Competition · AMC preparation · step 4 of 4

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

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)

(B)

(C)

(D)

(E)

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

Set up the movement rule

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

Track Alice's positions

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

Track Bob's positions

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

Compare the two columns

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 ;

Lining the two patterns side by side is exactly the Grade 5 standard's payoff — the first matching row gives the answer.

Scanning the two position columns together, the earliest turn where Alice and Bob share a point is the turn where the gap that has opened between them first stretches to one whole loop around the 12-point circle.

▸ Why?

Since both begin on the same point, they can stand together again only when the arc that has opened between them is a whole number of complete loops, and travelling one whole loop around the circle lands you back on the very same point.

A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.

▸ Why?

The gap between them widens by the same amount every single turn, a steady 2 points on the 12-point circle, so the running gap is easy to track.

▸ Why?

Each turn the separation grows by Alice's 5 forward points together with Bob's 9 backward points, because they move in opposite directions so the two distances add into one combined gap of 14 points.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

Opening a 14-point gap on a 12-point circle amounts to opening only a 2-point gap, because a full 12-point loop comes back to the same point and leaves just the extra 2 points showing.

A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.

▸ Why?

The first meeting comes when that steady 2-point-per-turn growth has piled up to exactly one full 12-point loop, and the number of turns that takes is found by undoing the repeated adding of 2.

▸ Why?

Piling up 2 points each turn until the total is a full 12-point loop is the same as splitting 12 into equal groups of 2, and division undoes that repeated adding to hand back the turn count.

Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.

5.OA.B.3Look For A Pattern

5STEP 5

Read the answer off the table

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

k = 6 → (A)

Answer

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.

Set up the movement rule
Track Alice's positions
Track Bob's positions
Compare the two columns
Read the answer off the table

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Set up the movement rule

Track Alice's positions

Track Bob's positions

Compare the two columns

Read the answer off the table

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