Continuous guessing games with two secret numbers

Clark, David; Layman, Nicholas

doi:10.2140/involve.2022.15.141

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Continuous guessing games with two secret numbers

David Clark and Nicholas Layman

Vol. 15 (2022), No. 1, 141–162

DOI: 10.2140/involve.2022.15.141

Abstract

A guessing game with two secret numbers is a game played between a questioner and a responder. The two players first agree upon the set, $N$ , in which the game will be played, as well as the number of questions, $Q$ , which will be asked by the questioner. The responder first chooses two distinct numbers from $N$ . The questioner then asks questions of the form “How many of your chosen numbers are in the set $A \subseteq N$ ?” to which the responder answers truthfully. The goal for the questioner is to determine the responder’s two numbers using at most $Q$ questions. We study a continuous version of this game where $N$ is the closed interval of real numbers from 0 to 1. We introduce tools to study this game and use them to examine strategies for the questioner using a geometric approach. We establish a condition that must be satisfied by optimal strategies and give a strategy that can be made arbitrarily close to optimal.

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Keywords

guessing games, search, combinatorics

Mathematical Subject Classification

Primary: 68P10

Secondary: 94D99

Milestones

Received: 5 April 2021

Revised: 20 August 2021

Accepted: 21 August 2021

Published: 14 March 2022

Communicated by Kenneth S. Berenhaut

Authors

David Clark
Department of Mathematics Grand Valley State University Allendale, MI United States

Nicholas Layman
Department of Mathematics Grand Valley State University Allendale, MI United States

Vol. 15, No. 1, 2022