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5

Part of 2023 Junior Macedonian Mathematical Olympiad

Problems(1)

Consider a $2023\times2023$ board split into unit squares.

Source: 2023 Junior Macedonian Mathematical Olympiad P5

10/6/2023
Consider a 2023×20232023\times2023 board split into unit squares. Two unit squares are called adjacent is they share at least one vertex. Mahler and Srecko play a game on this board. Initially, Mahler has one piece placed on the square marked M, and Srecko has a piece placed on the square marked by S (see the attachment). The players alternate moving their piece, following three rules: 1. A piece can only be moved to a unit square adjacent to the one it is placed on. 2. A piece cannot be placed on a unit square on which a piece has been placed before (once used, a unit square can never be used again). 3. A piece cannot be moved to a unit square adjacent to the square occupied by the opponent’s piece. A player wins the game if his piece gets to the corner diagonally opposite to its starting position (i.e. Srecko moves to sps_p, Mahler moves to mpm_p) or if the opponent has to move but has no legal move. Mahler moves first. Which player has a winning strategy?
combinatorics