Optimal strategies for the static black-peg AB game with two and three pegs

Abstract

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 [Formula: see text] colors on each of [Formula: see text] 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 [Formula: see text] and [Formula: see text], our goal is to find the smallest number [Formula: see text] of questions the codebreaker needs to win, regardless of the secret, and the corresponding list of questions, called a [Formula: see text]-strategy. We present a [Formula: see text]-strategy for [Formula: see text] for all [Formula: see text], and a [Formula: see text]-strategy for [Formula: see text] for all [Formula: see text] and show the optimality of both strategies, i.e., we prove that no [Formula: see text]-strategy for a smaller [Formula: see text] exists.