2024 Mock AMC 10 - Problem 6

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.

$\textbf{(A)}~39\qquad\textbf{(B)}~40\qquad\textbf{(C)}~77\qquad\textbf{(D)}~78\qquad\textbf{(E)}~\text{It is impossible for Ivan to win the game.}$