Playing Mastermind With Constant-Size Memory

We analyze the classic board game of Mastermind with $n$ holes and a constant number of colors. A result of Chvátal (Combinatorica 3 (1983), 325-329) states that the codebreaker can find the secret code with $Θ(n / \log n)$ 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.

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