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The Exact Query Complexity of Yes-No Permutation Mastermind

Author

Listed:
  • Mourad El Ouali

    (Polydisciplinary Faculty Ouarzazate, University Ibn Zohr, Agadir 80000, Morocco)

  • Volkmar Sauerland

    (Department of Mathematics, Kiel University, 24118 Kiel, Germany)

Abstract

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.

Suggested Citation

  • Mourad El Ouali & Volkmar Sauerland, 2020. "The Exact Query Complexity of Yes-No Permutation Mastermind," Games, MDPI, vol. 11(2), pages 1-11, April.
    • Handle: RePEc:gam:jgames:v:11:y:2020:i:2:p:19-:d:344780
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    References listed on IDEAS

    as
    1. Mourad El Ouali & Christian Glazik & Volkmar Sauerland & Anand Srivastav, 2018. "On the Query Complexity of Black-Peg AB-Mastermind," Games, MDPI, vol. 9(1), pages 1-12, January.
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