An $n$-in-a-row type game

TL;DR

This paper analyzes a positional game on the plane where Maker tries to align n points, and Breaker aims to prevent this, showing Breaker can delay Maker's win with relatively few points.

Contribution

It establishes bounds on Breaker's ability to prevent Maker from winning in an n-in-a-row game, including when Maker claims points at various growth rates.

Findings

Breaker can prevent Maker from winning until about Maker's n-th turn.

Breaker needs only (log t) points per turn to delay Maker.

Results extend to Maker claiming points at different growth rates.

Abstract

We consider a Maker-Breaker type game on the plane, in which each player takes $t$ points on their $t^\textrm{th}$ turn. Maker wins if he obtains $n$ points on a line (in any direction) without any of Breaker's points between them. We show that, despite Maker's apparent advantage, Breaker can prevent Maker from winning until about his $n^\textrm{th}$ turn. We actually prove a stronger result: that Breaker only needs to play $\omega(\log t)$ points on his $t^\textrm{th}$ turn to prevent Maker from winning until this time. We also consider the situation when the number of points claimed by Maker grows at other speeds, in particular, when Maker claims $t^\alpha$ points on his $t^\textrm{th}$ turn.