Mastermind with a Linear Number of Queries

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 Erdős and Rényi determining the optimal number of queries needed for two colors. For $k$ colors and $n$ positions, Chvátal found asymptotically optimal bounds when $k\le n^{1-\epsilon }$. Following a sequence of gradual improvements for $k\ge n$ 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 $n$.