#### Vol. 15, No. 1, 2022

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Continuous guessing games with two secret numbers

### David Clark and Nicholas Layman

Vol. 15 (2022), No. 1, 141–162
##### 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
Primary: 68P10
Secondary: 94D99