AMC 10 · 2022 · #9
Grade 7 geometry-2d
Pick an answer.
AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #1 (Draw a Diagram) — copy the figure and mark every pair of regions that shares an edge. This adjacency picture is the whole problem; once it is in front of us the counting is just bookkeeping. Tool #7 (Identify Subproblems) — split the count into five sub-counts, one per region, and combine with the multiplication principle. The choice of coloring order is the only craft involved: pick an order so that each new region is constrained by at most two already-colored neighbors, never three. Tool #2 (Systematic List) is implicit — listing the adjacencies as a small table makes the "at most two earlier neighbors" check easy.
Read the adjacencies off the figure: BC touches all four other regions; the only non-touching pairs are the diagonals TL–BR and BL–TR.
BC is the hub touching all four others; the only non-adjacencies are the diagonal pairs TL–BR and BL–TR.
5.G.A.2Draw A DiagramColor in the order TL, BL, BC, TR, BR — then each new region has at most two already-colored neighbors (counts 0, 1, 2, 2, 2).
An ordering where every step has ≤ 2 earlier neighbors lets each step subtract at most 2 from 5, keeping the choice-count clean.
5.OA.A.2Identify SubproblemsTL is free (5 choices); BL avoids TL (4); BC avoids both TL and BL, which differ, so exactly two are blocked, leaving 3.
Two earlier neighbors that are themselves adjacent occupy two distinct colors, forbidding exactly 2 from the palette.
7.SP.C.8Identify SubproblemsTR avoids TL and BC (adjacent, so different colors) — 3 left; BR avoids TR and BC (adjacent) — 3 left.
Same logic — two earlier adjacent neighbors block exactly two colors out of five.
7.SP.C.8Identify SubproblemsMultiply the per-region choice counts by the multiplication principle to reach the total of 540 colorings.
Independent stage-choices multiply — the fundamental counting principle.
7.SP.C.8Identify SubproblemsThis AMC 10 problem only needs Grade 7 counting-principle reasoning you already know — draw which regions touch which, color them in an order where each new region has at most two earlier neighbors, and multiply 5 · 4 · 3 · 3 · 3 = 540.