AMC 10 Step by Step
2022 AMC 10BProblem 19P16-20~8 minPrint

Each square in a 5×55 \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. figure Suppose the 5×55 \times 5 grid has a border of empty squares surrounding a 3×33 \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.) figure

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Problem © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. Solution and commentary are original to this site.