Question 19
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.)
• 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.)