{"status": "success", "data": {"description_md": "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:\n* Any filled square with two or three filled neighbors remains filled.\n* Any empty square with exactly three filled neighbors becomes a filled square.\n* All other squares remain empty or become empty.\nA sample transformation is shown in the figure below.\n\n
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\n\nSuppose 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.)\n\n
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\n\n$\\textbf{(A)}\\ 14 \\qquad\\textbf{(B)}\\ 18 \\qquad\\textbf{(C)}\\ 22 \\qquad\\textbf{(D)}\\ 26 \\qquad\\textbf{(E)}\\ 30$", "description_html": "

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:

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.)

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

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2022 AMC 10B Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc10B_p20", "prev": "/problem/22_amc10B_p18"}}