Mastermind with a Linear Number of Queries

TL;DR

This paper introduces a groundbreaking algorithm that solves the Mastermind game with a linear number of queries for the case where the number of colors equals the number of positions, resolving a long-standing open problem.

Contribution

The paper presents the first algorithm achieving a linear query complexity for Mastermind when the number of colors equals the number of positions, closing the gap in existing bounds.

Findings

Achieves linear query complexity for k=n Mastermind

Determines the query complexity for all parameters k and n

Resolves the open problem of optimal query bounds for k=n

Abstract

Since the 1960s Mastermind has been studied for the combinatorial and information theoretical interest the game has to offer. Many results have been discovered starting with Erd\H{o}s and R\'enyi determining the optimal number of queries needed for two colors. For $k$ colors and $n$ positions, Chv\'atal found asymptotically optimal bounds when $k \le n^{1-\epsilon}$. Following a sequence of gradual improvements for $k \geq n$ colors, the central open question is to resolve the gap between $\Omega(n)$ and $\mathcal{O}(n\log \log n)$ for $k=n$. In this paper, we resolve this gap by presenting the first algorithm for solving $k=n$ Mastermind with a linear number of queries. As a consequence, we are able to determine the query complexity of Mastermind for any parameters $k$ and $n$.