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 (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 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 non-adaptive strategy (deterministic or randomized) needs Θ(n log n) guesses.

Mastermind with a linear number of queries

“…In this paper, we resolve this problem after almost 40 years by showing how the n colour n slot Mastermind can be solved with O(n) guesses with high probability, matching the informationtheoretic lower bound up to a constant factor. By combining this with a result by Doerr, Doerr, Spöhel, and Thomas [4], we determine asymptotically the optimal number of guesses for all k and n.…”

“…As a comparison, we note that if k = k(n) is polynomial in n, then the entropy lower bound is simply (n). This gap is a very natural one as Doerr, Doerr, Spöhel, and Thomas [4] showed in a relatively recent paper that if one uses a non-adaptive strategy, there is in fact a lower bound of (n log n) when k = n. In the same paper, they also use an adaptive strategy to significantly narrow this gap, showing that O(n log log n) guesses suffice for k = n. Moreover, they present a randomised reduction from black-white-peg Mastermind to black-peg Mastermind which shows that bwmm…”

“…Similar to the algorithm by Doerr, Doerr, Spöhel, and Thomas [4], we will base our solution on coin-weighing schemes. Here, we think of the O(n log n) tasks described above as our coins, where a coin has the value 1 if the colour in the corresponding task is present in the left half of its interval, and 0 otherwise.…”

“…In particular, for k = n, the best known upper bound remained for a long time O(n log n) as shown by Chvátal, with only constant factor improvements given in [2], [8], and [9]. Only quite recently, this bound was improved to O(n log log n) in an article by Doerr, Doerr, Spöhel, and Thomas [4] published in Journal of the ACM in 2016. At the same time, no significant improvement on the information-theoretic lower bound of n queries has been obtained.…”

Optimal schemes for combinatorial query problems with integer feedback

“…In 2016, Doerr, Doerr, Spöhel and Thomas [DDST16] first managed to narrow this gap by showing that black-peg Mastermind with k = n colors can be solved in O(n log log n) queries. Finally, a recent work of the author together with Su [MS21] construct a randomized solutions for black-peg Mastermind with k = n colors using O(n) queries.…”