AMC 10 · 2025 · #16

Grade 7 counting
Archetype Casework by Position / Vertex Type └ Sub-archetype Arrangement Block Type
combinations-basiccaseworksymmetry-argument casework ↑ Prerequisites: systematic-enumeration
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A circle is cut into 6 sectors of different sizes, arranged in a ring. Use three colors, red, green, and blue, with each color painting exactly 2 sectors. No two sectors that touch may share a color. Count how many different colorings of the ring obey these rules.

Pick an answer.

(A)
12
(B)
16
(C)
18
(D)
24
(E)
28

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Make a Systematic List

The question is a 'how many ways' count, so a systematic list is the right engine. To keep the list short, first draw the ring and label the positions, then solve an easier version by fixing the color of one sector and multiplying at the end. The remaining count breaks into a few clean cases based on where the second same-colored sector sits.

1STEP 1

Label the ring positions

Number the sectors 1 to 6 clockwise, so each touches the two beside it and 6 touches 1. Different sizes make turned copies count separately.

neighbors: 1 - 2, 2 - 3, 3 - 4, 4 - 5, 5 - 6, 6 - 1
2STEP 2

Fix sector 1, multiply by 3 later

The three colors are interchangeable, so count only the colorings with sector 1 red, then multiply that count by 3 at the end.

total = 3 × (colorings with sector 1 = red)
3STEP 3

Where can the second red go?

Red is used twice, so the other red avoids the neighbors 2 and 6 and must sit at sector 3, 4, or 5. Count those three cases separately.

second red ∈ {3, 4, 5}
4STEP 4

Count each case by listing

Filling the rest with two greens and two blues: second red at 3 gives 2, at 4 (opposite) gives 4, at 5 gives 2, so 8 in all.

2 + 4 + 2 = 8
5STEP 5

Multiply by 3 for all top colors

Sector 1 green and sector 1 blue each give 8 too, and the three families never overlap, so the total is 8 x 3 = 24, choice (D).

8 × 3 = 24
Answer
24
A rough upper bound helps: filling positions in order gives 3 choices for the first, then about 2 for each neighbor, far more than 24, and the exact-twice rule plus the no-touch rule should trim it hard, so 24 sits in a believable range. A second check: the three cases 2, 4, 2 are symmetric front-to-back (Case A mirrors Case C), which is what you expect on a symmetric ring, and every case genuinely used two greens and two blues. The total 24 also equals 6 x 4, matching the count another way: 3! = 6 color orderings times 4 structural patterns.
💡Key takeaway

Pin one color in place, count the few ways the rest can fit, then multiply back by the colors you set aside.

  • Label the ring positions
  • Fix sector 1, multiply by 3 later
  • Where can the second red go?
  • Count each case by listing
  • Multiply by 3 for all top colors

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