Problem 23 · 2003 AMC 8
Stretch
Algebra & Patterns
mod-arithmeticfind-the-cycle
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Answer: A — the cat in the bottom-right square, the mouse on the bottom-left segment.
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Hint 1 of 2
247 is huge, but each animal just loops — so don't trace 247 steps, find each one's cycle length and use only the remainder.
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Hint 2 of 2
The cat returns home every 4 moves (4 squares); the mouse every 8 (8 segments). Divide 247 by each and keep the remainder.
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Approach: reduce 247 by each cycle length separately
- Both animals repeat, so after a full loop they're back to start — only the remainder of 247 matters, and the two can be handled independently.
- Cat: 4 squares per loop. 247 = 4·61 + 3, remainder 3, so the cat sits where it is after move 3 — the bottom-right square.
- Mouse: 8 segments per loop. 247 = 8·30 + 7, remainder 7, so the mouse sits where it is after move 7 — the bottom-left segment.
- The only picture with the cat bottom-right and mouse bottom-left is A.
- You'll see this again: for two things cycling at different rates, reduce the big move count mod each cycle separately — remainder by 4 for the cat, by 8 for the mouse — rather than tracking them together.
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