2022 AMC 10B Problems/Problem 19
Each square in a 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 grid has a border of empty squares surrounding a 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.)
Solution
There are two cases: 2\textbf{Case 2:} The center is not filled In this case, exactly three squares outside the center must be filled. Additionally, these three squares must all be adjacent to only one square, the center square. 316\textbf{Subcase 2:} No corners are filled In this case, there is only one way to do it (excluding rotations and reflections). There are a total of four ways to rotate this permutation, so this case has a count of . Our final answer is thus $2 + 16 + 4 = \boxed{\textbf{(C)}\ 22} ~mathboy100