "A7, B4, C4, D4, E8", "image_path": "image527.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 528, "question": "The goal is to fill a number into every empty cell of the grid. Rules: Orthogonally adjacent cells with the same number are considered part of the same region.The number of cells in every region must be equal to the number written in those cells. (For example, a region made of 4 cells must be filled with the number 4). As a consequence, two regions with the same number cannot be orthogonally adjacent. There is an additional constraint: No region is allowed to be a rectangle or a square. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Number]. NOTE: [Number] means that the cell is filled with the [Number], which also indicates that the cell belongs to a region with a total of [Number] cells. The example answer: A5 means the cell A is a part of 5 region which have 5 cells totally.", "answer": "A5, B6, C6, D8, E5", "image_path": "image528.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 529, "question": "The goal is to fill a number into every empty cell of the grid. Rules: Orthogonally adjacent cells with the same number are considered part of the same region.The number of cells in every region must be equal to the number written in those cells. (For example, a region made of 4 cells must be filled with the number 4). As a consequence, two regions with the same number cannot be orthogonally adjacent. There is an additional constraint: No region is allowed to be a rectangle or a square. Label some $1\\time 1$ cells (the smallest square unit) in the alphabetical order(from A to E). Give final answer in following format: [SquareID][Number]. NOTE: [Number] means that the cell is filled with the [Number], which also indicates that the cell belongs to a region with a total of [Number] cells. The example answer: A5 means the cell A is a part of 5 region which have 5 cells totally.", "answer": "A6, B4, C3, D3, E4", "image_path": "image529.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 530, "question": "Stan has five toys: a ball, a set of blocks, a game, a puzzle and a car. He puts each toy on a different shelf of the bookcase. The ball is higher than the blocks and lower than the car. The game is directly above the ball. On which shelf can the puzzle not be placed?\n", "answer": "3", "image_path": "image530.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 531, "question": "According to the rule given in the left picture below, we construct a numerical triangle with an integer number greater than 1 in each cell. Which of the numbers given in the answers cannot appear in the shaded cell?\n\nChoices: A. 154\nB. 100\nC. 90\nD. 88\nE. 60", "answer": "A", "image_path": "image531.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 532, "question": "Five cards are lying on the table in the order 1,3,5,4,2. You must get the cards in the order $1,2,3,4,5$. Per move, any two cards may be interchanged. How many moves do you need at least?\n", "answer": "2", "image_path": "image532.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 533, "question": "In how many ways can all the numbers $1,2,3,4,5,6$ be written in the squares of the figure (one in each square) so that there are no adjacent squares in which the difference of the numbers written is equal to 3? (Squares that share only a corner are not considered adjacent.)\n\nChoices: A. $3 \\cdot 2^{5}$\nB. $3^{6}$\nC. $6^{3}$\nD. $2 \\cdot 3^{5}$\nE. $3 \\cdot 5^{2}$", "answer": "A", "image_path": "image533.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error", "Divide-and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 534, "question": "Numbers 3,4 and two other unknown numbers are written in the cells of the $2 \\times 2$ table. It is known that the sums of numbers in the rows are equal to 5 and 10, and the sum of numbers in one of the columns is equal to 9. The larger number of the two unknown ones is\n", "answer": "6", "image_path": "image534.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 535, "question": "Diana wants to write whole numbers into each circle in the diagram, so that for all eight small triangles the sum of the three numbers in the corners is always the same. What is the maximum amount of different numbers she can use?\n", "answer": "3", "image_path": "image535.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 536, "question": "How many quadratic functions $y=a x^{2}+b x+c$ (with $a \\neq 0$ ) have graphs that go through at least 3 of the marked points?\n", "answer": "22", "image_path": "image536.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Algebra", "Logic" ] } }, { "id": 537, "question": "Nine whole numbers were written into the cells of a $3 \\times 3$-table. The sum of these nine numbers is 500. We know that the numbers in two adjacent cells (with a common sideline) differ by exactly 1. Which number is in the middle cell?\n", "answer": "56", "image_path": "image537.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 538, "question": "Three of the cards shown will be dealt to Nadia, the rest to Riny. Nadia multiplies the three values of her cards and Riny multiplies the two values of his cards. It turns out that the sum of those two products is a prime number. Determine the sum of the values of Nadia's cards.\n", "answer": "13", "image_path": "image538.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 539, "question": "The points of intersection of the network of bars shown are labelled with the numbers 1 to 10. The sums $S$ of the four numbers on the vertices of each square are\nall the same. What is the minimum value of $S$?\n", "answer": "20", "image_path": "image539.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic", "Geometry" ] } }, { "id": 540, "question": "Emma should colour in the three strips of the flag shown. She has four colours available. She can only use one colour for each strip and immediately adjacent strips are not to be of the same colour. How many different ways are there for her to colour in the flag? ", "answer": "36", "image_path": "image540.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 541, "question": "The numbers from 1 to 9 are to be distributed to the nine squares in the diagram according to the following rules: There is to be one number in each square. The sum of three adjacent numbers is always a multiple of 3 . The numbers 7 and 9 are already written in. How many ways are there to insert the remaining numbers? ", "answer": "24", "image_path": "image541.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic", "Combinatorics" ] } }, { "id": 542, "question": "Six numbers are written on the following cards, as shown.\n\nWhat is the smallest number you can form with the given cards?", "answer": "2309415687", "image_path": "image542.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 543, "question": "A square piece of paper is folded twice so that the result is a square again. In this square one of the corners is cut off. Then the paper is folded out. Which sample below cannot be obtained in this way?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image543.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 544, "question": "Johannes wrote the numbers 6,7 and 8 in the circles as shown. He wants to write the numbers 1, 2, 3, 4 and 5 in the remaining circles so that the sum of the numbers along each side of the square is 13. What will be the sum of the numbers in the grey circles?\n", "answer": "16", "image_path": "image544.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 545, "question": "Erwin has got the following paper pieces:\n\nWith these four pieces he must exactly cover a special shape. In which drawing will he manage this, if the piece is placed as shown?\n\nChoices: A. (A)\nB. (B)\nC. (C)\nD. (D)\nE. (E)", "answer": "C", "image_path": "image545.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 546, "question": "Elisabeth sorts the following cards:\n\nWith each move she is allowed to swap any two cards with each other. What is the smallest number of moves she needs in order to get the word KANGAROO.", "answer": "3", "image_path": "image546.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 547, "question": "In each box exactly one of the digits $0,1,2,3,4,5$ and 6 is to be written. Each digit will only be used once. Which digit has to be written in the grey box so that the sum is correct?\n", "answer": "5", "image_path": "image547.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Algebra", "Logic", "Number Theory" ] } }, { "id": 548, "question": "In the figure on the right a few of the small squares will be painted grey. In so doing no square that is made from four small grey squares must appear. At most how many of the squares in the figure can be painted grey?\n", "answer": "21", "image_path": "image548.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 549, "question": "Albin has put each of the digits from 1 to 9 in the fields of the table. In the diagram only 4 of these digits are visible. For the field containing the number 5, Albin noticed that the sum of the numbers in the neighbouring fields is 13. (neighbouring fields are fields which share a side). He noticed exactly the same for the field containing the digit 6 . Which digit had Albin written in the grey field?\n", "answer": "8", "image_path": "image549.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Algebra" ] } }, { "id": 550, "question": "The numbers $1,2,3,4$ and 9 are written into the squares on the following figure. The sum of the three numbers in the horizontal row, should be the same as the sum of the three numbers in the vertical column. Which number is written in the middle?\n", "answer": "9", "image_path": "image550.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 551, "question": "In this square there are 9 dots. The distance between the points is always the same. You can draw a square by joining 4 points. How many different sizes can such squares have?\n", "answer": "3", "image_path": "image551.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 552, "question": "Hannes has a game board with 11 spaces. He places one coin each on eight spaces that lie next to each other. He can choose on which space to place his first coin. No matter where Hannes starts some spaces will definitely be filled. How many spaces will definitely be filled?\n", "answer": "5", "image_path": "image552.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 553, "question": "Kate has four flowers, which have $6,7,8$ and 11 petals respectively. She now tears off one petal from each of three different flowers. She repeats this until it is no longer possible to tear off one petal from each of three different flowers. What is the minimum number of petals left over?\n", "answer": "2", "image_path": "image553.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Divide-and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 554, "question": "The rooms in Kanga's house are numbered. Eva enters the house through the main entrance. Eva has to walk through the rooms in such a way that each room that she enters has a number higher than the previous one. Through which door does Eva leave the house?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image554.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 555, "question": "Lea should write the numbers 1 to 7 in the fields of the given figure. There is only one number allowed in every field. Two consecutive numbers are not allowed to be in adjacent fields. Two fields are adjacent if they have one edge or one corner in common. Which numbers can she write into the field with the question mark?\n\nChoices: A. all 7 numbers\nB. only odd numbers\nC. only even numbers\nD. the number 4\nE. the numbers 1 or 7", "answer": "E", "image_path": "image555.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 556, "question": "Gaspar has these seven different pieces, formed by equal little squares.\n\nHe uses all these pieces to assemble rectangles with different perimeters, that is, with different shapes. How many different perimeters can he find?", "answer": "3", "image_path": "image556.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Perceive-and-Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics" ] } }, { "id": 557, "question": "Eva has the 5 stickers shown: . She stuck one of them on each of the 5 squares of this board so that is not on square 5, is on square 1, and is adjacent to and . On which square did Eva stick ?", "answer": "4", "image_path": "image557.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 558, "question": "Mia throws darts at balloons worth 3, 9, 13, 14 and 18 points. She scores 30 points in total. Which balloon does Mia definitely hit?\n", "answer": "3", "image_path": "image558.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 559, "question": "Elena wants to write the numbers from 1 to 9 in the squares shown. The arrows always point from a smaller number to a larger one. She has already written 5 and 7. Which number should she write instead of the question mark?\n", "answer": "6", "image_path": "image559.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic" ] } }, { "id": 560, "question": "Jan sends five postcards to his friends during his holiday.\nThe card for Michael does not have ducks.\nThe card for Lexi shows a dog.\nThe card for Clara shows the sun.\nThe card for Heidi shows kangaroos.\nThe card for Paula shows exactly two animals.\nWhich card does Jan send to Michael?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "A", "image_path": "image560.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 561, "question": "Hanni wants to colour in the circles in the diagram. When two circles are connected by a line they should have different colours. What is the minimum number of colours she needs?\n", "answer": "3", "image_path": "image561.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 562, "question": "Rebecca folds a square piece of paper twice. Then she cuts off one corner as you can see in the diagram.\n\nThen she unfolds the paper. What could the paper look like now?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "B", "image_path": "image562.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Geometry" ] } }, { "id": 563, "question": "The composite board shown in the picture consists of 20 fields $1 \\times 1$. How many possibilities are there exist to cover all 18 white fields with 9 rectangular stones $1 \\times 2$ ? (The board cannot be turned. Two possibilities are called different if at least one stone lies in another way.)\n", "answer": "4", "image_path": "image563.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 564, "question": "With how many ways one can get a number 2006 while following the arrows on the figure?\n", "answer": "8", "image_path": "image564.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Divide-and-Conquer", "Branch-and-Bound", "Trial-and-Error" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 565, "question": "Numbers 2, 3, 4 and one more unknown number are written in the cells of $2 \\times 2$ table. It is known that the sum of the numbers in the first row is equal to 9 , and the sum of the numbers in the second row is equal to 6 . The unknown number is\n", "answer": "6", "image_path": "image565.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Algebra", "Combinatorics", "Logic" ] } }, { "id": 566, "question": "Suppose you make a trip over the squared board shown, and you visit every square exactly once. Where must you start, if you can move only horizontally or vertically, but not diagonally?\n\nChoices: A. Only in the middle square\nB. Only at a corner square\nC. Only at an unshaded square\nD. Only at a shaded square\nE. At any square", "answer": "D", "image_path": "image566.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Branch-and-Bound", "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic" ] } }, { "id": 567, "question": "The numbers $1,4,7,10$ and 13 should be written into the squares so that the sum of the three numbers in the horizontal row is equal to the sum of the three numbers in the vertical column. What is the largest possible value of these sums?\n", "answer": "24", "image_path": "image567.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 568, "question": "Lydia draws a flower with 5 petals. She wants to colour in the flower using the colours white and black. How many different flowers can she draw with these two colours if the flower can also be just one colour?\n", "answer": "8", "image_path": "image568.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Branch-and-Bound", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 569, "question": "A few fields of a $4 \\times 4$ grid were painted red. The numbers in the bottom row and left column give the number of fields coloured red. The red was then rubbed away. Which of the following could grids could be a solution?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image569.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 570, "question": "Two buttons with smiling faces and two buttons with sad faces are in a row as shown in the picture. When you press a button the face changes, and so do the faces of the neighbouring buttons. What is the minimum number of button presses needed so that only smiling faces can be seen?\n", "answer": "3", "image_path": "image570.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 571, "question": "The kangaroos $A, B, C, D$ and $E$ sit in this order in a clockwise direction around a round table. After a bell sounds all but one kangaroo change seats with a neighbour. Afterwards they sit in the following order in a clockwise direction: A, E, B, D, C. Which kangaroo did not change places?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "B", "image_path": "image571.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic" ] } }, { "id": 572, "question": "The numbers $1,2,3,4$ and 5 have to be written into the five fields of this diagram according to the following rules: If one number is below another number, it has to be greater; if one number is to the right of another, it has to be greater. How many ways are there to place the numbers?\n", "answer": "6", "image_path": "image572.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 573, "question": "Instead of digits Hannes uses the letters A, B, C and D in a calculation. Different letters stand for different digits. Which digit does the letter B stand for?\n", "answer": "0", "image_path": "image573.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Branch-and-Bound" ], "sub_categories": [ "Algebra", "Combinatorics" ] } }, { "id": 574, "question": "The circles of the figure should be numbered from 0 to 10 , each with a different number. The five sums of the three numbers written on each diameter must be odd numbers. If one of these sums is the smallest possible, what will be the largest possible value of one of the remaining sums?\n", "answer": "21", "image_path": "image574.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 575, "question": "The map shows three bus stations at points $A, B$ and $C$. A tour from station $A$ to the Zoo and the Port and back to $A$ is $10 \\mathrm{~km}$ long. $A$ tour from station $B$ to the Park and the Zoo and back to B is $12 \\mathrm{~km}$ long. A tour from station C to the Port and the Park and back to $C$ is $13 \\mathrm{~km}$ long. Also, A tour from the Zoo to the Park and the Port and back to the Zoo is $15 \\mathrm{~km}$ long. How long is the shortest tour from A to B to $C$ and back to $A$?\n\nChoices: A. $18 \\mathrm{~km}$\nB. $20 \\mathrm{~km}$\nC. $25 \\mathrm{~km}$\nD. $35 \\mathrm{~km}$\nE. $50 \\mathrm{~km}$", "answer": "B", "image_path": "image575.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 576, "question": "Werner inserts numbers in various ways into the empty squares in such a way that the calculation is correct. He always uses four of the numbers 2,3, 4, 5 or 6 where in each calculation each number is only allowed to appear once. How many of the five numbers can Werner insert into the grey square?", "answer": "5", "image_path": "image576.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 577, "question": "Each animal in the picture on the right represents a natural number greater than zero. Different animals represent a different numbers. The sum of the two numbers of each column is written underneath each column. What is the maximum value the sum of the four numbers in the upper row can have?\n", "answer": "20", "image_path": "image577.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Algebra", "Number Theory", "Logic" ] } }, { "id": 578, "question": "Martin has three cards that are labelled on both sides with a number. Martin places the three cards on the table without paying attention to back or front. He adds the three numbers that he can then see. How many different sums can Martin get that way?\n\nChoices: A. 3\nB. 5\nC. 6\nD. 9\nE. A different amount.", "answer": "E", "image_path": "image578.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 579, "question": "The first row shows 11 cards, each with two letters. The second row shows rearangement of the cards. Which of the following could appear on the bottom line of the second row?\n\nChoices: A. ANJAMKILIOR\nB. RLIIMKOJNAA\nC. JANAMKILIRO\nD. RAONJMILIKA\nE. ANMAIKOLIRJ", "answer": "E", "image_path": "image579.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 580, "question": "In the picture any letter stands for some digit (different letters for different digits, equal letters for equal digits). Find the largest possible value of the number KAN.\n", "answer": "864", "image_path": "image580.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Algebra" ] } }, { "id": 581, "question": "In the figure there are nine regions inside the circles. The numbers 1 to 9 should be written in the regions so that the sum of the numbers in each circle is exactly 11. Which number has to go in the region with the question mark?\n", "answer": "6", "image_path": "image581.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 582, "question": "A piece of string is folded as shown in the diagram by folding it in the middle, then folding it in the middle again und finally folding it in the middle once more. Then this folded piece of string is cut so that several pieces emerge. Amongst the resulting pieces there are some with length $4 \\mathrm{~m}$ and some with length $9 \\mathrm{~m}$. Which of the following lengths cannot be the total length of the original piece of string?\n\nChoices: A. $52 \\mathrm{~m}$\nB. $68 \\mathrm{~m}$\nC. $72 \\mathrm{~m}$\nD. $88 \\mathrm{~m}$\nE. All answers are possible.", "answer": "C", "image_path": "image582.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 583, "question": "Four cars drive into a roundabout at the same point in time, each one coming from a different direction (see diagram). No car drives all the way around the roundabout, and no two cars leave at the same exit. In how many different ways can the cars exit the roundabout?\n", "answer": "9", "image_path": "image583.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 584, "question": "Natascha has some blue, red, yellow and green sticks of $1 \\mathrm{~cm}$ length. She wants to make a $3 \\times 3$ grid as shown in such a way that the four sides of each $1 \\times 1-$ square in the grid each are of a different colour. What is the minimum number of green sticks she can use?\n", "answer": "5", "image_path": "image584.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 585, "question": "Peter colours in each of the eight circles in one of the colours red, yellow or blue. Two circles that are directly connected by a line, are not allowed to be of the same colour. Which two circles does Peter definitely have to colour in the same colour?\n\nChoices: A. 5 and 8\nB. 1 and 6\nC. 2 and 7\nD. 4 and 5\nE. 3 and 6", "answer": "A", "image_path": "image585.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 586, "question": "In a tournament each of the 6 teams plays one match against every other team. In each round of matches, 3 take place simultaneously. A TV station has already decided which match it will broadcast for each round, as shown in the diagram. In which round will team D play against team F?\n", "answer": "1", "image_path": "image586.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 587, "question": "The digits 0 to 9 can be formed using matchsticks (see diagram). How many different positive whole numbers can be formed this way with exactly 6 matchsticks? ", "answer": "6", "image_path": "image587.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics", "Geometry" ] } }, { "id": 588, "question": "Unit squares of a squared board $2 \\times 3$ are coloured black and white like a chessboard (see picture). Determine the minimum number of steps necessary to achieve the reverse of the left board, following the rule: in each step, we must repaint two unit squares that have a joint edge, but we must repaint a black square with green, a green square with white and a white square with black.\n", "answer": "6", "image_path": "image588.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 589, "question": "How many ways are there to choose a white square and a black square from an $8 \\times 8$ chess-board so that these squares lie neither in the same row nor in the same column?\n", "answer": "768", "image_path": "image589.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test", "Divide-and-Conquer" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 590, "question": "In the picture any letter stands for some digit (different letters for different digits, equal letters for equal digits). Which digit is $\\mathrm{K}$?\n", "answer": "9", "image_path": "image590.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 591, "question": "In the diagram one should go from A to B along the arrows. Along the way calculate the sum of the numbers that are stepped on. How many different results can be obtained?\n", "answer": "2", "image_path": "image591.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 592, "question": "The cells of the $4 \\times 4$-table on the right should be coloured either in black or white. The numbers determine how many cells in each row/column should be black. How many ways are there to do the colouring in?\n", "answer": "5", "image_path": "image592.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 593, "question": "Three big boxes $P, Q$ and $R$ are stored in a warehouse. The upper picture on the right shows their placements from above. The boxes are so heavy that they can only be rotated $90^{\\circ}$ around a vertical edge as indicated in the pictures below. Now the boxes should be rotated to stand against the wall in a certain order. Which arrangement is possible?\n\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. All four arrangements are possible.", "answer": "B", "image_path": "image593.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 594, "question": "In the diagram Karl wants to add lines joining two of the marked points at a time, so that each of the seven marked points is joined to the same number of other marked points. What is the minimum number of lines he must draw?\n", "answer": "9", "image_path": "image594.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics", "Logic" ] } }, { "id": 595, "question": "The points $A$ and $B$ lie on a circle with centre $M$. The point $P$ lies on the straight line through $A$ and $M. P B$ touches the circle in $B$. The lengths of the segments $P A$ and $M B$ are whole numbers, and $P B=P A+6$. How many possible values for $M B$ are there?\n", "answer": "6", "image_path": "image595.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 596, "question": "Diana draws a rectangle made up of twelve squares onto a piece of squared paper. Some of the squares are coloured in black. She writes the number of adjacent black squares into every white square. The diagram shows an example of such a rectangle. Now she does the same with a rectangle made up of 2018 squares. What is the biggest number that she can obtain as the sum of all numbers in the white squares?\n", "answer": "3025", "image_path": "image596.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 597, "question": "In the addition beside, different letters represent different numbers and equal letters represent equal numbers. The resulting sum is a number of four digits, B being different from zero. What is the sum of the numbers of this number?\n\nChoices: A. AA\nB. BB\nC. AB\nD. BE\nE. EA", "answer": "B", "image_path": "image597.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Logic" ] } }, { "id": 598, "question": "Toninho wants to write strictly positive and consecutive whole numbers, in the nine places of the figure, so that the sum of the three numbers in each diameter is equal to 24. What is the largest possible sum for all the nine numbers?\n", "answer": "81", "image_path": "image598.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 599, "question": "Julia puts the nine chips on the right in a box. She then takes one chip at a time, without looking, and notes down its digit, obtaining, at the end, a number of nine different digits. What is the probability that the number written by Julia is divisible by 45?\n\nChoices: A. $\\frac{1}{9}$\nB. $\\frac{2}{9}$\nC. $\\frac{1}{3}$\nD. $\\frac{4}{9}$\nE. $\\frac{8}{9}$", "answer": "A", "image_path": "image599.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Probability & Statistics" ] } }, { "id": 600, "question": "Julia wrote four positive integers, one at each vertex of a triangular base pyramid. She calculated the sum of the numbers written on the vertices of one face and the product of the numbers written on the vertices of other two faces, obtaining 15, 20 and 30, respectively. What is the highest possible value of the product of the four numbers?\n", "answer": "120", "image_path": "image600.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Algebra", "Number Theory", "Combinatorics" ] } }, { "id": 601, "question": "Five cars participated in a race, starting in the order shown.\n. Whenever a car overtook another car, a point was awarded. The cars reached the finish line in the following order: . What is the smallest number of points in total that could have been awarded?", "answer": "6", "image_path": "image601.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 602, "question": "In the $4 \\times 4$ table some cells must be painted black. The numbers next to and below the table show how many cells in that row or column must be black. In how many ways can this table be painted?\n", "answer": "5", "image_path": "image602.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 603, "question": "A staircase has 2023 steps. Every third step is coloured in black. The first seven steps of this staircase can be fully seen in the diagram. Anita walks up the staircase and steps on each step exactly once. She can start with either the right or the left foot and then steps down alternately with the right or left foot. What is the minimum number of black steps she sets her right foot on? ", "answer": "337", "image_path": "image603.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 604, "question": "Natasha has many sticks of length 1 . Each stick is coloured blue, red, yellow or green. She wants to make a $3 \\times 3$ grid, as shown, so that each $1 \\times 1$ square in the grid has four sides of different colours. What is the smallest number of green sticks that she could use? ", "answer": "5", "image_path": "image604.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 605, "question": "The statements on the right give clues to the identity of a four-digit number.\n\nWhat is the last digit of the four-digit number?", "answer": "3", "image_path": "image605.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Logic" ] } }, { "id": 606, "question": "Evita wants to write the numbers 1 to 8 in the boxes of the grid shown, so that the sums of the numbers in the boxes in each row are equal and the sums of the numbers in the boxes in each column are equal. She has already written numbers 3,4 and 8 , as shown. What number should she write in the shaded box? ", "answer": "7", "image_path": "image606.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Logic" ] } }, { "id": 607, "question": "Bart wrote the number 1015 as a sum of numbers using only the digit 7 . He used a 7 a total of 10 times, including using the number 77 three times, as shown. Now he wants to write the number 2023 as a sum of numbers using only the digit 7, using a 7 a total of 19 times. How many times will the number 77 occur in the sum? ", "answer": "6", "image_path": "image607.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic", "Algebra" ] } }, { "id": 608, "question": "In the addition sum below, $a, b$ and $c$ stand for different digits.\n\nWhat is the value of $a+b+c$ ?", "answer": "16", "image_path": "image608.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Algebra", "Logic" ] } }, { "id": 609, "question": "Emily has two identical cards in the shape of equilateral triangles. She places them both onto a sheet of paper so that they touch or overlap and draws around the shape she creates. Which one of the following is it impossible for her to draw?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "E", "image_path": "image609.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 610, "question": "Tom throws two darts at the target shown in the diagram. Both his darts hit the target. For each dart, he scores the number of points shown in the region he hits. How many different totals could he score?\n", "answer": "9", "image_path": "image610.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 611, "question": "In the sum each letter stands for a different digit.\nWhat is the answer to the subtraction $ RN - KG $ ? ", "answer": "11", "image_path": "image611.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 612, "question": "Andrew wants to write the letters of the word KANGAROO in the cells of a $2 \\times 4$ grid such that each cell contains exactly one letter. He can write the first letter in any cell he chooses but each subsequent letter can only be written in a cell with at least one common vertex with the cell in which the previous letter was written. Which of the following arrangements of letters could he not produce in this way?\n\nChoices: A. A\nB. B\nC. C\nD. D\nE. E", "answer": "D", "image_path": "image612.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 613, "question": "What is the largest number of \" $\\mathrm{T}$ \" shaped pieces, as shown, that can be placed on the $4 \\times 5$ grid in the diagram, without any overlap of the pieces? ", "answer": "4", "image_path": "image613.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 614, "question": "Emily makes four identical numbered cubes using the net shown. She then glues them together so that only faces with the same number on are glued together to form the $2 \\times 2 \\times 1$ block shown. What is the largest possible total of all the numbers on the faces of the block that Emily could achieve? ", "answer": "68", "image_path": "image614.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 615, "question": "Each letter in the sum shown represents a different digit and the digit for $\\mathrm{A}$ is odd. What digit does $\\mathrm{G}$ represent?\n", "answer": "9", "image_path": "image615.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Algebra", "Logic" ] } }, { "id": 616, "question": "In the calculation alongside, different letters represent different digits.\n\nFind the least possible answer to the subtraction shown.", "answer": "110", "image_path": "image616.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Algebra", "Logic" ] } }, { "id": 617, "question": "The diagram shows a special die. Each pair of numbers on opposite faces has the same sum. The numbers on the hidden faces are all prime numbers. Which number is opposite to the 14 shown?\n", "answer": "23", "image_path": "image617.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 618, "question": "On a balance scale, three different masses were put at random on each pan and the result is shown in the picture. The masses are of 101, 102, 103, 104, 105 and 106 grams. What is the probability that the 106 gram mass stands on the heavier pan?\n\nChoices: A. $75 \\%$\nB. $80 \\%$\nC. $90 \\%$\nD. $95 \\%$\nE. $100 \\%$", "answer": "B", "image_path": "image618.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 619, "question": "The points $G$ and $I$ are on the circle with centre $H$, and $F I$ is tangent to the circle at $I$. The distances $F G$ and $H I$ are integers, and $F I=F G+6$. The point $G$ lies on the straight line through $F$ and $H$. How many possible values are there for $H I$ ? ", "answer": "6", "image_path": "image619.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 620, "question": "In the calculations shown, each letter stands for a digit. They are used to make some two-digit numbers. The two numbers on the left have a total of 79. What is the total of the four numbers on the right? ", "answer": "158", "image_path": "image620.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Algebra", "Logic" ] } }, { "id": 621, "question": "Vumos wants to write the integers 1 to 9 in the nine boxes shown so that the sum of the integers in any three adjacent boxes is a multiple of 3 . In how many ways can he do this? \nChoices: A. $6 \\times 6 \\times 6 \\times 6$\nB. $6 \\times 6 \\times 6$\nC. $2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2 \\times 2$\nD. $6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1$\nE. $9 \\times 8 \\times 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1$", "answer": "A", "image_path": "image621.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 622, "question": "A barcode of the type shown in the two examples is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either colour) has a width of 1 or 2 . The total width of the barcode is 12 . The barcodes are always read from left to right. How many distinct barcodes are possible?\n", "answer": "116", "image_path": "image622.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 623, "question": "At each of the vertices of a cube sits a Bunchkin. Two Bunchkins are said to be adjacent if and only if they sit at either end of one of the cube's edges. Each Bunchkin is either a 'truther', who always tells the truth, or a 'liar', who always lies. All eight Bunchkins say 'I am adjacent to exactly two liars'. What is the maximum number of Bunchkins who are telling the truth?\n", "answer": "4", "image_path": "image623.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Geometry" ] } }, { "id": 624, "question": "Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct.\n\nACROSS\n1. A square\n3. The answer to this Kangaroo question\n5. A square\nDOWN\n1. 4 down minus eleven\n2. One less than a cube\n4. The highest common factor of 1 down and 4 down is greater than one", "answer": "829", "image_path": "image624.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Geometry", "Logic" ] } }, { "id": 625, "question": "Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct. What is the answer to 3 ACROSS?\n\n\\section*{ACROSS}\n1. A composite factor of 1001\n3. Not a palindrome\n5. $p q^{3}$ where $p, q$ prime and $p \\neq q$\n\\section*{DOWN}\n1. One more than a prime, one less than a prime\n2. A multiple of 9\n4. $p^{3} q$ using the same $p, q$ as 5 ACROSS", "answer": "295", "image_path": "image625.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 626, "question": "Margot writes the numbers $1,2,3,4,5,6,7$ and 8 in the top row of a table, as shown. In the second row she plans to write the same set of numbers, in any order.\nEach number in the third row is obtained by finding the sum of the two numbers above it.\n\nIn how many different ways can Margot complete row 2 so that every entry in row 3 is even?", "answer": "576", "image_path": "image626.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 627, "question": "Each cell in this cross-number can be filled with a non-zero digit so that all of the conditions in the clues are satisfied. The digits used are not necessarily distinct.\n\n\\section*{ACROSS}\n1. Four less than a factor of 105.\n3. One more than a palindrome.\n5. The square-root of the answer to this Kangaroo question.\n\\section*{DOWN}\n1. Two less than a square.\n2. Four hundred less than a cube.\n4. Six less than the sum of the answers to two of the other clues.\nWhat is the square of the answer to 5 ACROSS?", "answer": "841", "image_path": "image627.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 628, "question": "In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes (\"eats'') all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\\times$. (The squares with two or more dotted edges have been removed form the original board in previous moves.)\n\n\nThe object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.", "answer": "792", "image_path": "image628.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 629, "question": "A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \\leq k \\leq 11$. With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a $0$ or a $1$ is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of $0$'s and $1$'s in the bottom row is the number in the top square a multiple of $3$?\n", "answer": "640", "image_path": "image629.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Divide-and-Conquer" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 630, "question": "The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and will paint each of the six sections a solid color. Find the number of ways you can choose to paint each of the six sections if no two adjacent section can be painted with the same color.\n\n", "answer": "732", "image_path": "image630.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 631, "question": "The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point \\(A\\). At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path \\(AJABCHCHIJA\\), which has \\(10\\) steps. Let \\(n\\) be the number of paths with \\(15\\) steps that begin and end at point \\(A\\). Find the remainder when \\(n\\) is divided by \\(1000\\).\n\n", "answer": "4", "image_path": "image631.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 632, "question": "There are $5$ yellow pegs, $4$ red pegs, $3$ green pegs, $2$ blue pegs, and $1$ orange peg on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?\n\n\nChoices: A. $0$\nB. $1$\nC. $5!\\cdot4!\\cdot3!\\cdot2!\\cdot1!$\nD. $\\frac{15!}{5!\\cdot4!\\cdot3!\\cdot2!\\cdot1!}$\nE. $15!$", "answer": "B", "image_path": "image632.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Divide-and-Conquer", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 633, "question": "The $ 5\\times 5$ grid shown contains a collection of squares with sizes from $ 1\\times 1$ to $ 5\\times 5$. How many of these squares contain the black center square?\n", "answer": "19", "image_path": "image633.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Geometry", "Combinatorics" ] } }, { "id": 634, "question": "In the five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $\\overline{AB}$,$\\overline{BC}$,$\\overline{CD}$,$\\overline{DE}$, and $\\overline{EA}$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?\n\n", "answer": "12", "image_path": "image634.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 635, "question": "The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In how many ways can this be done? \n", "answer": "1152", "image_path": "image635.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 636, "question": "As shown in the figure below a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?\n\n", "answer": "810", "image_path": "image636.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 637, "question": "A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?\n", "answer": "84", "image_path": "image637.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 638, "question": "Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2)$.\n\n\nArjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?\nChoices: A. (6,1,1)\nB. (6,2,1)\nC. (6,2,2)\nD. (6,3,1)\nE. (6,3,2)", "answer": "B", "image_path": "image638.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Branch-and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 639, "question": "A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?\n", "answer": "540", "image_path": "image639.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 640, "question": "Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \\cup S$, and $T$ contains $\\frac{1}{4}$ of the lattice points contained in $R \\cup S$. See the figure (not drawn to scale).\n\n\nThe fraction of lattice points in $S$ that are in $S \\cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \\cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?", "answer": "337", "image_path": "image640.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 641, "question": "Each square in a $5 \\times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:\n\nAny filled square with two or three filled neighbors remains filled.\nAny empty square with exactly three filled neighbors becomes a filled square.\nAll other squares remain empty or become empty.\n\nA sample transformation is shown in the figure below.\n\n\nSuppose the $5 \\times 5$ grid has a border of empty squares surrounding a $3 \\times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)\n", "answer": "22", "image_path": "image641.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 642, "question": "Each square in a $3\\times 3$ grid of squares is colored red, white, blue, or green so that every $2\\times 2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?\n", "answer": "72", "image_path": "image642.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 643, "question": "Twenty cubical blocks are arranged as shown. First, $10$ are arranged in a triangular pattern; then a layer of $6$, arranged in a triangular pattern, is centered on the $10$; then a layer of $3$, arranged in a triangular pattern, is centered on the $6$; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered $1$ through $10$ in some order. Each block in layers $2, 3$ and $4$ is assigned the number which is the sum of the numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.\n", "answer": "114", "image_path": "image643.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Branch-and-Bound" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 644, "question": "Several figures can be made by attaching two equilateral triangles to the regular pentagon $ ABCDE$ in two of the five positions shown. How many non-congruent figures can be constructed in this way?\n", "answer": "2", "image_path": "image644.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 645, "question": "\n\nFive cards are lying on a table as shown. Each card has a letter on one side and a whole number on the other side. Jane said, \"If a vowel is on one side of any card, then an even number is on the other side.\" Mary showed Jane was wrong by turning over one card. Which card did Mary turn over?\nChoices: A. $3$\nB. $4$\nC. $6$\nD. $\\text{P}$\nE. $\\text{Q}$", "answer": "A", "image_path": "image645.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 646, "question": "What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $ 4,5,6,7,8,9 $ in one of the six boxes in this addition problem?\n\n", "answer": "1047", "image_path": "image646.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 647, "question": "How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different.\n\n", "answer": "8", "image_path": "image647.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 648, "question": "In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then $C=$\n\n", "answer": "1", "image_path": "image648.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Algebra", "Logic" ] } }, { "id": 649, "question": "Eight $1\\times 1$ square tiles are arranged as shown so their outside edges form a polygon with a perimeter of $14$ units. Two additional tiles of the same size are added to the figure so that at least one side of each tile is shared with a side of one of the squares in the original figure. Which of the following could be the perimeter of the new figure?\n\n\nChoices: A. 15\nB. 17\nC. 18\nD. 19\nE. 20", "answer": "C", "image_path": "image649.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 650, "question": "How many triangles are in this figure? (Some triangles may overlap other triangles.)\n", "answer": "5", "image_path": "image650.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error", "Divide-and-Conquer" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 651, "question": "Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is\n\n", "answer": "24", "image_path": "image651.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 652, "question": "Three-digit powers of 2 and 5 are used in this ''cross-number'' puzzle. What is the only possible digit for the outlined square?\n\\begin{tabular}{lcl}\n\\textbf{ACROSS} & & \\textbf{DOWN} \\\\\n\\textbf{2}. $2^m$ & & \\textbf{1}. $5^n$\n\\end{tabular}\n", "answer": "6", "image_path": "image652.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 653, "question": "Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?\n\n", "answer": "5", "image_path": "image653.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 654, "question": "Three $\\text{A's}$, three $\\text{B's}$, and three $\\text{C's}$ are placed in the nine spaces so that each row and column contain one of each letter. If $\\text{A}$ is placed in the upper left corner, how many arrangements are possible?\n", "answer": "4", "image_path": "image654.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 655, "question": "In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path allows only moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.\n", "answer": "24", "image_path": "image655.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 656, "question": "In a sign pyramid a cell gets a \"+\" if the two cells below it have the same sign, and it gets a \"-\" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a \"+\" at the top of the pyramid?\n\n", "answer": "8", "image_path": "image656.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 657, "question": "The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number $x$ in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of $x$?\n", "answer": "8", "image_path": "image657.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Algebra", "Logic" ] } }, { "id": 658, "question": "Quadrilateral $ABCD$ (with $A, B, C$ not collinear and $A, D, C$ not collinear) has $AB = 4$, $BC = 7$, $CD = 10$, and $DA = 5$. Compute the number of possible integer lengths $AC$.\\n", "answer": "5", "image_path": "image658.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Mathematics", "answer_type": "Free-form questions", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 659, "question": "Sam spends his days walking around the following $2\\times 2$ grid of squares. Say that two squares are adjacent if they share a side. He starts at the square labeled $1$ and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal to $20$ (not counting the square he started on)?\\n", "answer": "167", "image_path": "image659.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Free-form questions", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic", "Algebra" ] } }, { "id": 660, "question": "Each unit square of a $4 \\times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)\\n", "answer": "18", "image_path": "image660.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 661, "question": "How many circles is the red point inside?", "answer": "4", "image_path": "image661.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 662, "question": "Write the numbers 1, 2, 3, 4, 5 in the square 5x5 in such a way that every row and every column has each number. How many kinds of number can be put to replace 'x'? ", "answer": "3", "image_path": "image662.png", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics" ] } }, { "id": 663, "question": "Tom writes down two five-digit number. He places different shapes on different digits. He places the same shape on the same digits. Find the value of first five-digit number. ", "answer": "34844", "image_path": "image663.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 664, "question": "As shown in the figure, use five different colors to color the six points $O$, $A$, $B$, $C$, $D$, and $E$ in the graph (the five colors are not necessarily all used). Each point must be colored with one color, and the two endpoints of each line segment in the graph must be colored with different colors. What is the number of different coloring methods?\n A. 480\n B. 720\n C. 1080\n D. 1200\n Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "D", "image_path": "image664.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic", "Geometry" ] } }, { "id": 665, "question": "As shown in the figure, 5 different colors are provided to color the 5 small regions in the diagram. Each region is required to be painted with only one color, and adjacent regions must have different colors. Define event $A$: 'The colors of regions 1 and 3 are different,' and event $B$: 'All regions are painted with different colors.' What is the value of $P(B|A)$?\n A. $\\frac{2}{7}$\n B. $\\frac{1}{2}$\n C. $\\frac{2}{3}$\n D. $\\frac{3}{4}$\n Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "B", "image_path": "image665.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 666, "question": "There are 4 different colors of paint. The 6 regions in the diagram are to be painted, requiring that adjacent regions have different colors. How many different coloring methods are there?\n A. 1512\n B. 1346\n C. 912\n D. 756\n Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "D", "image_path": "image666.png", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 667, "question": "A botanical garden plans to plant fruit trees in the 5 regions shown in the diagram. There are 5 different types of fruit trees available. Adjacent regions cannot be planted with the same type of fruit tree. How many different planting methods are there?\n A. 120\n B. 360\n C. 420\n D. 480\n Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "C", "image_path": "image667.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 668, "question": "As shown in the picture, there are 4 bottles (id from 1 to 4) filled with\n A. sugar water\n B. salt water\n C. white water\n D. wine\n of the same color, and each bottle has a different label. However, the label on the bottle of wine is fake, while the labels on the other bottles are real. What are the contents of the 4 bottles? Please answer in the format of 'bottle id (1-4):water type (A-D)'.", "answer": "{'1': 'C', '2': 'A', '3': 'D', '4': 'B'}", "image_path": "image668.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 669, "question": "Five cards, 1-5, are placed side by side on a plane as shown in the picture. There are 2 cards larger than 1 to the left, and 1 card larger than 2 to the right. The sum of all the numbers to the right of 3 is 7. What is the number on the rightmost card? Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end. ", "answer": "4", "image_path": "image669.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 670, "question": "There are 2 pieces each of blue, yellow, and transparent glass, making a total of 6 pieces. When blue and yellow are overlapped, the glass appears green. Three individuals, A, B, and C, each take 2 pieces from the 6 glasses and overlap them. After overlapping, the colors they see are as follows: A sees blue. B sees yellow. C also sees blue. Next, A gives one piece of glass to B, B gives one piece to C, and C gives one piece to A. After this exchange, each overlaps the 2 pieces they now hold, and the resulting colors are: A sees blue. B sees yellow. C now sees yellow. Question: What color glass did A give to B?\n A. blue\n B. yellow\n C. transparent", "answer": "C", "image_path": "image670.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 671, "question": "Two players, A and B, are playing a number-guessing game. Game Rules: 1. Player A selects 5 numbers she likes from the numbers 1 to 10. 2. Player B guesses the selected numbers and tells A the guesses. 3. If the guess is correct, Player A tells the sum of the correctly guessed numbers to Player B. 4. Repeat steps 2 and 3. As shown in the image, What is the sum of the 5 numbers selected by player A? Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end. ", "answer": "24", "image_path": "image671.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 672, "question": "There are 16 obstacles in the diagram. How many different routes can be drawn from A to C, moving only upwards and to the right? Please answer the question requiring an integer answer and provide the final value, e.g., 1, 2, 3, at the end. ", "answer": "70", "image_path": "image672.png", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 673, "question": "Replace the letters with numbers from 0 to 9 so that the equation is correct. You can only use each number once. Remember, both T's must be the same number! Answer in the form of 'letter:number'.", "answer": "VALIDATION RULES:\n1. Letters A,C,T,D,O,G,P,E,S each can only appear once. It's also ALLOWED to directly provide the nine digits in order, which each corresponds to the letters A,C,T,D,O,G,P,E,S.\n2. Only digits 0–9 may be used, and exactly 9 distinct digits must appear (no repeats).\n3. The substitution must satisfy CAT + DOG = PETS, where CAT and DOG are three-digit numbers (C×100 + A×10 + T, etc.) and PETS is a four-digit number (P×1000 + E×100 + T×10 + S).", "image_path": "image673.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Others", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Algebra", "Logic" ] } }, { "id": 674, "question": "What is the missing color if the part denoted with the question mark has the number 7?\n A.purple\n B.red\n C.yellow\n D.green\n Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "A", "image_path": "image674.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 675, "question": "Skyscrapers also known as \"Towers\" is a logic puzzle with simple rules and challenging solutions.\n\nThe rules are simple. The objective to place numbers representing skyscraper heights in all empty cells of the grid according to the rules:\n1. The height of the skyscrapers is from 1 to the size of the grid. i.e. 1 to 3 for a 3x3 puzzle.\n2. You cannot have two skyscrapers with the same height on the same row or column.\n3. The numbers on the sides of the grid indicate how many skyscrapers would you see if you look in the direction of the arrow. Remember, higher skyscrapers will block the skyscrapers behind them. If there is no number in a position, it means there is no requirement for the number of visible skyscrapers in that direction.\n4. Place numbers in each cell to indicate the height of the skyscrapers.\n5. You cannot change any numbers that already exist in the grid.\n\nPlease complete the solution for the 3x3 Skyscrapers puzzle in the diagram. \nAt the end of your response, summary your answer **as a single filled matrix** (list of lists or equivalent). ", "answer": "VALIDATION RULES:\n1. Provide a 3×3 matrix of digits 1–3 (e.g. [[2,1,3],[3,2,1],[1,3,2]]). \n2. Do not change any pre-filled numbers. \n3. Each row and column must contain 1, 2, 3 exactly once.\n4. The grid must satisfy all non-None visibility clues: from each clue's edge, count buildings that are strictly taller than all previous ones; the total must equal the clue value.", "image_path": "image675.png", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Others", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 676, "question": "You are given a partially filled 6×6 grid. Your task is to fill in the empty cells with digits 1–6 so that:\n\n1. Each row contains each digit 1–6 exactly once. \n2. Each column contains each digit 1–6 exactly once. \n3. Each 2×3 sub-grid contains each digit 1–6 exactly once. \n\nPlease complete the solution for the Jigsaw Sudoku puzzle in the diagram. \nAt the end of your response, summary your answer **as a single filled matrix** (list of lists or equivalent).", "answer": "VALIDATION RULES:\n1. Provide a 6×6 matrix of digits 1–6 (e.g. [[1,2,3,4,5,6],…]).\n2. Do not change any pre-filled numbers.\n3. Each row and each column must contain 1–6 exactly once.\n4. Each 2×3 box must contain 1–6 exactly once: rows 1–2 cols 1–3 ; rows 1–2 cols 4–6 ; rows 3–4 cols 1–3 ; rows 3–4 cols 4–6 ; rows 5–6 cols 1–3 ; rows 5–6 cols 4–6.", "image_path": "image676.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Others", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 677, "question": "Each number in the pyramid is the sum of the two numbers immediately below it. Fill in the pyramid with the missing numbers. Please answer in the format of 'block id:number'.", "answer": "{'A': 100, 'B': 59, 'C': 22, 'D': 23, 'E': 14, 'F': 8, 'G': 15, 'H': 8, 'I': 2, 'J': 9}", "image_path": "image677.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 678, "question": "Complete the grid by placing numbers in the empty squares so that the calculations are correct both across and down. Please answer in the format of 'grid id:number'.", "answer": "{'A': 6, 'B': 2, 'C': 2, 'D': 6, 'E': 6, 'F': 2}", "image_path": "image678.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Algebra", "Logic" ] } }, { "id": 679, "question": "What is the minimum number of different colours required to paint he given figure such that no two adjacent regions have the same colour?\n A. 3\n B. 4\n C. 5\n D. 6\n Please answer the question and provide the correct option letter, e.g., A, B, C, D, at the end.", "answer": "A", "image_path": "image679.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 680, "question": "Each one of the four keys locks exactly one padlock. Every letter on a padlock stands for exactly one digit. Same letters mean same digits.\nWhich letters must be written on the fourth padlock?\n Options: A. GDA, B. ADG, C. GAD, D. GAG, E. DAD", "answer": "D", "image_path": "image680.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 681, "question": "There are four cards on the table as in the picture. Every card has a letter on one side and a number on the other side. Peter said: \"For every card on the table it is true that if there is a vowel on one side, there is an even number on the other side.\" What is the smallest number of cards Alice must turn in order to check whether Peter said the truth?\n", "answer": "2", "image_path": "image681.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error" ], "sub_categories": [ "Logic" ] } }, { "id": 682, "question": "Wanda chooses some of the following shapes. She says: \"I have chosen exactly 2 grey, 2 big and 2 round shapes.\" What is the minimum number of shapes Wanda has chosen?\n", "answer": "3", "image_path": "image682.jpg", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Logic", "Combinatorics" ] } }, { "id": 683, "question": "There are 5 boxes and each box contains some cards labeled K, M, H, P, T, as shown below. Peter wants to remove cards out of each box so that at the end each box contained only one card, and different boxes contained cards with different letters. Which card remains in the first box?\n Options: A. It is impossible to do this, B. T, C. M, D. H, E. P", "answer": "D", "image_path": "image683.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 684, "question": "Sofie wants to write the word KENGU by using letters from the boxes. She can only take one letter from each box. What letter must Sofie take from box 4?\n Options: A. K, B. E, C. N, D. G, E. U", "answer": "D", "image_path": "image684.jpg", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 685, "question": "The diagram below shows five rectangles, each containing some of the letters $\\mathrm{P}, \\mathrm{R}, \\mathrm{I}, \\mathrm{S}$ and $\\mathrm{M}$.\n\nHarry wants to cross out letters so that each rectangle contains only one letter and each rectangle contains a different letter. Which letter does he not cross out in rectangle 2? Options: A. P, B. R, C. I, D. S, E. M", "answer": "B", "image_path": "image685.jpg", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Logic" ] } }, { "id": 686, "question": "If individual #1 in the following pedigree is a heterozygote for a rare, AR disease, what is the probability that individual #7 will be affected by the disease? Assume that #2 and the spouses of #3 and #4 are not carriers. Options: A. 1/64, B. 1/32, C. 1/16, D. 1/8, E. 1/4", "answer": "A", "image_path": "image686.png", "annotated": { "difficulty_tier": "Medium", "subject": "Biology", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Branch-and-Bound" ], "sub_categories": [ "Probability & Statistics", "Logic" ] } }, { "id": 687, "question": "In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes (\"eats\") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\\times.$ (The squares with two or more dotted edges have been removed from the original board in previous moves.) The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.", "answer": "792", "image_path": "image687.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 688, "question": "In a given circle, n > 2 arbitrary chords are drawn such that no three are concurrent within the interior of the circle. Suppose m is the number of points of intersection of the chords within the interior. Find, in termsof n and m, the number r of line segments obtained through dividing the chords by their points of intersection.", "answer": "r = n + 2m", "image_path": "image688.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Divide-and-Conquer", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 689, "question": "Consider an $a \\times b$ board, with $a$ and $b$ integers greater than or equal to two. Initially all the squares are painted white and black as a chessboard. The allowed operation is to choose two unit squares that share one side and recolor them in the following way: Any white square is painted black, any black square is painted green and any green square is painted white. Determine for which values of $a$ and $b$ it is possible, using this operation several times, to get all the original black squares to be painted white and all the original white squares to be painted black. Note: Initially there are no green squares, but these appear after the first time we use the operation.", "answer": "one of a and b is a multiple of 3", "image_path": "image689.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Others", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 690, "question": "At the conference of the party of liars and the party of fans of the truth, a presidency of 32 members was chosen. Members of the presidency are arranged to sit on 32 chairs (four rows with eight chairs each, and eight columns with four chairs each). Members of the party of liars always lie, and the fans of the truth always tell the truth. By definition, B is adjacent to A if B is arranged to sit on a chair which is to the left of A, or to the right of A, or infrontofA, or in back of A. At the coffee break every member of the presidency said that he had a liar and a fan of the truth among his neighbors. Determine the minimal number of liars in the presidency for which the situation described above is possible.", "answer": "8", "image_path": "image690.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 691, "question": "Given a positive integer k and a positive real number a. For any partition k₁ + k₂ + ... + kᵣ = k (kᵢ is a positive integer, 1 ≤ r ≤ k), find the maximum of F = aᵏ¹ + aᵏ² + ... + aᵏᵣ", "answer": "max{aᵏ, kα}", "image_path": "image691.png", "annotated": { "difficulty_tier": "Hard", "subject": "Mathematics", "answer_type": "Formula", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 692, "question": "Consider some positive integers whose sum is 1976. Find the maximum value of the product of these positive integers.", "answer": "3^658 * 2", "image_path": "image692.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 693, "question": "There are 1989 points in the space, no three being collinear. Divide them into 30 groups, the number of points in each group being different. Select three points from three different groups respectively, and form a triangle using these three points as vertices. What are the numbers of elements in these groups when the number of possible triangles is maximized? (4th China Mathematics Olympiad)", "answer": "The numbers of points of each group are 51, 52,..., 56, 58, 59 ..., 81.", "image_path": "image693.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Others", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 694, "question": "In a $7 \\times 8$ chessboard, each grid is placed with a stone. If two grids share a vertex, then the two stones placed on these two grids are called connected. Now remove $r$ stones from the chessboard, so that there are no five stones connected one by one in a same straight line (horizontal, vertical or in diagonal direction). Find the minimum value of $r$. ", "answer": "11", "image_path": "image694.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 695, "question": "Ten people go to a bookstore to buy books. We know that each person has bought three books, and that for any two people, there is at least one book bought by both of them. Consider the book that is the most popular (that is, bought by the largest number of people). At least how many people bought this book? ", "answer": "5", "image_path": "image695.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 696, "question": "There are 16 students participating in an exam, where all problems are multiple choice problems with four choices. After the exam, it is found that for any two students there is at most one problem to which their answers are identical. At most how many problems can there be? (1992 China National Team Selection)", "answer": "5", "image_path": "image696.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 697, "question": "There are n (n > 3) actors in a group. They have made some shows, each of which involves three actors performing. In one occasion they found that it is possible to arrange some shows, so that each pair of actors has exactly one chance to perform in the same show. Find the minimum value of n.", "answer": "7", "image_path": "image697.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 698, "question": "There are r stones in an n × n chessboard. Each cell has at most one stone. Suppose that the r stones have the following property P: each row and each column of the chessboard has at least one stone, and if any stone is removed, the property P will not hold. Find the maximum value of r_n of r.", "answer": "r_n = 2n - 2", "image_path": "image698.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 699, "question": "Fix a positive integer n ≥ 3, let \\(a_1, a_2, \\dots, a_n\\) be n different real numbers, whose sum is positive. If a permutation of these numbers is such that for any \\( k = 1, 2, \\dots, n \\), \\( b_1 + b_2 + \\dots + b_k > 0 \\), then the permutation is called 'good'. Find the minimum number of 'good' permutations.", "answer": "(n-1)!", "image_path": "image699.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Formula", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 700, "question": "Given a positive integer a, let X = {a1, a2, ..., an} be a set of positive integers, where a1 <= a2 <= a3 <= ... <= an. If for any integer p(1 <= p <= a), there is a subset of X such that S(A) = p, where S(A) is the sum of elements in set A, find the minimum value of n.", "answer": " log₂a +1", "image_path": "image700.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 701, "question": "There are n delegates at a conference, each of them knowing at most k languages. Among any three delegates, at least two speak a common language. Find the least number n such that for any distribution of the languages satisfying the above properties, it is possible to find a language spoken by at least three delegates.", "answer": "2k+3", "image_path": "image701.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 702, "question": "Suppose that $a_1, a_2, \\dots, a_{2004}$ are nonnegative integers such that $a_1^n + a_2^n + \\dots + a_{2004}^n$ is a perfect square for all positive integers $n$. What is the least number of such integers that must equal 0?", "answer": "68", "image_path": "image702.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 703, "question": "A test has 3 problems, and the score for each problem is an integer from 0 to 7. A student's result is represented by a score vector $(s_1, s_2, s_3)$, where each $s_i \\in \\{0, 1, \\dots, 7\\}$. We say that student $A$ **dominates** student $B$ if student $A$'s score is greater than or equal to student $B$'s score for each of the 3 problems. An 'elite club' is a group of students in which no student in the group dominates another student in the same group. What is the maximum possible number of students in such an 'elite club'?", "answer": "48", "image_path": "image703.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 704, "question": "Consider the rectangle in the plane with vertices $(0, 0)$, $(m, 0)$, $(0, n)$, and $(m, n)$, where $m$ and $n$ are odd positive integers. This rectangle is partitioned into a finite number of smaller triangles, satisfying the following conditions: \\begin{itemize} \\item[1)] The vertices of all triangles in the partition are points with integer coordinates. \\item[2)] Each triangle has at least one side (called a 'good side') which lies on a grid line of the form $x = j$ or $y = k$ for some integers $j, k$. The height of the triangle with respect to this 'good side' must be exactly 1. \\item[3)] Any side of a triangle that is not a 'good side' is an interior edge, shared by exactly two triangles of the partition. \\end{itemize} A triangle in the partition is called a \textit{special triangle} if it has two 'good' sides. What is the minimum number of such \\textit{special triangles} that are guaranteed to exist in any such partition?", "answer": "2", "image_path": "image704.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 705, "question": "Let n be a positive integer. Suppose that n airline companies offer trips to citizens of N cities such that for any two cities there exists a direct flight in both directions. Find the least N such that we can always find a company which can offer a trip in a cycle with an odd number of landing points.", "answer": "$2^n + 1$", "image_path": "image705.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Formula", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 706, "question": "In each square of an $8 \times 8$ chessboard, a positive real number is written. The numbers satisfy the following two conditions: \\begin{itemize} \\item[1)] The sum of the numbers in each row is exactly 1. \\item[2)] For any set of 8 squares, where no two are in the same row or column, the product of the numbers in these squares does not exceed the product of the numbers on the main diagonal. \\end{itemize} What is the minimum possible value for the sum of the numbers on the main diagonal?", "answer": "1", "image_path": "image706.png", "annotated": { "difficulty_tier": "Hard", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 707, "question": "Let $n$ and $m$ be positive integers satisfying the condition $n < m-1$. Let $a_1, a_2, \\dots, a_m$ be a sequence of nonzero integers such that for all integers $k$ in the range $0 \\le k \\le n$, the following equality holds: $$ \\sum_{j=1}^{m} a_j \\cdot j^k = 0 $$ What is the minimum number of pairs of consecutive terms with opposite signs that is guaranteed to exist in the sequence $a_1, a_2, \\dots, a_m$? Express your answer in terms of $n$ and/or $m$.", "answer": "n+1", "image_path": "image707.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Formula", "main_category": [ "Trial-and-Error", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 708, "question": "Points A, R, M , and L are consecutively the midpoints of the sides of a square whose area is 650. The coordinates of point A are (11, 5). If points R, M , and L are all lattice points, and R is in Quadrant I, Compute the number of possible ordered pairs (x, y) of coordinates for point R", "answer": "10", "image_path": "image708.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 709, "question": " The \\textit{taxicab distance} between points \\((x_1, y_1, z_1)\\) and \\((x_2, y_2, z_2)\\) is given by\\n\\[ d\\left((x_1, y_1, z_1), (x_2, y_2, z_2)\\right) = |x_1 - x_2| + |y_1 - y_2| + |z_1 - z_2|. \\]\\nThe region \\(\\mathcal{R}\\) is obtained by taking the cube \\(\\{(x, y, z) : 0 \\le x, y, z \\le 1\\}\\) and removing every point whose taxicab distance to any vertex of the cube is less than \\(\\frac{3}{5}\\). Compute the volume of \\(\\mathcal{R}\\). ", "answer": "\\frac{179}{250}", "image_path": "image709.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 710, "question": "Complete the following \"cross-number puzzle\", where each \"Across\" answer represents a fourdigit number, and each \"Down\" answer represents a three-digit number. No answer begins with the digit 0. Across: 1. A B C D is the cube of the sum of the digits in the answer to 1 Down. 5. From left to right, the digits in E F G H are strictly decreasing. 6. From left to right, the digits in I J K L are strictly decreasing. Down: 1. A E I is a perfect fourth power. 2. B F J is a perfect square. 3. The digits in C G K form a geometric progression. 4. D H L has a two-digit prime factor.", "answer": "2197, 5431, 6410", "image_path": "image710.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 711, "question": "The six sides of convex hexagon A_1 A_2 A_3 A_4 A_5 A_6 are colored red. Each of the diagonals of the hexagon is colored either red or blue. Compute the number of colorings such that every triangle A_i A_j A_k has at least one red side.", "answer": "392", "image_path": "image711.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 712, "question": "Given a regular 16-gon, extend three of its sides to form a triangle none of whose vertices lie on the 16-gon itself. Compute the number of noncongruent triangles that can be formed in this manner. ", "answer": "11", "image_path": "image712.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 713, "question": "We would hope that voters are a bit more nuanced than this and perhaps base their preferences on two issues! Assume there are three candidates A, B, and C with views on two issues (again, on a 0–10 scale) as given in the graph below. A vecter sets their preference ranking depending on the (Cartesian) distance between their views on these two topics and those of the candidate. If voters’ views on both topics were independent and evenly distributed throughout the 0–10 scales, then the fraction of the votes received by a candidate would be equal to the fraction of the square [0, 10] × [0, 10] that is closest to the point corresponding to the candidate’s views. A voter in this election with views (xv, yv) is indifferent between all three candidates. Compute the ordered pair (x_v, y_v).", "answer": "(5, 6)", "image_path": "image713.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 714, "question": "Compute the number of ways to place the integers 1 through 7 in the blanks below so that the chain of inequalities is satisfied.", "answer": "90", "image_path": "image714.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 715, "question": "Let T = 5. A1A2A3 ... AT and A1A2AT +1 ... A2T −2 are distinct regular T -gons in the plane. Let S be the set of all real numbers that are distances between distinct vertices. Compute the number of elements in S. (In the example to the left, there are two distinct non-zero distances, A1A2 and A3A4.)", "answer": "6", "image_path": "image715.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Geometry", "Logic" ] } }, { "id": 716, "question": "On a five-function calculator +, −, ×, ÷, √ that can display 10 digits you perform the following procedure: Step 1. Enter a 5-digit number. Step 2. Multiply the number by 2010. Step 3. Take the fourth root of the result (by hitting the square root button twice). Step 4. Repeat Steps 2 and 3, getting a new result each time. After a while, you notice that the results on the screen are not changing after applying Steps 2 and 3 (that is, the results are constant, rounded to 10 digits). If N is this unchanging result, compute $\\lfloor N\\rfloor$, the greatest integer less than or equal to N.", "answer": "12", "image_path": "image716.png", "annotated": { "difficulty_tier": "Basic", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Algebra", "Logic" ] } }, { "id": 717, "question": "Consider the following two-player game. Andy chooses an integer between 2 and 12, inclusive. Bailes, knowing Andy’s choice, chooses a different integer between 2 and 12. Two six-sided dice are thrown, and the rolls are summed. The player who has picked the closer number to the sum wins $1. If both numbers are equidistant from the sum, the players split the $1. Compute Andy’s expected winnings, assuming an optimal strategy by both players.", "answer": "frac{7}{12}", "image_path": "image717.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Probability & Statistics", "Logic" ] } }, { "id": 718, "question": "In the figure below, distinct non-zero digits are placed in the nine circles such that the sums of the digits along each side of the triangle are equal. Compute the smallest possible value of the four-digit number A B C D.", "answer": "1297", "image_path": "image718.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 719, "question": "A n-sided die has the integers between 1 and n (inclusive) on its faces. All values on the faces of the die are equally likely to be rolled. Consider the following game. A 20-sided die is rolled, and the player can either receive the number rolled in dollars, or can discard the 20-sided die and roll a 4-sided and an 8-sided die and receive the product of the two rolls in dollars. Assuming the player chooses a strategy to keep or pass their first roll that maximizes their expected winnings, compute the expected winnings by the player on a single play of the game.", "answer": "\\frac{1071}{80} or 13.3875", "image_path": "image719.png", "annotated": { "difficulty_tier": "Medium", "subject": "Mathematics", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Probability & Statistics", "Logic" ] } }, { "id": 720, "question": "Let S be the set {1, 2, 3, 4}. Compute the number of functions f : S → S such that this picture is satisfied.", "answer": "10", "image_path": "image720.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 721, "question": "A hexaknight can move either two squares horizontally, or two squares vertically and one square horizontally. The graphic below shows the six possible locations to which the hexaknight in the center of the board can move. In the 3 × 10 board below, two of the eight non-corner white squares are colored gray. If all eight squares are equally likely to be colored gray, compute the probability (as a fraction) that there exists a path for the hexaknight in the upper left corner to reach the lower right corner of the board while traveling only on white squares.", "answer": "\\frac{3}{4}", "image_path": "image721.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 722, "question": "For integers m and n with 0 ≤ m ≤ n, a (m, n)-restricted knight can only move either up m squares and right n squares or up n squares and right m squares. The possible moves of a (1, 2)-restricted knight are shown below. Compute the number of distinct paths a (1, 2)-restricted knight can take from the lower left corner to the upper right corner of a 10 × 10 board.", "answer": "20", "image_path": "image722.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 723, "question": "For integers m and n with 0 ≤ m ≤ n, a (m, n)-restricted knight can only move either up m squares and right n squares or up n squares and right m squares. The possible moves of a (1, 2)-restricted knight are shown below. Compute the number of ordered pairs (m, n) with 0 ≤ m ≤ n such that there exist at least 20 distinct paths for a (m, n)-restricted knight from the lower left corner to the upper right corner of a 17 × 17 board.", "answer": "4", "image_path": "image723.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 724, "question": "Compute the number of ways to color the cells of a 3 × 3 grid red, green, or blue such that each color appears in at least two cells and no cells that share an edge have the same color. Note that reflections and rotations of a grid are considered distinct.", "answer": "186", "image_path": "image724.png", "annotated": { "difficulty_tier": "Easy", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 725, "question": "Place the integers from 1 through 6 in the small circles so that the sums of the four numbers connected by each of the large circles are equal. How many unique solutions to this puzzle are there? A solution that is obtained by merely rotating or reflecting a known solution is NOT considered another unique solution.", "answer": "8", "image_path": "image725.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 726, "question": "Place the integers from 1 through 13 in each of the circles so that the sums of the five numbers connected by each of the three lines are equal. a. What numbers can go in the center circle? b. If only the odd numbers from 1 through 25 were used, what numbers could go in the center circle? A solution that is obtained by merely rotating or reflecting a known solution is NOT considered another unique solution.", "answer": "a. {1, 4, 7, 10, 13} b. {1, 7, 13, 19, 25}", "image_path": "image726.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 727, "question": "In the provided diagrams for number puzzles: Place the integers from 1 through 8 in the circles in Figure so that the sums, S, of the three numbers along each edge are equal. How many unique values for S are there? A solution that is obtained by merely rotating or reflecting a known solution is NOT considered another unique solution.", "answer": "4", "image_path": "image727.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 728, "question": "Place the integers from 1 through 12 in the circles in Figure 4b so that the sums, S, of the four numbers along each edge and the sum of the numbers in the four corners is also equal to S. How many unique values of S are there? A solution that is obtained by merely rotating or reflecting a known solution is NOT considered another unique solution.", "answer": "1", "image_path": "image728.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Free-form questions", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 729, "question": "For the triangular puzzle using integers 1 through 9 where the sum of the four numbers along each edge is S: For which values of S does this puzzle have a solution? A solution that is obtained by merely rotating or reflecting a known solution is NOT considered another unique solution.", "answer": "[17, 19, 20, 21, 23]", "image_path": "image729.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Structure", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 730, "question": "Place the integers from 1 through 9 in the squares of the diagram to the right so that the 3-digit number formed in the second row is twice the 3-digit number formed in the first row and the 3-digit number formed in the third row is three times the 3-digit number formed in the first row. How many unique solutions are there to this puzzle? A solution that is obtained by merely rotating or reflecting a known solution is NOT considered another unique solution.", "answer": "4", "image_path": "image730.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 731, "question": "Magic Multiplicative Square: The numbers a, b, c, d, e, f, g, h, and i are unique (but not necessarily consecutive) integers with the products, abc = def = ghi = adg = beh = cfi = aei = ceg = P. For P = 216, how many unique solutions to the puzzle are there? A solution that is obtained by merely rotating or reflecting a known solution is NOT considered another unique solution.", "answer": "1", "image_path": "image731.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Combinatorics", "Logic" ] } }, { "id": 732, "question": "An Anti-Magic Square: An anti-magic square is an n by n array of integers from 1 to n² such that the n numbers in each row, column, and diagonal sum to a different sum and the sums form a sequence of 2n + 2 consecutive integers. The following anti-magic square has been started for you. How many solutions are there? A solution that is obtained by merely rotating or reflecting a known solution is NOT considered another unique solution.", "answer": "3", "image_path": "image732.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 733, "question": "Replace a, b, c, d, e, f, g, h, and i with digits 1 through 9 and place a minus sign in front of some of them so that the products acbfe = 320, dhg = 162, acdhi = 6048, bfg = −60, bcd = −240, and efghi = 756. How many unique solutions are there? A solution that is obtained by merely rotating or reflecting a known solution is NOT considered another unique solution.", "answer": "8", "image_path": "image733.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Trial-and-Error", "Perceive-and-Comprehend", "Hypothesize-and-Test" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 734, "question": "Eight students attend a Harper Valley ARML practice. At the end of the practice, they decide to take selfies to celebrate the event. Each selfie will have either two or three students in the picture. Compute the minimum number of selfies so that each pair of the eight students appears in exactly one selfie.", "answer": "12", "image_path": "image734.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Number Theory", "Combinatorics" ] } }, { "id": 735, "question": "In this game, there is an area ten squares wide and a number of squares tall. Pieces chosen randomly from among the seven “tetrominoes” made up of four squares glued together, as shown below, fall from the top of the screen. As the pieces fall, the player may rotate them or slide them left or right, but once they touch a piece below them they stick in place. If the player is able to fit the pieces together so as to leave no gaps in a row, that row disappears and all the blocks above fall to leave more room for new blocks. Otherwise the screen fills up with blocks and the game ends. ...what if the player could choose the piece to come next? Can any pattern be left on the screen? The answer to this is easily seen to be no. For instance, there can never be just a single square—or any odd number of squares—because the pieces all are made of four squares, and squares disappear from the screen only in complete rows of ten squares. Since both four and ten are even numbers, the player can never have an odd number of squares remaining. The puzzle is ten spaces wide and the pattern to be obtained consists of exactly two squares (that is, after complete rows disappear, only two squares are left that didn’t disappear), must the number of rows that were cleared be odd or even?", "answer": "odd", "image_path": "image735.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Others", "main_category": [ "Hypothesize-and-Test", "Trial-and-Error", "Perceive-and-Comprehend" ], "sub_categories": [ "Number Theory", "Logic" ] } }, { "id": 736, "question": "In this game, there is an area ten squares wide and a number of squares tall. Pieces chosen randomly from among the seven “tetrominoes” made up of four squares glued together, as shown below, fall from the top of the screen. As the pieces fall, the player may rotate them or slide them left or right, but once they touch a piece below them they stick in place. If the player is able to fit the pieces together so as to leave no gaps in a row, that row disappears and all the blocks above fall to leave more room for new blocks. Otherwise the screen fills up with blocks and the game ends. ...what if the player could choose the piece to come next? Can any pattern be left on the screen? The answer to this is easily seen to be no. For instance, there can never be just a single square—or any odd number of squares—because the pieces all are made of four squares, and squares disappear from the screen only in complete rows of ten squares. Since both four and ten are even numbers, the player can never have an odd number of squares remaining. Consider If you are playing on a screen that is eight squares wide, and the blocks that fell were straight “triominoes”—three squares glued together along their edges with their centers in a straight line. Is it possible to either create or clear the pattern with just a single square in the corner.", "answer": "No", "image_path": "image736.png", "annotated": { "difficulty_tier": "Basic", "subject": "Logic", "answer_type": "Others", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 737, "question": "In this game, there is an area ten squares wide and a number of squares tall. Pieces chosen randomly from among the seven “tetrominoes” made up of four squares glued together, as shown below, fall from the top of the screen. As the pieces fall, the player may rotate them or slide them left or right, but once they touch a piece below them they stick in place. If the player is able to fit the pieces together so as to leave no gaps in a row, that row disappears and all the blocks above fall to leave more room for new blocks. Otherwise the screen fills up with blocks and the game ends. ...what if the player could choose the piece to come next? Can any pattern be left on the screen? The answer to this is easily seen to be no. For instance, there can never be just a single square—or any odd number of squares—because the pieces all are made of four squares, and squares disappear from the screen only in complete rows of ten squares. Since both four and ten are even numbers, the player can never have an odd number of squares remaining. we are playing on a nine-square-wide screen, with the regular tetrominoes. Let’s say we want to create the pattern with just a single square in the lower left-hand corner. Which tetromino can create this pattern by itself? (Choose from A, B, C, D, E, F or G) ", "answer": "F", "image_path": "image737.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Multiple-choice questions", "main_category": [ "Hypothesize-and-Test", "Perceive-and-Comprehend", "Trial-and-Error" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 738, "question": "A hexaknight can move either two squares horizontally, or two squares vertically and one square horizontally. The graphic below shows the six possible locations to which the hexaknight in the center of the board can move. In the 3 × 10 board below, three of the eight non-corner white squares are colored gray. If all eight squares are equally likely to be colored gray, compute the probability (as a fraction) that there exists a path for the hexaknight in the upper left corner to reach the lower right corner of the board while traveling only on white squares.", "answer": "\\frac{5}{14}", "image_path": "image738.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 739, "question": "In chess, a knight can move either two squares horizontally and one square vertically, or two squares vertically and one square horizontally. The graphic below shows the eight possible locations to which the knight in the center of the 5 × 5 board can move. Unlike all other standard chess pieces, the knight can ‘jump over’ all other pieces (of either color) to its destination square. On a board, a square is considered attacked if it contains a knight or can be reached by a knight in one move (on the 5 × 5 board showing the possible moves of a knight, nine squares are attacked). Two knights are randomly placed on distinct squares of a 3 × 3 board, with all squares equally likely. Compute the expected number of squares that are attacked.", "answer": "\\frac{44}{9}", "image_path": "image739.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Divide-and-Conquer", "Hypothesize-and-Test", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } }, { "id": 740, "question": "In chess, a knight can move either two squares horizontally and one square vertically, or two squares vertically and one square horizontally. The graphic below shows the eight possible locations to which the knight in the center of the 5 × 5 board can move. Unlike all other standard chess pieces, the knight can ‘jump over’ all other pieces (of either color) to its destination square. On a board, a square is considered attacked if it contains a knight or can be reached by a knight in one move (on the 5 × 5 board showing the possible moves of a knight, nine squares are attacked). Compute the minimum number of knights to attack every square of a 4 × 6 board.", "answer": "4", "image_path": "image740.png", "annotated": { "difficulty_tier": "Medium", "subject": "Logic", "answer_type": "Numerical", "main_category": [ "Branch-and-Bound", "Perceive-and-Comprehend" ], "sub_categories": [ "Combinatorics", "Logic" ] } } ]