Variations on Narrow Dots-and-Boxes and Dots-and-Triangles

TL;DR

This paper proves specific strategic outcomes in narrow Dots-and-Boxes and Dots-and-Triangles games, confirming a conjecture and analyzing first-player guarantees in various game configurations.

Contribution

It verifies a conjecture about first-player wins in closed 1×n Dots-and-Triangles and analyzes tie guarantees in open and closed 1×n Dots-and-Boxes for even n.

Findings

Closed 1×n Dots-and-Triangles is a first-player win when n ≠ 2

First player can guarantee a tie in open and closed 1×n Dots-and-Boxes for even n

Confirmed conjecture of Nowakowski and Ottaway

Abstract

We verify a conjecture of Nowakowski and Ottaway that closed $1 \times n$ Dots-and-Triangles is a first-player win when $n \neq 2$. We also prove that in both the open and closed $1 \times n$ Dots-and-Boxes games where $n$ is even, the first player can guarantee a tie.