SAGT 2018: Algorithmic Game Theory pp 261-266

# An Optimal Strategy for Static Black-Peg Mastermind with Three Pegs

• Gerold Jäger
• Frank Drewes
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11059)

## 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 $$\lfloor 3c/2 \rfloor +1$$ questions.

A reformulation of our result is that the metric dimension of $$\mathbb {Z}_n \times \mathbb {Z}_n \times \mathbb {Z}_n$$ is equal to $$\lfloor 3n/2 \rfloor$$.

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