Solving Static Permutation Mastermind using $O(n \log n)$ Queries

TL;DR

This paper proves that the static permutation Mastermind game can be solved with an optimal number of queries proportional to n log n, improving previous bounds and answering an open question.

Contribution

The paper demonstrates that the query complexity for solving static permutation Mastermind is O(n log n), matching the theoretical lower bound and resolving a longstanding open problem.

Findings

Established an O(n log n) query bound for static permutation Mastermind

Provided a simple probabilistic proof for the improved bound

Resolved the open question from previous research

Abstract

Permutation Mastermind is a version of the classical mastermind game in which the number of positions $n$ is equal to the number of colors $k$, and repetition of colors is not allowed, neither in the codeword nor in the queries. In this paper we solve the main open question from Glazik, J\"ager, Schiemann and Srivastav (2021), who asked whether their bound of $O(n^{1.525})$ for the static version can be improved to $O(n \log n)$, which would be best possible. By using a simple probabilistic argument we show that this is indeed the case.