Question

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.

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.)

$\ 14$

$\ 18$

$\ 22$

$\ 26$

$\ 30$

Solution:

Suppose the center is initially filled. Then, there are either either $2$ or $3$ other filled squares, each of which can't have $2$ or $3$ filled neighbors.

This means that there are at most $4$ filled squares, so each square has at most $3$ neighbors. Since they don't have $2$ or $3$ neighbors, they must have at most $1$ neighbor. The center square is a neighbor, so they can't have any other neighbor.

Suppose I have a filled square on an edge. Since there is some filled square that isn't a neighbor of the square, we can examine the two edges which are neighbors of the filled edge. If I have a filled edge on the corner, the edge on the same side as the corner would have three neighbors. If I choose the opposite edge, the adjacent edges would have three neighbors.

Suppose I choose a corner. Then, I need to choose another corner. If I choose the adjacent corner, then the edge between would have three neighbors, making it filled. Therefore, it must be the adjacent corner. This has $2$ configurations.

Suppose the center is initially empty. Then, there are $3$ filled neighbors of the center, each with at most $1$ neighbors. This means no square has two filled neighbors. This makes it only possible to do in the following ways:

The first three can rotated making $4$ configurations, and the last one can be rotated and reflected making $8$ configurations. There are $20$ configurations with the center being empty. This means there are $20+2=22$ different configurations.

Thus, the answer is C .

2022 AMC 10B Problem 19

Solution:

Problem 19 in Other Years