Problem 21 · 2025 AMC 8
Hard
Logic & Word Problems
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Answer: A — 12.
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Hint 1 of 2
The hardest pressure is where many pods are mutually connected — tackle that knot first. Find the largest group of pods that are all directly linked to each other; their grades are squeezed into a tiny set.
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Hint 2 of 2
Pods A, B, C, F are pairwise connected (a 4-clique), so their four grades must be mutually ≥ 2 apart. From {1,…,7}, the only such quadruple is {1, 3, 5, 7} — the most spread-out choice.
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Approach: find the clique to lock down four grades, then radiate outward
- Spot the tightest constraint: A, B, C, F are all mutually connected, so their four grades must each differ by ≥ 2. Packing 4 grades into 1–7 that far apart forces exactly {1, 3, 5, 7} — there's no slack.
- G connects to A and F. If G = 2, then A, F ≠ 1 or 3, forcing {A, F} ⊂ {5, 7} and {B, C} = {1, 3}.
- D and E only touch C and F. The extreme grades 1 and 7 each have just one neighbor in this clique, so place 1 at C and 7 at F. The remaining {4, 6} go to D, E.
- Filling in: D = 6 (avoids 7 enough), E = 4 (avoids 1 and 7), B = 3 (next to C = 1), A = 5. All constraints hold.
- C + E + F = 1 + 4 + 7 = 12.
- Why this transfers: in any constraint/coloring puzzle on a network, attack the densest cluster (the clique) first — it has the fewest possibilities, so it locks in the most and leaves the loose pods easy to finish.
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