An optimal strategy for static black-peg mastermind with three pegs

Abstract

Mastermind is a famous game played by a codebreaker against a codemaker. We investigate its static (also called non-adaptive) black-peg variant. Given c colors and p pegs, the codemaker has to choose a secret, a p-tuple of c colors, and the codebreaker asks a set of questions all at once. Like the secret, a question is a p-tuple of c colors. The codemaker then tells the codebreaker how many pegs in each question are correct in position and color. Then the codebreaker has one final question to find the secret. His aim is to use as few of questions as possible. Our main result is an optimal strategy for the codebreaker for p=3 pegs and an arbitrary number c of colors using [3c/2] +1 questions. A reformulation of our result is that the metric dimension of Zn × Zn × Zn is equal to [3n/2].