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Bounds for Static Black-Peg AB Mastermind

  • Christian Glazik
  • Gerold JägerEmail author
  • Jan Schiemann
  • Anand Srivastav
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10628)

Abstract

Mastermind is a famous two-player game introduced by M. Meirowitz (1970). Its combinatorics has gained increased interest over the last years for different variants.

In this paper we consider a version known as the Black-Peg AB Game, where one player creates a secret code consisting of c colors on \(p \le c\) pegs, where each color is used at most once. The second player tries to guess the secret code with as few questions as possible. For each question he receives the number of correctly placed colors. In the static variant the second player doesn’t receive the answers one at a time, but all at once after asking the last question. There are several results both for the AB and the static version, but the combination of both versions has not been considered so far. The most prominent case is \(n:=p=c\), where the secret code and all questions are permutations. The main result of this paper is an upper bound of \(\mathcal {O}(n^{1.525})\) questions for this setting. With a slight modification of the arguments of Doerr et al. (2016) we also give a lower bound of \(\varOmega (n\log n)\). Furthermore, we complement the upper bound for \(p=c\) by an optimal \((\lceil 4c/3 \rceil -1)\)-strategy for the special case \(p=2\) and arbitrary \(c\ge 2\) and list optimal strategies for six additional pairs (pc) .

Notes

Acknowledgments

The second author’s research was supported by the Kempe Foundation Grant No. SMK-1354 (Sweden).

Furthermore, we would like to thank the anonymous referees for their valuable comments which significantly helped to improve the paper.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Christian Glazik
    • 1
  • Gerold Jäger
    • 2
    Email author
  • Jan Schiemann
    • 1
  • Anand Srivastav
    • 1
  1. 1.Department of Computer ScienceKiel UniversityKielGermany
  2. 2.Department of Mathematics and Mathematical StatisticsUniversity of UmeåUmeåSweden

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