An n-in-a-row Game

TL;DR

This paper analyzes a variant of the n-in-a-row game where players claim an increasing number of points each turn, showing that the game duration is approximately proportional to n, specifically (1-o(1))n.

Contribution

It introduces and studies a new variant of the n-in-a-row game with increasing claims per turn, providing bounds on the game's duration.

Findings

Game duration is approximately (1-o(1))n turns.

The increasing claim rule significantly affects game length.

The result is somewhat unexpected given the game's structure.

Abstract

The usual $n$-in-a-row game is a positional game in which two player alternately claim points in $\bb{Z}^2$ with the winner being the first player to claim $n$ consecutive points in a line. We consider a variant of the game, suggested by Croft, where the number of points claimed increases by 1 each turn, and so on turn $t$ a player claims $t$ points. Croft asked how long it takes to win this game. We show that, perhaps surprisingly, the time needed to win this game is $(1-o(1))n$.