# 2003 AMC 8 Problems/Problem 23

## Problem

In the pattern below, the cat (denoted as a cat in the figures below) moves clockwise through the four squares, and the mouse (denoted as a mouse in the figures below) 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?

## Solution

Break this problem into two parts: where the cat will be after the $247^{th}$ move, and where the mouse will be.

The cat has four possible configurations 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 configurations that repeat every eight moves. $247$ has a remainder of $7$ when divided by $8$. This corresponds to the position the rat has after the 7th move, which can easily be found by writing two more steps to be the bottom edge on the left side of the grid.

The only configuration with the mouse in that position and the cat in the bottom right square is $\boxed{\textbf{(A)}}$.

## Video Solution

https://youtu.be/RCUzhVOi7XI ~DSA_Catachu

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 