Algorithm to Prove Formulas for the Expected Number of Questions in Mastermind Games

TL;DR

This paper presents a new, generalized algorithmic model to prove formulas for the expected number of questions needed in Mastermind and AB games, extending previous proofs limited to two pegs.

Contribution

The authors introduce a versatile model and algorithm that automate proofs for the expected number of questions in Mastermind with any number of pegs, surpassing prior two-peg limitations.

Findings

Successfully closed the proof gap for two-peg games.

Developed a parametrized model for any number of pegs.

Automated proof process for expected questions in Mastermind.

Abstract

We close the gap in the proof (published by Chen and Lin) of formulas for the minimum number of questions required in the expected case for Mastermind and its variant called AB game, where both games are played with two pegs and $n$ colors. For this purpose, we introduce a new model to represent the game guessing process and we develop an algorithm with automatizes the proof. In contrary to the model used by Chen and Lin, called graph-partition approach, which is limited to two pegs, our model and algorithm are parametrized with the number of pegs and they could potentially be used for any number of pegs.