{"status": "success", "data": {"description_md": "Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$<br><center><img class=\"problem-image\" alt='[asy] unitsize(4mm); real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33}; for(real i:boxes){ \tdraw(box((i,0),(i+1,3))); } draw((8,1.5)--(12,1.5),Arrow()); defaultpen(fontsize(20pt)); label(\",\",(20,0)); label(\",\",(29,0)); label(\",...\",(35.5,0)); [/asy]' class=\"latexcenter\" height=\"68\" src=\"https://latex.artofproblemsolving.com/1/a/5/1a5938c90db67cc867d798bb9693e149b9438071.png\" width=\"695\"/></center><br>Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?\n\n$\\textbf{(A) }(6,1,1) \\qquad \\textbf{(B) }(6,2,1) \\qquad \\textbf{(C) }(6,2,2)\\qquad \\textbf{(D) }(6,3,1) \\qquad \\textbf{(E) }(6,3,2)$\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>Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one &#8220;wall&#8221; among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes  <span class=\"katex--inline\">4</span>  and  <span class=\"katex--inline\">2</span>  can be changed into any of the following by one move:  <span class=\"katex--inline\">(3,2),(2,1,2),(4),(4,1),(2,2),</span>  or  <span class=\"katex--inline\">(1,1,2).</span> <br/><center><img class=\"latexcenter\" alt=\"[asy] unitsize(4mm); real[] boxes = {0,1,2,3,5,6,13,14,15,17,18,21,22,24,26,27,30,31,32,33}; for(real i:boxes){ &#9;draw(box((i,0),(i+1,3))); } draw((8,1.5)--(12,1.5),Arrow()); defaultpen(fontsize(20pt)); label(&#34;,&#34;,(20,0)); label(&#34;,&#34;,(29,0)); label(&#34;,...&#34;,(35.5,0)); [/asy]\" height=\"68\" src=\"https://latex.artofproblemsolving.com/1/a/5/1a5938c90db67cc867d798bb9693e149b9438071.png\" width=\"695\"/></center><br/>Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }(6,1,1) \\qquad \\textbf{(B) }(6,2,1) \\qquad \\textbf{(C) }(6,2,2)\\qquad \\textbf{(D) }(6,3,1) \\qquad \\textbf{(E) }(6,3,2)</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": 5, "problem_name": "2021 AMC 12B Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc12B_p23", "prev": "/problem/21_amc12B_p21"}}