Maths Olympiad Prep

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Combinatorics Difficulty 5.2 AIME, harder Find the answer

7. (6 points) According to the pattern of numbers in the first four figures, fill in the appropriate number in the fifth figure:

The above text has been translated into English, retaining the original text's line breaks and format.

Solution

【Analysis】(1) According to the problem, the movement pattern of position 1 in the figure is: it moves one grid clockwise in sequence;
(2) The number arrangement pattern is: arranged in the order of 1,3,5,2,4,61, 3, 5, 2, 4, 6, and the odd-numbered figures are arranged clockwise, while the even-numbered figures are arranged counterclockwise; the fifth figure is the odd-numbered one, so it is arranged clockwise.

【Solution】According to the analysis of the problem, we can get: (1) The movement pattern of 1 in the figure is: it moves one grid clockwise in sequence; so we first determine the position of 1 as shown in the figure below;
(2) The fifth figure is the odd-numbered one, so it is arranged clockwise, so we can add the correct numbers in the figure as shown below:

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.