Dots & Boxes is PSPACE-complete

TL;DR

This paper proves that the game Dots & Boxes is PSPACE-complete, resolving a long-standing open problem by providing a complexity classification that was previously unknown.

Contribution

It establishes the PSPACE-completeness of Dots & Boxes, a significant advancement in understanding the game's computational complexity.

Findings

Dots & Boxes is PSPACE-complete.

The proof involves a reduction from a game on propositional formulas.

This resolves a 20-year-old open problem about the game's complexity.

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.