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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

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

David Molnar

Princeton University Press

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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