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Optimal strategies for the static black-peg AB game with two and three pegs
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Computing Science. (Foundations of Language Processing)ORCID iD: 0000-0001-7349-7693
2023 (English)In: Discrete Mathematics, Algorithms and Applications (DMAA), ISSN 1793-8309, E-ISSN 1793-8317Article in journal (Refereed) Epub ahead of print
##### Abstract [en]

The AB Game is a game similar to the popular game Mastermind. We study a version of this game called Static Black-Peg AB Game. It is played by two players, the codemaker and the codebreaker. The codemaker creates a so-called secret by placing a color from a set of c colors on each of p ≤ c pegs, subject to the condition that every color is used at most once. The codebreaker tries to determine the secret by asking questions, where all questions are given at once and each question is a possible secret. As an answer the codemaker reveals the number of correctly placed colors for each of the questions. After that, the codebreaker only has one more try to determine the secret and thus to win the game.

For given p and c, our goal is to find the smallest number k of questions the codebreaker needs to win, regardless of the secret, and the corresponding list of questions, called a (k + 1)-strategy. We present a (⌈4c/3⌉ − 1)-strategy for p = 2 for all c ≥ 2, and a ⌊(3c − 1)/2⌋-strategy for p = 3 for all c ≥ 4 and show the optimality of both strategies, i.e., we prove that no (k + 1)-strategy for a smaller k exists.

##### Place, publisher, year, edition, pages
World Scientific, 2023.
##### Keywords [en]
Game theory, mastermind, AB game, optimal strategy
##### National Category
Discrete Mathematics
Mathematics
##### Identifiers
ISI: 001034748600002Scopus ID: 2-s2.0-85165934499OAI: oai:DiVA.org:umu-210346DiVA, id: diva2:1771561
##### Funder
The Kempe Foundations, JCK-2022.1Available from: 2023-06-20 Created: 2023-06-20 Last updated: 2023-09-08

#### Open Access in DiVA

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Publisher's full textScopus

#### Authority records

Jäger, GeroldDrewes, Frank

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Jäger, GeroldDrewes, Frank
##### By organisation
Department of Mathematics and Mathematical StatisticsDepartment of Computing Science
##### In the same journal
Discrete Mathematics, Algorithms and Applications (DMAA)
##### On the subject
Discrete Mathematics

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Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
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