The Worst Case Number of Questions in Generalized AB Game with and without White-peg Answers

TL;DR

This paper investigates the maximum number of questions needed to solve generalized AB and Black-peg AB games with distinct-color pegs, providing exact values and bounds for small parameters using computational and theoretical methods.

Contribution

It extends previous work by establishing tight bounds and exact values for the worst-case questions in generalized AB games with and without white-peg answers for small code lengths.

Findings

Confirmed exact values of (2,c) and (3,c).

Proved tight bounds for (4,c).

Provided exact values and bounds for (p,c) and (b,p,c).

Abstract

The AB game is a two-player game, where the codemaker has to choose a secret code and the codebreaker has to guess it in as few questions as possible. It is a variant of the famous Mastermind game, with the only difference that all pegs in both, the secret and the questions must have distinct colors. In this work, we consider the Generalized AB game, where for given arbitrary numbers $p$, $c$ with $p \le c$ the secret code consists of $p$ pegs each having one of $c$ colors and the answer consists only of a number of black and white pegs. There the number of black pegs equals the number of pegs matching in the corresponding question and the secret in position and color, and the number of white pegs equals the additional number of pegs matching in the corresponding question and the secret only in color. We consider also a variant of the Generalized AB game, where the information of white pegs is omitted. This variant is called Generalized Black-peg AB game. Let $\ab(p,c)$ and $\abb(p,c)$ be the worst case number of questions for Generalized AB game and Generalized Black-peg AB game, respectively. Combining a computer program with theoretical considerations, we confirm known exact values of $\ab(2,c)$ and $\ab(3,c)$ and prove tight bounds for $\ab(4,c)$. Furthermore, we present exact values for $\abb(2,c)$ and $\abb(3,c)$ and tight bounds for $\abb(4,c)$.