{"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:<br>* Any filled square with two or three filled neighbors remains filled.<br>* Any empty square with exactly three filled neighbors becomes a filled square.<br>* All other squares remain empty or become empty.<br>A sample transformation is shown in the figure below.<br><center><img class=\"problem-image\" alt='[asy]         import geometry;         unitsize(0.6cm);          void ds(pair x) {             filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,invisible);         }          ds((1,1));         ds((2,1));         ds((3,1));         ds((1,3));          for (int i = 0; i &lt;= 5; ++i) {             draw((0,i)--(5,i));             draw((i,0)--(i,5));         }          label(\"Initial\", (2.5,-1));         draw((6,2.5)--(8,2.5),Arrow);          ds((10,2));         ds((11,1));         ds((11,0));          for (int i = 0; i &lt;= 5; ++i) {             draw((9,i)--(14,i));             draw((i+9,0)--(i+9,5));         }          label(\"Transformed\", (11.5,-1)); [/asy]' class=\"latexcenter\" height=\"182\" src=\"https://latex.artofproblemsolving.com/0/0/3/00333423a7ddd483233c145023df02bdb469a3b9.png\" width=\"398\"/></center><br>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.)<br><center><img class=\"problem-image\" alt='[asy]         import geometry;         unitsize(0.6cm);          void ds(pair x) {             filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,invisible);         }          ds((1,1));         ds((2,1));         ds((3,1));         ds((1,3));          for (int i = 0; i &lt;= 5; ++i) {             draw((0,i)--(5,i));             draw((i,0)--(i,5));         }          label(\"Initial\", (2.5,-1));         draw((6,2.5)--(8,2.5),Arrow);          ds((10,2));         ds((11,1));         ds((11,0));          for (int i = 0; i &lt;= 5; ++i) {             draw((9,i)--(14,i));             draw((i+9,0)--(i+9,5));         }          label(\"Transformed\", (11.5,-1)); [/asy]' class=\"latexcenter\" height=\"182\" src=\"https://latex.artofproblemsolving.com/0/0/3/00333423a7ddd483233c145023df02bdb469a3b9.png\" width=\"398\"/></center>\n\n$\\textbf{(A)}\\ 14 \\qquad\\textbf{(B)}\\ 18 \\qquad\\textbf{(C)}\\ 22 \\qquad\\textbf{(D)}\\ 26 \\qquad\\textbf{(E)}\\ 30$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "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:<br/>* Any filled square with two or three filled neighbors remains filled.<br/>* Any empty square with exactly three filled neighbors becomes a filled square.<br/>* All other squares remain empty or become empty.<br/>A sample transformation is shown in the figure below.<br/><center><img class=\"latexcenter\" alt=\"[asy]         import geometry;         unitsize(0.6cm);          void ds(pair x) {             filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,invisible);         }          ds((1,1));         ds((2,1));         ds((3,1));         ds((1,3));          for (int i = 0; i &lt;= 5; ++i) {             draw((0,i)--(5,i));             draw((i,0)--(i,5));         }          label(&#34;Initial&#34;, (2.5,-1));         draw((6,2.5)--(8,2.5),Arrow);          ds((10,2));         ds((11,1));         ds((11,0));          for (int i = 0; i &lt;= 5; ++i) {             draw((9,i)--(14,i));             draw((i+9,0)--(i+9,5));         }          label(&#34;Transformed&#34;, (11.5,-1)); [/asy]\" height=\"182\" src=\"https://latex.artofproblemsolving.com/0/0/3/00333423a7ddd483233c145023df02bdb469a3b9.png\" width=\"398\"/></center><br/>Suppose the  <span class=\"katex--inline\">5 \\times 5</span>  grid has a border of empty squares surrounding a  <span class=\"katex--inline\">3 \\times 3</span>  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.)<br/><center><img class=\"latexcenter\" alt=\"[asy]         import geometry;         unitsize(0.6cm);          void ds(pair x) {             filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,mediumgray,invisible);         }          ds((1,1));         ds((2,1));         ds((3,1));         ds((1,3));          for (int i = 0; i &lt;= 5; ++i) {             draw((0,i)--(5,i));             draw((i,0)--(i,5));         }          label(&#34;Initial&#34;, (2.5,-1));         draw((6,2.5)--(8,2.5),Arrow);          ds((10,2));         ds((11,1));         ds((11,0));          for (int i = 0; i &lt;= 5; ++i) {             draw((9,i)--(14,i));             draw((i+9,0)--(i+9,5));         }          label(&#34;Transformed&#34;, (11.5,-1)); [/asy]\" height=\"182\" src=\"https://latex.artofproblemsolving.com/0/0/3/00333423a7ddd483233c145023df02bdb469a3b9.png\" width=\"398\"/></center></p>&#10;<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>&#10;<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 12B Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc12B_p19", "prev": "/problem/22_amc12B_p17"}}