Dots & Boxes is PSPACE-complete

Abstract

Exactly 20 years ago at MFCS, Demaine posed the open problem whether the game of Dots&Boxes is PSPACE-complete. Dots&Boxes has been studied extensively, with for instance a chapter in Berlekamp et al."Winning Ways for Your Mathematical Plays", a whole book on the game"The Dots and Boxes Game: Sophisticated Child's Play"by Berlekamp, and numerous articles in the"Games of No Chance"series. While known to be NP-hard, the question of its complexity remained open. We resolve this question, proving that the game is PSPACE-complete by a reduction from a game played on propositional formulas.