AMC 10 · 2022 · #18
Grade 3 geometry-2d
Pick an answer.
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.
Count the neighbours
Each position has a different neighbour count.
Draw the 3 × 3 once and count neighbors for each role — corners (sparse), edges (medium), center (everyone).
Cells that the grid's symmetries carry onto each other behave identically, so counting splits by role.
▸ Why?
A symmetry moves the grid onto itself without stretching, so neighbours are carried to neighbours.
▸ Why?
Each cell belongs to exactly one role, so the roles cover everything without overlapping.
When the centre survives
Write the condition when the centre started filled.
Treat the center separately because it's the only cell every other cell sees.
1.OA.A.2Identify SubproblemsCount the first sub-case
The first sub-case gives two.
Opposite-corner pairs are the only non-adjacent pairs with disjoint peripheral neighborhoods.
3.OA.D.8Make A Systematic ListCount the second sub-case
The second sub-case gives none.
Three mutually non-adjacent peripherals always leave a 'common neighbor' empty cell that gets born.
3.OA.D.8Make A Systematic ListTotal for the first case
Total up the first case.
Two opposite-corner pairs, nothing else.
1.OA.A.2Identify SubproblemsWhen the centre started empty
Now exactly three must be filled.
Three filled cells, no cell touches both others, no outside cell touches all three.
1.OA.A.2Identify SubproblemsCount the first shape
Count the placements of the first shape.
Three of the four corners — pick which corner to leave out.
3.OA.A.3Make A Systematic ListCount the second shape
Count the second shape too.
Two corners flanking one side, with the lonely opposite edge filling in.
3.OA.A.3Make A Systematic ListCount the third shape
Count the third shape as well.
Two adjacent edges + the far-diagonal corner — the V-and-dot pattern.
3.OA.A.3Make A Systematic ListCount the fourth shape
The fourth shape has more placements.
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 ListAdd them all
Adding them gives 22.
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).