AMC 8 2003 Problem 23
AMC8 2003 年第 23 题
余数与同余★★★☆☆
In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through the eight exterior segments of the four squares.
If the pattern is continued, where would the cat and mouse be after the 247th move?
在下面的图案里，猫顺时针在四个正方形里移动，老鼠逆时针沿着四个正方形外围的 8 根线段移动。如果照此规律继续移动，第 247 步后猫和老鼠分别在什么位置？
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正确答案 · Correct Answer
A
解析 · Solution
Solution 1
Break this problem into two parts: where the cat will be after the move, and where the mouse will be. The cat has four possible positions in 1 cycle which are repeated every four moves. 247 has a remainder of 3 when divided by 4. This corresponds to the position the cat has after the 3rd move, which is the bottom right corner. Similarly, the mouse has eight possible positions in 1 cycle that repeat every eight moves. 247 has a remainder of 7 when divided by 8. This corresponds to the position the mouse has after the 7th move which is bottom left corner. The only arrangement with the mouse in that position and the cat in the bottom right square is.
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