Problem 12 · 2026 AMC 8
Hard
Logic & Word Problems
constraint-propagation
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Answer: D — 5.
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Hint 1 of 2
Don't guess the top circle — attack the most constrained edge first. With only the digits 1–6, which edge-sum can be made in only one way?
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Hint 2 of 2
The edge labeled 10 is the tightest: among 1–6, the only pair adding to 10 is 4 + 6. Pin those two circles down, then each neighboring sum forces the next digit like dominoes.
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Approach: start at the most-constrained edge, then let the sums force each digit
- Scan for the edge with the fewest options. The sum 10 is the giveaway: out of 1–6, only 4 + 6 reaches 10, so that left edge's two circles are 4 and 6 in some order.
- Test which way round: the edge above it sums to 9. If the upper circle is 4, the top is 9 − 4 = 5; the lower one is then 6. (The other way, top = 9 − 6 = 3, collides later.)
- Now the dominoes fall: bottom-left 6 with edge 8 → bottom-middle 2; edge 5 → bottom-right 3; right edge 4 → upper-right 1; and the last check 5 + 1 = 6 matches the top-right edge. Digits used: {5,4,6,2,3,1} = 1–6 exactly once. ✓
- So the top circle is 5.
- Why this transfers: in any fill-the-grid / fill-the-graph puzzle, begin at the cell with the fewest legal choices — one forced value usually triggers a chain that solves the rest with no guessing.
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