AMC 12 2022b Problem 18

Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:

• Any filled square with two or three filled neighbors remains filled.

• Any empty square with exactly three filled neighbors becomes a filled square.

• All other squares remain empty or become empty.

A sample transformation is shown in the figure below.

import geometry;
unitsize(0.6cm);

void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,invisible);
}

ds((1,1));
ds((2,1));
ds((3,1));
ds((1,3));

for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
}

label("Initial", (2.5,-1));
draw((6,2.5)--(8,2.5),Arrow);

ds((10,2));
ds((11,1));
ds((11,0));

for (int i = 0; i <= 5; ++i) {
draw((9,i)--(14,i));
draw((i+9,0)--(i+9,5));
}

label("Transformed", (11.5,-1));

Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)

import geometry;
unitsize(0.6cm);

void ds(pair x) {
filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,invisible);
}

for (int i = 1; i < 4; ++ i) {
for (int j = 1; j < 4; ++j) {
label("?",(i + 0.5, j + 0.5));
}
}

for (int i = 0; i <= 5; ++i) {
draw((0,i)--(5,i));
draw((i,0)--(i,5));
}

label("Initial", (2.5,-1));
draw((6,2.5)--(8,2.5),Arrow);

ds((11,2));

for (int i = 0; i <= 5; ++i) {
draw((9,i)--(14,i));
draw((i+9,0)--(i+9,5));
}

label("Transformed", (11.5,-1));

\textbf{(A)}\ 14 \textbf{(B)}\ 18 \textbf{(C)}\ 22 \textbf{(D)}\ 26 \textbf{(E)}\ 30