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# The Exact Query Complexity of Yes-No Permutation Mastermind

by 1,* and
1
Polydisciplinary Faculty Ouarzazate, University Ibn Zohr, Agadir 80000, Morocco
2
Department of Mathematics, Kiel University, 24118 Kiel, Germany
*
Author to whom correspondence should be addressed.
Games 2020, 11(2), 19; https://doi.org/10.3390/g11020019
Received: 14 February 2020 / Revised: 30 March 2020 / Accepted: 4 April 2020 / Published: 13 April 2020
Mastermind is famous two-player game. The first player (codemaker) chooses a secret code which the second player (codebreaker) is supposed to crack within a minimum number of code guesses (queries). Therefore, the codemaker’s duty is to help the codebreaker by providing a well-defined error measure between the secret code and the guessed code after each query. We consider a variant, called Yes-No AB-Mastermind, where both secret code and queries must be repetition-free and the provided information by the codemaker only indicates if a query contains any correct position at all. For this Mastermind version with n positions and $k ≥ n$ colors and $ℓ : = k + 1 − n$ , we prove a lower bound of $∑ j = ℓ k log 2 j$ and an upper bound of $n log 2 n + k$ on the number of queries necessary to break the secret code. For the important case $k = n$ , where both secret code and queries represent permutations, our results imply an exact asymptotic complexity of $Θ ( n log n )$ queries. View Full-Text
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• Externally hosted supplementary file 1
Doi: 10.5281/zenodo.3723659
Description: GitHub repositry "yn-ab-mastermindv1.0": Codebreaker strategies for Yes-No AB-Mastermind (Matlab/Octave)
MDPI and ACS Style

El Ouali, M.; Sauerland, V. The Exact Query Complexity of Yes-No Permutation Mastermind. Games 2020, 11, 19. https://doi.org/10.3390/g11020019

AMA Style

El Ouali M, Sauerland V. The Exact Query Complexity of Yes-No Permutation Mastermind. Games. 2020; 11(2):19. https://doi.org/10.3390/g11020019

Chicago/Turabian Style

El Ouali, Mourad, and Volkmar Sauerland. 2020. "The Exact Query Complexity of Yes-No Permutation Mastermind" Games 11, no. 2: 19. https://doi.org/10.3390/g11020019

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