{"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<center>\n<img class=\"problem-image\" height=\"182\" src=\"https://latex.artofproblemsolving.com/0/0/3/00333423a7ddd483233c145023df02bdb469a3b9.png\" width=\"398\"/>\n</center>\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<center>\n<img class=\"problem-image\" height=\"182\" src=\"https://latex.artofproblemsolving.com/0/0/3/00333423a7ddd483233c145023df02bdb469a3b9.png\" width=\"398\"/>\n</center>\n\n$\\textbf{(A)}\\ 14 \\qquad\\textbf{(B)}\\ 18 \\qquad\\textbf{(C)}\\ 22 \\qquad\\textbf{(D)}\\ 26 \\qquad\\textbf{(E)}\\ 30$", "description_html": "<p>Each square in a  <span class=\"katex--inline\">5 \\times 5</span>  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:</p>\n<ul>\n<li>Any filled square with two or three filled neighbors remains filled.</li>\n<li>Any empty square with exactly three filled neighbors becomes a filled square.</li>\n<li>All other squares remain empty or become empty.<br/>\nA sample transformation is shown in the figure below.</li>\n</ul>\n<center>\n<img class=\"problem-image\" height=\"182\" src=\"https://latex.artofproblemsolving.com/0/0/3/00333423a7ddd483233c145023df02bdb469a3b9.png\" width=\"398\"/>\n</center>\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<center>\n<img class=\"problem-image\" height=\"182\" src=\"https://latex.artofproblemsolving.com/0/0/3/00333423a7ddd483233c145023df02bdb469a3b9.png\" width=\"398\"/>\n</center>\n<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 14 \\qquad\\textbf{(B)}\\ 18 \\qquad\\textbf{(C)}\\ 22 \\qquad\\textbf{(D)}\\ 26 \\qquad\\textbf{(E)}\\ 30</span> </p>\n<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "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"}}