AMC8 2003

AMC8 2003 · Q23

AMC8 2003 · Q23. It mainly tests Patterns & sequences (misc), Remainders & modular arithmetic.

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?

在下面的图案中，猫顺时针通过四个正方形的四个位置，鼠标逆时针通过四个正方形的八个外部边段。如果图案继续，247步后猫和鼠标在哪里？

(A)

(B)

(C)

(D)

(E)

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) There are four different positions for the cat in the $2 \times 2$ array, so after every fourth move, the cat will be in the same location. Because $247 = 4 \times 61 + 3$, the cat will be in the $3$rd position clockwise from the first, or the lower right quadrant. There are eight possible positions for the mouse. Because $247 = 8 \times 30 + 7$, the mouse will be in the $7$th position counterclockwise from the first, or the left-hand side of the lower left quadrant.

（A）猫在$2 \times 2$的阵列中有四个不同的位置，所以每移动四步，猫就会回到同一位置。因为$247 = 4 \times 61 + 3$，猫将位于从第一个位置起顺时针数的第$3$个位置，也就是右下象限。老鼠有八个可能的位置。因为$247 = 8 \times 30 + 7$，老鼠将位于从第一个位置起逆时针数的第$7$个位置，也就是左下象限的左侧。

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