Playing Mastermind with Many Colors

Doerr, Benjamin; Spöhel, Reto; Thomas, Henning; Winzen, Carola

doi:10.1137/1.9781611973105.50

Cited by 23 publications

(39 citation statements)

References 6 publications

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“…It starts with the questions (0, 1), (2, 3), (4,5), i.e., we use questions of the pattern (2i, 2i + 1). At most two of them can receive a non-empty answer.…”

Section: Algorithm For Two Pegsmentioning

confidence: 99%

“…Furthermore, we obtained additional values f (p, c) by a computer search. Recently, Doerr et al have shown that f (c, c) = O(c log log c) [4], improving the classical result f (c, c) = O(c log c) of Chvátal [3]. Also many variants of Mastermind have been investigated in the literature.…”

mentioning

confidence: 96%

“…The first candidates are (4,4) for the original variant, as f 0 (4, 4) > f (4,4), and (4, 3) for the black-peg variant, as b 0 (4, 3) > b (4,3). If we have found such a case, it would be interesting to consider strategies with size-two or larger memory.…”

mentioning

confidence: 99%

“…Thus, we receive the additional pairs (3,2), (3,3), (3,4), (3,5), (4,2), (4, 3), (4,5), (5,2), (5,3), (5,4), (6,2), (6,3) for which it holds that…”

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confidence: 99%

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Bounding memory for Mastermind might not make it harder

Jäger

Peczarski

2015

Theoretical Computer Science

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We investigate a version of the Mastermind game, where the codebreaker may only store a constant number of questions and answers, called Constant-Size Memory Mastermind, which was recently introduced by Doerr and Winzen. We concentrate on the most difficult case, where the codebreaker may store only one question and one answer, called Size-One Memory Mastermind. We consider two variants of the game: the original one, where the answer is coded with white and black pegs, and the simplified one, where only black pegs are used in the answer. We show that for two pegs and an arbitrary number of colors, the number of questions needed by the codebreaker for an optimal strategy in the worst case for these games is equal to the corresponding number of questions in the games without a memory restriction. We show that this also holds for the black-peg variant and three pegs. In other words, for these cases restricting the memory size does not make the game harder for the codebreaker. This is a continuation of a result of Doerr and Winzen, who showed that this holds asymptotically for a fixed number of colors and an arbitrary number of pegs. Furthermore, by computer search we determine additional pairs (p, c), where again the numbers of questions for an optimal strategy in the worst case for Size-One Memory Mastermind and original Mastermind are equal.

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“…It starts with the questions (0, 1), (2, 3), (4,5), i.e., we use questions of the pattern (2i, 2i + 1). At most two of them can receive a non-empty answer.…”

Section: Algorithm For Two Pegsmentioning

confidence: 99%

mentioning

confidence: 96%

mentioning

confidence: 99%

“…Thus, we receive the additional pairs (3,2), (3,3), (3,4), (3,5), (4,2), (4, 3), (4,5), (5,2), (5,3), (5,4), (6,2), (6,3) for which it holds that…”

mentioning

confidence: 99%

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Bounding memory for Mastermind might not make it harder

Jäger

Peczarski

2015

Theoretical Computer Science

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“…We could stop at that and call MasterMind a worthwhile challenge, but it so happens that solving MasterMind can also be used to break ATMs guessing PINs [6] and even for an operation in biotechnology called selective phenotyping [7]. Not only that, but since it is a problem which is NP-Hard [8] the issue of finding bounds to the number of guesses needed is still open [9]. Therefore, there is room for making new venues into finding better or faster solutions to the game of MasterMind such as the heuristic solution we present in this paper.…”

Section: Introduction and State Of The Artmentioning

confidence: 99%

A search for scalable evolutionary solutions to the game of MasterMind

Merelo

Mora

Castillo

et al. 2013

2013 IEEE Congress on Evolutionary Computation

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MasterMind is a puzzle in which a hidden string of symbols must be discovered by producing query strings which are compared with the hidden one; the result of this comparison (in terms of number of correct positions and colors) is fed back to the player that is trying to crack the code (codebreaker). Methods for solving this puzzle are usually compared in terms of the number of query strings (guesses) made and the total time needed to produce those strings. In this paper we focus on the latter by trying to find a combination of parameters that is, first, uniform and independent of the problem size, and second, adequate to find a fast solution that is, at the same time, good enough. The key to this combination of parameters will be the consistent set size, that is, the maximum number of combinations that are sought before being scored and played as a guess. Having found in previous papers that the consistent set size has an influence on speed, we will concentrate on small sizes and test them through two different scoring methods from literature: most parts and entropy to find out the influence of that parameter on the outcome and which method scales better. With this we try to find out which method and size yield the best results an are effectively able to? find solutions for sizes not approached so far in a reasonable time.

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Complexity Theory for Discrete Black-Box Optimization Heuristics

Doerr

2019

Natural Computing Series

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A predominant topic in the theory of evolutionary algorithms and, more generally, theory of randomized black-box optimization techniques is running time analysis. Running time analysis aims at understanding the performance of a given heuristic on a given problem by bounding the number of function evaluations that are needed by the heuristic to identify a solution of a desired quality. As in general algorithms theory, this running time perspective is most useful when it is complemented by a meaningful complexity theory that studies the limits of algorithmic solutions.In the context of discrete black-box optimization, several black-box complexity models have been developed to analyze the best possible performance that a black-box optimization algorithm can achieve on a given problem. The models differ in the classes of algorithms to which these lower bounds apply. This way, black-box complexity contributes to a better understanding of how certain algorithmic choices (such as the amount of memory used by a heuristic, its selective pressure, or properties of the strategies that it uses to create new solution candidates) influences performance.In this chapter we review the different black-box complexity models that have been proposed in the literature, survey the bounds that have been obtained for these models, and discuss how the interplay of running time analysis and black-box complexity can inspire new algorithmic solutions to well-researched problems in evolutionary computation. We also discuss in this chapter several interesting open questions for future work.

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Playing Mastermind with Many Colors

Cited by 23 publications

References 6 publications

Bounding memory for Mastermind might not make it harder

Bounding memory for Mastermind might not make it harder

A search for scalable evolutionary solutions to the game of MasterMind

Complexity Theory for Discrete Black-Box Optimization Heuristics

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