On the Existence of Optimal Strategies in a Combinatorial Game

TL;DR

This paper analyzes a combinatorial game based on a mathematical competition problem, identifying winning strategies depending on set size and parameters, and demonstrates results with a web implementation.

Contribution

It introduces a generalized version of the game and characterizes winning strategies for various game variants based on set size and parameters.

Findings

Second player wins for even-sized sets

First player wins for many odd-sized sets

Web implementation validates theoretical results

Abstract

We study a combinatorial game derived from a problem in the German National Mathematics Competition. In this game, two players take turns removing numbers from a finite set of natural numbers, aiming to satisfy a certain divisibility condition. We introduce a generalized version of the original game, which depends on two parameters: the size of the initial number set and a fixed divisor. For both players, we identify a broad range of game variants in which they can force a win. In particular, we show that for even-sized sets, the second player to move can always win, while for many odd-sized cases, the first player to move has a winning strategy. A web implementation of the game demonstrates some of our results in practice.