Searching for knights and spies: a majority/minority game

TL;DR

This paper investigates the problem of identifying knights and spies in a group using the minimum number of questions, considering different spy behaviors, and provides optimal strategies and computational results.

Contribution

It introduces optimal questioning strategies for identifying spies and knights under various conditions, including when spies always lie or answer freely.

Findings

Optimal question counts for identifying spies and knights.

Strategies that solve the problem with minimal questions.

Computational results for identifying all identities.

Abstract

There are n people, each of whom is either a knight or a spy. It is known that at least k knights are present, where n/2 < k < n. Knights always tell the truth. We consider both spies who always lie and spies who answer as they see fit. This paper determines the number of questions required to find a spy or prove that everyone in the room is a knight. We also determine the minimum number of questions needed to find at least one person's identity, or a nominated person's identity, or to find a spy (under the assumption that a spy is present). For spies who always lie, we prove that these searching problems, and the problem of finding a knight, can be solved by a simultaneous optimal strategy. We also give some computational results on the problem of finding all identities when spies always lie, and end by stating some open problems.