Euclid, Calkin \& Wilf --- playing with rationals
type: Item
instance of: scholarly article
title: Euclid, Calkin \& Wilf --- playing with rationals
zbMATH Open document ID: 1193.91033
DOI: 10.4171/EM/95
author: Stefanie Hofmann
author: Gertraud Schuster
author: Jörn Steuding
published in: Elemente der Mathematik
publication date: 2008-09-08T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: Summary (translated from the German): The combinatorial game Euclid was introduced by \textit{A. J. Cole} and \textit{A. J. T. Davie} [Math. Gaz. 53, 354--357 (1969; Zbl 0186.25303)]. Two players take turns drawing from a position \((a, b)\in\mathbb N^2\), where the drawing player has to subtract a multiple of the smaller number from the larger number. The player making the move has to subtract a multiple of the smaller number from the larger number, so that the result is still in \(\mathbb N^2\). The first player who can no longer move has lost. Up to now, we know elementary (Cole \& Davie), geometrical (Lengyel) and number (Lengyel) and number-theoretic (Schwartz) descriptions of the winning positions for Euclid (optimal play of both players is assumed). The authors add a graph-theoretic winning strategy by relating Euclid to the Calkin-Wilf. This remarkable tree, introduced by \textit{N. J. Calkin} and {it H. S. Wilf} [Am. Math. Mon. 107, No. 4, 360--363 (2000; Zbl 0983.11009)], generates all positive rational numbers in abbreviated form by a simple rule exactly once.
Mathematics Subject Classification ID: 91A46
Mathematics Subject Classification ID: 91A43
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 05C05
zbMATH DE Number: 5322346
zbMATH Keywords: Calkin Wilf tree
zbMATH Keywords: game Euclid
zbMATH Keywords: winning strategy
zbMATH Keywords: continued fractions
zbMATH Keywords: playing Euclid on the Calkin Wilf tree
describes a project that uses: OEIS
MaRDI profile type: Publication
cites work: FUNCTIONAL PEARL: Enumerating the rationals
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description: scientific article; zbMATH DE number 5322346