AMC 8 · 2003 · #23

Grade 4 pattern

Archetype Modular Cycle Frequency Count └ Sub-archetype Position-in-Cycle Lookup

modular-arithmeticpattern-recognitionspatial-visualizationsequences-arithmetic pattern-recognitionmodular-arithmetic ↑ Prerequisites: modular-arithmeticpattern-recognition

📏 Medium solution 💡 3 insights 📊 Diagram

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Problem

Four unit squares are arranged in a 2 × 2 block. A cat marches clockwise through the four interior squares, one square per move; a mouse marches counterclockwise through the eight exterior edge segments, one segment per move. Both start in the configuration shown in the first figure. After 247 moves, which of the five labeled diagrams shows their new positions?

Pick an answer.

(A)

Cat in bottom-right square; mouse on bottom segment of bottom-left square

(B)

Cat in top-right square; mouse on left segment of bottom-left square

(C)

Cat in bottom-left square; mouse on top segment of top-left square

(D)

Cat in top-left square; mouse on right segment of top-right square

(E)

Cat in bottom-right square; mouse on right segment of bottom-right square

AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The cat and mouse each repeat their orbit on a fixed schedule — Tool #5 (Look for a Pattern) is exactly the "find the 247th term of a repeating cycle" move. The cycles have different lengths (4 for the cat, 8 for the mouse), so we handle each animal as its own easier problem (Tool #9): find the position after the first cycle's worth of moves, then jump to move 247 using the remainder modulo the cycle length. Finally, Tool #3 (Eliminate Possibilities) checks the five diagrams against the cat square and mouse edge we computed and picks the unique match.

1STEP 1

The cat loops clockwise through the four squares, so its cycle length is 4.

cat cycle length = 4

2STEP 2

247 divided by 4 leaves remainder 3, matching the cat's spot after move 3.

247 = 4 × 61 + 3 → 247 mod 4 = 3

3STEP 3

Tracing the loop, move 3 lands the cat in the bottom-right square.

cat path: 1 → 2 → 3 → 4 → 1 → …, so move 3 = bottom-right square

4STEP 4

The mouse loops counterclockwise through the eight outer edges, so its cycle length is 8.

mouse cycle length = 8

5STEP 5

247 divided by 8 leaves remainder 7, matching the mouse's edge after move 7.

247 = 8 × 30 + 7 → 247 mod 8 = 7

6STEP 6

Following the loop, edge 7 puts the mouse on the bottom edge of the bottom-left square.

mouse path through 8 edges; move 7 = bottom edge of bottom-left square

7STEP 7

Only one diagram shows the cat bottom-right and the mouse on that bottom-left edge — choice (A).

cat = bottom-right ∧ mouse = bottom edge of bottom-left → (A)

Answer

Cat in bottom-right square; mouse on bottom segment of bottom-left square

Sanity check with a smaller cycle target. After 4 moves the cat is back where it started, and after 8 moves the mouse is back where it started — so move 248 = 8 × 31 would reset the mouse to its starting edge, and move 244 = 4 × 61 would reset the cat to its starting square. Move 247 is one move before the mouse resets and three moves after the cat's last reset, exactly matching remainders 7 and 3. The cat and mouse positions we picked (bottom-right square; bottom edge of bottom-left square) are consistent with a snapshot taken just one tick before both animals would round the southwest corner of their loops, which is the configuration drawn in (A).

💡Key takeaway

Big move numbers in a repeating loop reduce to a small remainder. Once you know the cat repeats every 4 moves and the mouse repeats every 8, 247 collapses to the 3rd cat position (bottom-right) and the 7th mouse position (bottom edge of bottom-left) — diagram (A).

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