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The Mathematics of Various Entertaining SubjectsResearch in Recreational Math$
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Jennifer Beineke and Jason Rosenhouse

Print publication date: 2015

Print ISBN-13: 9780691164038

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691164038.001.0001

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Connection Games and Sperner’s Lemma

Connection Games and Sperner’s Lemma

Chapter:
(p.213) 15 Connection Games and Sperner’s Lemma
Source:
The Mathematics of Various Entertaining Subjects
Author(s):

David Molnar

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691164038.003.0015

This chapter provides an introduction to connection games in general. It also recounts how Sperner's Lemma, a result about labeling a triangulation of a simplex, can be used to prove that someone must win at Hex—the best-known connection game—as well as The Game of Y®, (or simply, Y) another well-known connection game. Moreover, the chapter proves a generalization of Sperner's Lemma and uses it to show that there is always a winner in the many variations of Atoll and Begird, two connection games which can be played on a variety of boards and include Hex and Y, respectively, as special cases. These “must-win” results have significant strategic implications—if one prevents the opponent from making the desired connection, one would be able to make this connection by necessity.

Keywords:   connection games, Sperner's Lemma, simplex, Hex, The Game of Y, Atoll, Begird

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