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- W4300783872 abstract "We analyze the general version of the classic guessing game Mastermind with $n$ positions and $k$ colors. Since the case $k le n^{1-varepsilon}$, $varepsilon>0$ 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 Chv'atal (Combinatorica 3 (1983), 325--329). We also show that if both black and white answer-pegs are used, then the $O(n loglog n)$ bound holds for up to $n^2 loglog n$ colors. These bounds are almost tight as the known lower bound of $Omega(n)$ shows. Unlike for $k le n^{1-varepsilon}$, simply guessing at random until the secret code is determined is not sufficient. In fact, we show that an optimal non-adaptive strategy (deterministic or randomized) needs $Theta(n log n)$ guesses." @default.
- W4300783872 created "2022-10-04" @default.
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- W4300783872 date "2013-01-06" @default.
- W4300783872 modified "2023-09-27" @default.
- W4300783872 title "Playing mastermind with many colors" @default.
- W4300783872 hasPublicationYear "2013" @default.
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