Playing Mastermind With Many Colors

Abstract

We analyze the general version of the classic guessing game Mastermind with n positions and k colors. Since the case k ≤ n 1 − ε , ε > 0 a constant, is well understood, we concentrate on larger numbers of colors. For the most prominent case k = n , our results imply that Codebreaker can find the secret code with O ( n log log n ) guesses. This bound is valid also when only black answer pegs are used. It improves the O ( n log n ) bound first proven by Chvátal. We also show that if both black and white answer pegs are used, then the O ( n log log n ) bound holds for up to n 2 log log n colors. These bounds are almost tight, as the known lower bound of Ω( n ) shows. Unlike for k ≤ n 1 − ε , simply guessing at random until the secret code is determined is not sufficient. In fact, we show that an optimal nonadaptive strategy (deterministic or randomized) needs Θ( n log n ) guesses.