AMC 8 · 2005 · #20

Grade 5 number-theory
Archetype Modular Cycle Frequency Count └ Sub-archetype Power-Cycle Residue
modular-arithmeticlcmequal-spacing modular-arithmeticidentify-subproblems ↑ Prerequisites: modular-arithmetic
📏 Medium solution 💡 3 insights
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Problem
A circle has 12 equally-spaced points numbered 1 through 12. Alice and Bob both start at point 12. Each turn, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. After how many turns do they first land on the same point?

Pick an answer.

(A)
6
(B)
8
(C)
12
(D)
14
(E)
24

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Make a Systematic List

The answer choices are small (6, 8, 12, 14, 24), and after each turn both positions are easy to update by adding or subtracting on a 12-point clock. That is the classic setup for Tool #2 (Systematic List): build a table of (turn, Alice's point, Bob's point) and watch for the first row where they agree. Tool #5 (Look for a Pattern) backs it up — each turn Alice gains 5 points and Bob loses 9, so the gap between them changes by a fixed amount every turn, and a constant change makes the wrap-around easy to predict instead of recomputing from scratch.

1STEP 1

Track both on the clock from turn 0: each turn Alice's point goes up 5 (wrapping past 12), Bob's goes down 9 (wrapping past 1).

Alice next = Alice now + 5 (mod 12), Bob next = Bob now - 9 (mod 12)
2STEP 2

Add 5 each turn from 12: Alice's points run 5, 10, 3, 8, 1, and then 6.

Turn & Alice's point ; 0 & 12 ; 1 & 12 + 5 = 5 ; 2 & 5 + 5 = 10 ; 3 & 10 + 5 = 15 → 3 ; 4 & 3 + 5 = 8 ; 5 & 8 + 5 = 13 → 1 ; 6 & 1 + 5 = 6 ;
3STEP 3

Subtract 9 each turn from 12: Bob's points cycle 3, 6, 9, 12 over and over, landing on 6 again at turn six.

Turn & Bob's point ; 0 & 12 ; 1 & 12 - 9 = 3 ; 2 & 3 - 9 = -6 → 6 ; 3 & 6 - 9 = -3 → 9 ; 4 & 9 - 9 = 0 → 12 ; 5 & 12 - 9 = 3 ; 6 & 3 - 9 = -6 → 6 ;
4STEP 4

Line the two columns up turn by turn; the first matching row is turn six, where both sit on point 6.

Turn & Alice & Bob & Same? ; 1 & 5 & 3 & no ; 2 & 10 & 6 & no ; 3 & 3 & 9 & no ; 4 & 8 & 12 & no ; 5 & 1 & 3 & no ; 6 & 6 & 6 & yes ;
5STEP 5

So the earliest turn they share a point is turn six, giving answer (A).

k = 6 → (A)
Answer
6
Quick sanity check on the meeting point. Alice's total clockwise travel after 6 turns is 5 × 6 = 30 points. On a 12-point circle, 30 = 2 × 12 + 6, so she ends 6 points clockwise of her start at 12 — that lands on point 6. Bob's total counterclockwise travel is 9 × 6 = 54 points; 54 = 4 × 12 + 6, so he ends 6 points counterclockwise of 12 — which is also point 6 (counting 11, 10, 9, 8, 7, 6). Both land on 6, confirming turn 6 and answer (A). None of the smaller turns (1 through 5) produced a match, so 6 really is the first.
💡Key takeaway

On any "when do they meet on a circle?" question, a side-by-side table of each player's position usually finds the answer in fewer turns than the answer choices suggest — here it takes just 6 rows to see Alice and Bob both land on point 6.

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