English

Misere Connect Four is Solved

Combinatorics 2025-02-18 v2 Computer Science and Game Theory

Abstract

Connect Four is a two-player game where each player attempts to be the first to create a sequence of four of their pieces, arranged horizontally, vertically, or diagonally, by dropping pieces into the columns of a grid of width seven and height six, in alternating turns. Misere Connect Four is played by the same rules, but with the opposite objective: do not connect four. This paper announces that Misere Connect Four is solved: perfect play by both sides leads to a second-player win. More generally, this paper also announces that Misere Connect kk played on a w×hw \times h board is also solved, but the outcome depends on the game's parameters kk, ww, and hh, and may be a first-player win, a second-player win, or a draw. These results are constructive, meaning that we provide explicit strategies, thus enabling readers to impress their friends and foes alike with provably optimal play in the misere form of a table-top game for children.

Keywords

Cite

@article{arxiv.2410.05551,
  title  = {Misere Connect Four is Solved},
  author = {Robert Steele and Daniel B. Larremore},
  journal= {arXiv preprint arXiv:2410.05551},
  year   = {2025}
}

Comments

Revised. 13 pages, 4 figures, 2 tables, and a handful of proofs. Title also changed to remove diacritic so that it is searchable under Misere

R2 v1 2026-06-28T19:12:14.617Z