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An Optimal Strategy for Static Black-Peg Mastermind with Two Pegs

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Book cover Combinatorial Optimization and Applications (COCOA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10043))

Abstract

Mastermind is a famous two-player game which has attracted much attention in literature within the last years. In this work we investigate the static (also called non-adaptive) variant of Mastermind. The principal rule is that the codemaker has to choose a secret consisting of p pegs and c colors for each peg and the codebreaker may give a number of guesses at once, where for each guess he receives information from the codemaker. Using this information he has a final guess for the correct secret. The aim of the game is to minimize the number of guesses. Whereas Goddard has investigated the static version of original Mastermind in 2003, we do such an investigation of its black-peg variant, where the received information consists only of a number of black pegs which corresponds to the number of pegs matching in the corresponding question and the secret. As main result we present a strategy for this game for \(p=2\) pegs and arbitrarily many colors \( c\ge 3 \) colors and prove its feasibility and optimality. Furthermore, by computer search we found optimal strategies for 9 other pairs (p, c).

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Notes

  1. 1.

    Note that the parameter c is reserved for the number of colors. So we use the parameters \(a,b,d,e,\dots \) here.

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Authors and Affiliations

  1. Department of Mathematics and Mathematical Statistics, University of Umeå, 90187, Umeå, Sweden

    Gerold Jäger

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  1. Gerold Jäger
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Correspondence to Gerold Jäger .

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Editors and Affiliations

  1. University of Hong Kong, Hong Kong, China

    T-H. Hubert Chan

  2. City University of Hong Kong, Hong Kong, China

    Minming Li

  3. City University of Hong Kong, Hong Kong, China

    Lusheng Wang

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Jäger, G. (2016). An Optimal Strategy for Static Black-Peg Mastermind with Two Pegs. In: Chan, TH., Li, M., Wang, L. (eds) Combinatorial Optimization and Applications. COCOA 2016. Lecture Notes in Computer Science(), vol 10043. Springer, Cham. https://doi.org/10.1007/978-3-319-48749-6_48

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  • DOI: https://doi.org/10.1007/978-3-319-48749-6_48

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-48748-9

  • Online ISBN: 978-3-319-48749-6

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