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Computer Science, Data Structures and Algorithms, cs.DS,Computer Science, Discrete Mathematics, cs.DM
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 $O(n
\log\log n)$ bound holds for up to $n^2 \log\log n$ colors. These bounds are
almost tight as the known lower bound of $\Omega(n)$ 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.
