A Non-Terminating Game of Beggar-My-Neighbor

Abstract

We demonstrate the existence of a non-terminating game of Beggar-My-Neighbor discovered by lead author Brayden Casella. We detail the method for constructing this game and identify a cyclical structure of 62 tricks that is reached by 30 distinct starting hands. We further present a short history of the search for this solution since the problem was posed, and a record of previously found longest terminating games. The existence of this non-terminating game provides a solution to a long-standing question which John H. Conway called an “anti-Hilbert problem.”

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Item Type:	Article
Authors/Creators:	Casella, B. Anderson, P.M. Kleber, M. Mann, R.P. https://orcid.org/0000-0003-0701-1274 Nessler, R. Rucklidge, W. Williams, S.G. Wu, N.
Copyright, Publisher and Additional Information:	© 2025 The Author(s). Published with license by Taylor & Francis Group, LLC. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent.
Dates:	Accepted: 25 November 2024 Published (online): 20 October 2025 Published: 26 November 2025
Institution:	The University of Leeds
Academic Units:	The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds)
Date Deposited:	26 Nov 2024 11:00
Last Modified:	15 Apr 2026 12:26
Status:	Published
Publisher:	Taylor & Francis
Identification Number:	10.1080/00029890.2025.2558484
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:220112

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A Non-Terminating Game of Beggar-My-Neighbor

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