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Optimal play in 'Guess Who?'

David Cushing, Stuart Gipp, Ezra Levick, Em Rickinson, David I. Stewart

TL;DR

An optimal strategy for the children's game Guess Who?

Abstract

We prove an optimal strategy for the children's game Guess Who? assuming the official rules are in use and that both players ask `classical' questions with a bipartite response. Applying a technique described in [Rabern, B \& Rabern, L 2008, 'A simple solution to the hardest logic puzzle ever', \textit{Analysis}, vol. 68, no. 2, pp.~105-112.] allows for questions with tripartite responses; we explain this innovation and give an optimal strategy for two players applying it.

Optimal play in 'Guess Who?'

TL;DR

An optimal strategy for the children's game Guess Who?

Abstract

We prove an optimal strategy for the children's game Guess Who? assuming the official rules are in use and that both players ask `classical' questions with a bipartite response. Applying a technique described in [Rabern, B \& Rabern, L 2008, 'A simple solution to the hardest logic puzzle ever', \textit{Analysis}, vol. 68, no. 2, pp.~105-112.] allows for questions with tripartite responses; we explain this innovation and give an optimal strategy for two players applying it.

Paper Structure

This paper contains 4 sections, 21 theorems, 66 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The official rules with classical questions
  3. With tripartite questions
  4. Proof of the optimal strategy for tripartite Guess Who?

Key Result

theorem 1

Define where $n_<$ is defined to be $1$ if $n=2$ and is $\left\lfloor\frac{n}{4}\right\rfloor+\left\lfloor\frac{n+1}{4}\right\rfloor$ for $n\geq 3$. Then $\mathcal{S}(n,m)$ is an optimal decision at board state $(n,m)$.

Theorems & Definitions (51)

  • theorem 1
  • lemma 1
  • proof
  • lemma 2
  • lemma 3
  • proof : Proof of \ref{['mainthm']}
  • lemma 4
  • proof
  • theorem 2
  • proof : Sketch proof.
  • ...and 41 more