2022 AMC 10B Problem 19

Label the inner $3 \times 3$ like this, with $X$ the center:

Call $a, c, f, h$ the corners and $b, d, e, g$ the edge cells. Every one of these eight touches $X$. Besides $X$:

• a corner touches only its two edge cells (for example, $a$ touches $b$ and $d$);
• an edge cell touches every other edge cell except the opposite one, plus the two corners beside it (for example, $b$ touches $d$, $e$, $a$, $c$).

What must happen. Afterwards $X$ is filled and every other cell is empty. So:

• if $X$ starts filled, it needs $2$ or $3$ filled neighbours; if it starts empty, exactly $3$;
• a filled cell other than $X$ must not have $2$ or $3$ filled neighbours;
• an empty cell must not have exactly $3$ filled neighbours;
• a border cell can see at most three inner cells, and all three only if an entire side of the $3 \times 3$ is filled, so no full side is allowed.

Case 1: $X$ starts filled, with $2$ or $3$ filled outer cells.

1. Every filled outer cell already sees $X$. If it touched even one other filled outer cell it would have $2$ or more filled neighbours and survive. So no two filled outer cells may touch.
2. An empty outer cell that touches two filled outer cells would also see $X$, making $3$, and would fill up. So no empty cell may touch two of them.
3. Two filled. Two corners on the same side (like $a$, $c$) both touch $b$: fails. A corner and an edge cell that do not touch also fail: $a$ and $e$ both touch $b$, and $a$ and $g$ both touch $d$. Opposite edge cells (like $b$, $g$) both touch $d$: fails. Opposite corners ($a$ and $h$, or $c$ and $f$) share no neighbour: this works. That gives $2$.
4. Three filled, none touching. The only such groups are three corners, or an edge cell with the two corners on the far side (like $b$, $f$, $h$). In both, an edge cell between two of the filled corners ($g$ in the second example) sees two corners plus $X$, which is $3$. Fails. That gives $0$.

Case 2: $X$ starts empty, with exactly $3$ filled outer cells.

1. A filled cell now sees only outer cells, so it must not touch both of the other two. An empty cell must not touch all three. (A full side, like $a, b, c$, already breaks the first rule, since $b$ touches $a$ and $c$.)
2. Three corners. No two corners touch, and each edge cell touches only two corners. Works: $4$ ways.
3. Two corners on the same side plus an edge cell. Take $a, c$. Adding $b$: $b$ touches both, fails. Adding $d$: then $b$ is empty and touches $a$, $c$, $d$, fails ($e$ is the same). Adding $g$: $g$ touches neither corner, and no empty cell touches all three. Works. There are $4$ sides, so $4$ ways.
4. Two opposite corners plus an edge cell. Take $a, h$ and add $b$. Only $a$ and $b$ touch each other; the empty cells $d$ and $e$ each see just $2$. Works, and the same holds for any of the $4$ edge cells. With $2$ pairs of opposite corners: $2 \times 4 = 8$ ways.
5. One corner plus two edge cells. Take $a$. Opposite edge cells fail ($b, g$: then $d$ touches $a$, $b$, $g$; and $d, e$: then $b$ touches all three). Any pair that includes $b$ or $d$ also touches $a$ and the other edge cell, so that cell touches both others: fails. The only pair left is $e, g$, which touch each other but not $a$. Check the empty cells: $b$ sees $a, e$; $d$ sees $a, g$; $h$ sees $e, g$. All $2$ or fewer. Works: $1$ way per corner, $4$ ways.
6. Three edge cells. Two of them are opposite, and the third touches both. Fails: $0$ ways.

Total. $2 + 4 + 4 + 8 + 4 = 22$.

The answer is $\boxed{\textbf{(C)}\ 22}$.

The rules only let a cell be filled afterwards if it has two or three filled neighbours, so "only the center is filled afterwards" limits the start to very few filled cells. After that it is careful bookkeeping of which cells touch which. Sorting by how many corners are used, and using the symmetry of the square (count one corner, then multiply by $4$), keeps each case short.