Playing Mastermind With Constant-Size Memory

TL;DR

This paper proves that in the game of Mastermind with a fixed number of colors, the optimal number of questions remains proportional to n / log n even when the codebreaker has only constant memory, challenging previous conjectures.

Contribution

It demonstrates that the known bounds for Mastermind's complexity hold under constant memory constraints, disproving a prior conjecture on memory-restricted black-box complexity.

Findings

The codebreaker can find the secret with Θ(n / log n) questions despite constant memory limits.

The result applies to the classic Mastermind game with a fixed number of colors.

It disproves a conjecture on the black-box complexity of the OneMax function class.

Abstract

We analyze the classic board game of Mastermind with $n$ holes and a constant number of colors. A result of Chv\'atal (Combinatorica 3 (1983), 325-329) states that the codebreaker can find the secret code with $\Theta(n / \log n)$ questions. We show that this bound remains valid if the codebreaker may only store a constant number of guesses and answers. In addition to an intrinsic interest in this question, our result also disproves a conjecture of Droste, Jansen, and Wegener (Theory of Computing Systems 39 (2006), 525-544) on the memory-restricted black-box complexity of the OneMax function class.