Square Achievement Game

The square achievement game is a two-player game in which the players alternately claim cells of an square grid. The first player to occupy all four corner cells of an axis-parallel square wins. The side length may be any positive integral number of grid spacings. A full board with no winner is a draw, and tilted squares do not count (Reid 2026).

Reid (2026) proved that the first player has a winning strategy iff . For , optimal play gives a draw. A single explicit strategy works for every . Its proof combines finite checks with a simulation theorem extending the checks to arbitrarily large boards. Reid (2026) supplies a Lean formalization of the game and the winning strategies.