Misere Connect Four is Solved

TL;DR

This paper proves that Misere Connect Four, where players aim to avoid connecting four, is solved with perfect play leading to a second-player win, and extends the results to generalized boards and connect lengths.

Contribution

It provides the first complete solution for Misere Connect Four and its generalizations, including explicit optimal strategies for all game states.

Findings

Misere Connect Four is solved with a second-player win under perfect play.

The outcome of Misere Connect $k$ on $w imes h$ boards varies with parameters, including first-player win, second-player win, or draw.

Explicit strategies are provided for optimal play in Misere Connect Four and its variants.

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 $k$ played on a $w \times h$ board is also solved, but the outcome depends on the game's parameters $k$, $w$, and $h$, 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.