{"status": "success", "data": {"description_md": "Ivan and Owen are playing a game, where the goal is to move a cart from the top-left cell to the bottom-right cell of an $80\\times 80$ board without moving the cart out of the board. Without changing columns, Ivan can move the cart down $1, 2$, or $3$ rows. Owen can move the cart right $1, 2$, or $3$ columns without changing rows. If a player can't move, they lose. Given that Ivan moves first and both play optimally, find the minimum number of moves that Ivan can make to guarantee his win.\n\n$\\textbf{(A)}~39\\qquad\\textbf{(B)}~40\\qquad\\textbf{(C)}~77\\qquad\\textbf{(D)}~78\\qquad\\textbf{(E)}~\\text{It is impossible for Ivan to win the game.}$", "description_html": "<p>Ivan and Owen are playing a game, where the goal is to move a cart from the top-left cell to the bottom-right cell of an <span class=\"katex--inline\">80\\times 80</span> board without moving the cart out of the board. Without changing columns, Ivan can move the cart down <span class=\"katex--inline\">1, 2</span>, or <span class=\"katex--inline\">3</span> rows. Owen can move the cart right <span class=\"katex--inline\">1, 2</span>, or <span class=\"katex--inline\">3</span> columns without changing rows. If a player can&#8217;t move, they lose. Given that Ivan moves first and both play optimally, find the minimum number of moves that Ivan can make to guarantee his win.</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}~39\\qquad\\textbf{(B)}~40\\qquad\\textbf{(C)}~77\\qquad\\textbf{(D)}~78\\qquad\\textbf{(E)}~\\text{It is impossible for Ivan to win the game.}</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 2, "problem_name": "2024 Mock AMC 10 - Problem 6", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}