On the Query Complexity of Black-Peg AB-Mastermind

TL;DR

This paper investigates the query complexity of the Black-Peg Mastermind game with no repeated colors, providing improved bounds on the number of guesses needed to identify the secret code.

Contribution

It presents new upper and lower bounds on the number of guesses required, improving previous results and extending understanding of the game's complexity.

Findings

Upper bound of (n-3)*log(n)+2.5n guesses for k=n

Upper bound of (n-1)*log(n)+k+1 guesses for k>n

Lower bound of n guesses for k=n

Abstract

Mastermind game is a two players zero sum game of imperfect information. The first player, called codemaker, chooses a secret code and the second player, called codebreaker, tries to break the secret code by making as few guesses as possible, exploiting information that is given by the codemaker after each guess. In this paper, we consider the so called Black-Peg variant of Mastermind, where the only information concerning a guess is the number of positions in which the guess coincides with the secret code. More precisely, we deal with a special version of the Black-Peg game with n holes and k<=n colors where no repetition of colors is allowed. We present upper and lower bounds on the number of guesses necessary to break the secret code. We first come back to the upper bound results introduced by El Ouali and Sauerland (2013). For the case k=n the secret code can be algorithmically identified within less than (n-3)*ld(n)+5n/2 queries. That result improves the result of Ker-I Ko and Shia-Chung Teng (1985) by almost a factor of 2. For the case k>n we prove an upper bound for the problem of (n-1)*ld(n)+k+1. Furthermore we prove a new lower bound for (a generalization of) the case k=n that improves the recent result of Berger et al. (2016) from n-log(log(n)) to n. We also give a lower bound of k queries for the case k>n.