AMC 10 · 2022 · #19
Grade 3 geometry-2d
Pick an answer.
AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #7 (Subproblems): split on whether the center starts filled or empty — the two cases need very different reasoning. If center starts filled, it must keep 2 or 3 filled peripheral neighbors and no peripheral can survive or be born. If center starts empty, it must gain exactly 3 filled neighbors and again all peripherals must end empty. Tool #1 (Diagram): use a labeled 3 × 3 picture with corners C and edges E to track neighbor counts. Tool #2 (Systematic List): once each sub-case has its constraint, walk through the small finite set of peripheral patterns. Tool #10 (Physical): coins on graph paper let you check each pattern by counting neighbors with your finger.
Label the inner 3 × 3: four corners, four edges, center M. Neighbor counts — corner: 3, edge: 5, center: 8.
Draw the 3 × 3 once and count neighbors for each role — corners (sparse), edges (medium), center (everyone).
K.G.A.1Draw A DiagramCase 1 — center M starts filled: it survives only with 2 or 3 filled peripherals, and every other peripheral must end empty.
Treat the center separately because it's the only cell every other cell sees.
1.OA.A.2Identify SubproblemsSub-case 1.1 (M has 2 filled peripherals): only the two opposite-corner pairs avoid a reborn shared empty neighbor → 2 configs.
Opposite-corner pairs are the only non-adjacent pairs with disjoint peripheral neighborhoods.
3.OA.D.8Make A Systematic ListSub-case 1.2 (M has 3 filled peripherals): any non-adjacent triple leaves an empty cell touching two of them, which is reborn → 0 configs.
Three mutually non-adjacent peripherals always leave a 'common neighbor' empty cell that gets born.
3.OA.D.8Make A Systematic ListCase 1 total: only the two opposite-corner pairs work → 2 configurations.
Two opposite-corner pairs, nothing else.
1.OA.A.2Identify SubproblemsCase 2 — center M starts empty: it needs exactly 3 filled peripherals, their induced graph max degree ≤ 1, and no empty cell touching all 3.
Three filled cells, no cell touches both others, no outside cell touches all three.
1.OA.A.2Identify SubproblemsShape A — three corners (mutually non-adjacent): choose which corner to omit → 4 configs.
Three of the four corners — pick which corner to leave out.
3.OA.A.3Make A Systematic ListShape B — two corners on one side plus the opposite middle edge: pick which side holds the two corners → 4 configs.
Two corners flanking one side, with the lonely opposite edge filling in.
3.OA.A.3Make A Systematic ListShape C — two edges meeting at a corner plus the far diagonal corner: place the shared corner in 4 spots → 4 configs.
Two adjacent edges + the far-diagonal corner — the V-and-dot pattern.
3.OA.A.3Make A Systematic ListShape D — an L (corner + one adjacent edge) plus the diagonally opposite corner: 4 corners × 2 edge choices → 8 configs.
L-shape (corner + one of its two adjacent edges) anchored at one corner of the grid, plus the diagonally opposite corner.
3.OA.A.3Make A Systematic ListCase 2: 4 + 4 + 4 + 8 = 20; with Case 1's 2, the total = 22, choice (C).
Add the two disjoint cases — done.
2.OA.A.1Identify SubproblemsSplit by what the center starts as. Center filled: only two opposite-corner pairs survive (2 configs). Center empty: exactly 3 filled peripherals in one of four geometric shapes (4 + 4 + 4 + 8 = 20 configs). Total = 22, choice (C).