Vol. 7, No. 5, 2014

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ISSN: 1944-4184 (e-only)
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The sock matching problem

Sarah Gilliand, Charles Johnson, Sam Rush and Deborah Wood

Vol. 7 (2014), No. 5, 691–697
Abstract

When matching socks after doing the laundry, how many unmatched socks can appear in the process of drawing one sock at a time from the basket? By connecting the problem of sock matching to the Catalan numbers, we give the probability that k unmatched socks appear. We also show that, for each fixed k, this probability approaches 1 as the number of socks becomes large enough. The relation between the number of socks and the k for which a given probability is first reached is also discussed, but a complete answer is open.

Keywords
Catalan numbers, sock matching, Dyck paths
Mathematical Subject Classification 2010
Primary: 05A15, 05A16
Secondary: 03B48, 00A69
Milestones
Received: 21 August 2013
Accepted: 25 November 2013
Published: 1 August 2014

Communicated by Jim Haglund
Authors
Sarah Gilliand
Department of Biology
The College of William & Mary
College Station Unit 3011
P.O. Box 8793
Williamsburg, VA 23187
United States
Charles Johnson
Department of Mathematics
The College of William & Mary
P.O. Box 8795
Williamsburg, VA 23187
United States
Sam Rush
Department of Computer Science
California Institute of Technology
1200 East California Boulevard, MS 305-16
Pasadena, CA 91125
United States
Deborah Wood
Department of Mathematics
The College of William & Mary
College Station Unit 4085
P.O. Box 8793
Williamsburg, VA 23187
United States