Abstract:Mastermind is a two players zero sum game of imperfect information. Starting with Erdős and Rényi (1963), its combinatorics have been studied to date by several authors, e.g., Knuth (1977), Chvátal (1983, . The first player, called "codemaker", chooses a secret code and the second player, called "codebreaker", tries to break the secret code by making as few guesses as possible, exploiting information that is given by the codemaker after each guess. For variants that allow color repetition, Doerr et al. (2016) showed optimal results. In this paper, we consider the so called Black-Peg variant of Mastermind, where the only information concerning a guess is the number of positions in which the guess coincides with the secret code. More precisely, we deal with a special version of the Black-Peg game with n holes and k ≥ n colors where no repetition of colors is allowed. We present upper and lower bounds on the number of guesses necessary to break the secret code. For the case k = n, the secret code can be algorithmically identified within less than (n − 3) log 2 n + 5 2 n − 1 queries. This result improves the result of Ker-I Ko and Shia-Chung Teng (1985) by almost a factor of 2. For the case k > n, we prove an upper bound of (n − 2) log 2 n + k + 1. Furthermore, we prove a new lower bound of n for the case k = n, which improves the recent n − log log(n) bound of Berger et al. (2016). We then generalize this lower bound to k queries for the case k ≥ n.
No abstract
The triangle game introduced by Chvátal and Erdős (1978) is one of the most famous combinatorial games. For n, q ∈ N, the (n, q)-triangle game is played by two players, called Maker and Breaker, on the complete graph Kn. Alternately Maker claims one edge and thereafter Breaker claims q edges of the graph. Maker wins the game if he can claim all three edges of a triangle, otherwise Breaker wins. Chvátal and Erdős (1978) proved that for q < √ 2n + 2 − 5/2 ≈ 1.414 √ n Maker has a winning strategy, and for q ≥ 2 √ n Breaker has a winning strategy. Since then, the problem of finding the exact leading constant for the threshold bias of the triangle game has been one of the famous open problems in combinatorial game theory. In fact, the constant is not known for any graph with a cycle and we do not even know if such a constant exists. Balogh and Samotij (2011) slightly improved the Chvátal-Erdős constant for Breaker's winning strategy from 2 to 1.935 with a randomized approach. Since then no progress was made. In this work, we present a new deterministic strategy for Breaker's win whenever n is sufficiently large and q ≥ (8/3 + o(1))n ≈ 1.633 √ n, significantly reducing the gap towards the lower bound. In previous strategies Breaker chooses his edges such that one node is part of the last edge chosen by Maker, whereas the remaining node is chosen more or less arbitrarily. In contrast, we introduce a suitable potential function on the set of nodes. This allows Breaker to pick edges that connect the most 'dangerous' nodes. The total potential of the game may still increase, even for several turns, but finally Breaker's strategy prevents the total potential of the game from exceeding a critical level and leads to Breaker's win.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.