2003 AMC 8 Problems/Problem 23

Problem

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

We'll break down this problem into two parts: where the cat will be after the 247th move and where the mouse will be after the 247th move.

Instead of using formulas two solve this problem we'll just look for patterns. Let's start off with the cat. After 4 rotations clockwise the cat will be back in its starting position. So the cat will be back in its starting position in the 5th move. Similarly, we can find that the cat will be in its starting position if the number of moves is divisible by 5. By dividing 247 by 5 we get a remainder of 2. So after two more moves the cat will be in the bottom right square.

Similarly, we can find that the mouse will be back on its starting position for every move which is divisible by 9. Dividing 247 by 9 we get a remainder of 5. After 5 moves the mouse will be on the top segment of the left hand side of the big square.

So the answer is $\boxed{\bolded{\text{C}}}$ (Error compiling LaTeX. Unknown error_msg).