{"status": "success", "data": {"description_md": "Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbaras turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jennas turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with $2013$ coins and when the game starts with $2014$ coins?\n\n$\\textbf{(A)}$ Barbara will win with $2013$ coins and Jenna will win with $2014$ coins. \n\n$\\textbf{(B)}$ Jenna will win with $2013$ coins, and whoever goes first will win with $2014$ coins. \n\n$\\textbf{(C)}$ Barbara will win with $2013$ coins, and whoever goes second will win with $2014$ coins.\n\n$\\textbf{(D)}$ Jenna will win with $2013$ coins, and Barbara will win with $2014$ coins.\n\n$\\textbf{(E)}$ Whoever goes first will win with $2013$ coins, and whoever goes second will win with $2014$ coins.\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>Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara&#8217;s turn, she must remove  <span class=\"katex--inline\">2</span>  or  <span class=\"katex--inline\">4</span>  coins, unless only one coin remains, in which case she loses her turn. When it is Jenna&#8217;s turn, she must remove  <span class=\"katex--inline\">1</span>  or  <span class=\"katex--inline\">3</span>  coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with  <span class=\"katex--inline\">2013</span>  coins and when the game starts with  <span class=\"katex--inline\">2014</span>  coins?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}</span>  Barbara will win with  <span class=\"katex--inline\">2013</span>  coins and Jenna will win with  <span class=\"katex--inline\">2014</span>  coins.</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(B)}</span>  Jenna will win with  <span class=\"katex--inline\">2013</span>  coins, and whoever goes first will win with  <span class=\"katex--inline\">2014</span>  coins.</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(C)}</span>  Barbara will win with  <span class=\"katex--inline\">2013</span>  coins, and whoever goes second will win with  <span class=\"katex--inline\">2014</span>  coins.</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(D)}</span>  Jenna will win with  <span class=\"katex--inline\">2013</span>  coins, and Barbara will win with  <span class=\"katex--inline\">2014</span>  coins.</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(E)}</span>  Whoever goes first will win with  <span class=\"katex--inline\">2013</span>  coins, and whoever goes second will win with  <span class=\"katex--inline\">2014</span>  coins.</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": "2013 AMC 12B Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/13_amc12B_p19", "prev": "/problem/13_amc12B_p17"}}