AMC 10 · 2022 · #19

• Label the $3 \times 3$ inner grid: four corners $C_1, C_2, C_3, C_4$, four edges $E_N, E_S, E_E, E_W$, and center $M$.
• Two facts to remember: every corner has $3$ peripheral neighbors (two edges and the center); every edge has $5$ peripheral neighbors (two corners, the two adjacent edges across the center are actually neighbors via diagonals — wait, count by adjacency: edge $E_N$ at $(1,2)$ has neighbors $(1,1), (1,3), (2,1), (2,2), (2,3)$, so $5$ neighbors).
• Center $M$ at $(2,2)$ has all $8$ peripheral cells as neighbors.

💡 Draw the $3 \times 3$ once and count neighbors for each role — corners (sparse), edges (medium), center (everyone).

• Case 1: center $M$ is initially filled.
• For $M$ to survive it needs $2$ or $3$ filled peripherals.
• Every other peripheral must end empty: if filled, its neighbor count (which includes $M = 1$) must not be $2$ or $3$; if empty, it must not see exactly $3$ filled neighbors.

💡 Treat the center separately because it's the only cell every other cell sees.

• Sub-case 1.1: $M$ has exactly $2$ filled peripherals, call them $P_1, P_2$.
• Each $P_i$ has $1$ (from $M$) plus any filled neighbors among other peripherals; for $P_i$ to die, this total must not be $2$ or $3$, so $P_1, P_2$ must not be neighbors.
• Also any empty peripheral neighbor of both $P_1$ and $P_2$ would see $1 + 2 = 3$ filled and be born — forbidden.
• Enumerating non-adjacent peripheral pairs with no common empty neighbor leaves exactly two cases: the two pairs of opposite corners $\{C_1, C_3\}$ and $\{C_2, C_4\}$.
• Other non-adjacent pairs (e.g., two corners on the same row) share the middle edge as common neighbor.

💡 Opposite-corner pairs are the only non-adjacent pairs with disjoint peripheral neighborhoods.

• Sub-case 1.2: $M$ has $3$ filled peripherals $P_1, P_2, P_3$.
• For each $P_i$ to die, it must have $0$ or $1$ other-peripheral filled neighbor (so $\#\text{nbrs} = 1, 2$, but we need $\ne 2$, so $0$ peripheral neighbors).
• All three must be mutually non-adjacent.
• But any independent set of $3$ peripheral squares produces an empty peripheral neighbor adjacent to two of them — that cell would see $1 + 2 = 3$ filled and be born.
• So this sub-case is empty: $0$ configurations.

💡 Three mutually non-adjacent peripherals always leave a 'common neighbor' empty cell that gets born.

Case 1 total: $2$ configurations.

💡 Two opposite-corner pairs, nothing else.

• Case 2: $M$ is initially empty.
• For $M$ to be born it needs exactly $3$ filled peripherals $P_1, P_2, P_3$.
• For each filled $P_i$ to die it must have $0$ or $1$ peripheral neighbor in $\{P_1, P_2, P_3\}$ (since center is empty, $\#\text{nbrs} = $ peripheral count; must $\ne 2, 3$).
• For each empty peripheral, $\#$ filled peripheral neighbors $\ne 3$.
• Classify the valid $\{P_1, P_2, P_3\}$ shapes by adjacency.

💡 Three filled cells, no cell touches both others, no outside cell touches all three.

• Shape A — three corners (mutually non-adjacent): $\binom{4}{3} = 4$ ways.
• Each remaining empty cell sees at most $2$ filled neighbors.
• Valid.
• $\Rightarrow 4$ configs.

💡 Three of the four corners — pick which corner to leave out.

• Shape B — two corners on the same side plus the opposite middle edge (e.g., $C_1, C_2, E_S$): mutually non-adjacent, no empty cell is adjacent to all three.
• Choose which side has two corners: $4$ ways.
• $\Rightarrow 4$ configs.

💡 Two corners flanking one side, with the lonely opposite edge filling in.

• Shape C — one filled edge plus the diagonally opposite corner (e.g., $E_N, E_W, C_3$ where $E_N$ and $E_W$ share corner $C_1$ that is empty)?
• Re-examine: this shape is "two edges sharing a corner plus the opposite corner." Two edges sharing a corner have adjacency degree $1$ each (they touch each other), and the lone opposite corner is non-adjacent to both.
• The shared common-neighbor corner sees both edges plus possibly more — needs $\#$ filled neighbors $\ne 3$.
• The shared corner sees the two filled edges ($2$), the corner across ($1$ via diagonal?
• no, opposite corner is not adjacent to the shared corner) — wait, the empty corner at $C_1$ has $E_N$ and $E_W$ as neighbors and $M$ as neighbor; with $M$ empty and the opposite corner $C_3$ not adjacent to $C_1$, that's $2$ filled neighbors — fine.
• Count: $4$ corners to position the empty shared corner, giving $4$ configs.

💡 Two adjacent edges + the far-diagonal corner — the V-and-dot pattern.

• Shape D — an L (one corner + an adjacent edge) plus the diagonally opposite corner.
• E.g., $C_1, E_N, C_3$: $C_1$ and $E_N$ are adjacent (degree $1$ each), $C_3$ is non-adjacent to both.
• Check empty-cell rule: the middle edge $E_S$ has $C_3$ as a neighbor and $M$ empty — only $1$ filled neighbor.
• The empty corner adjacent to $C_3$ has $1$ filled neighbor.
• No empty cell sees $3$ filled.
• $4$ corners $\times 2$ choices of which adjacent edge $= 8$ configs.

💡 L-shape (corner + one of its two adjacent edges) anchored at one corner of the grid, plus the diagonally opposite corner.

• Case 2 total: $4 + 4 + 4 + 8 = 20$ configurations.
• Combined with Case 1: total $= 2 + 20 = 22$.
• Answer $\textbf{(C)}\ 22$.

💡 Add the two disjoint cases — done.