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A Non-Terminating Game of Beggar-My-Neighbor

Brayden Casella, Philip M. Anderson, Michael Kleber, Richard P. Mann, Reed Nessler, William Rucklidge, Samuel G. Williams, Nicolas Wu

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.'

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.'
Paper Structure (12 sections, 5 equations, 3 figures)

This paper contains 12 sections, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. History of the search
  3. Constructing a non-terminating game
  4. The first non-terminating game and its family
  5. Conclusion
  6. Appendix A: Historical record of longest known terminating games
  7. Appendix B: Example of games which lead to the same state after one trick
  8. Appendix C: Found pieces and non-terminating standard games from them
  9. Appendix D: A trace of the non-terminating cycle

Figures (3)

  • Figure 1: Frequency distribution of game lengths from $10^7$ randomly-dealt games, either with uniformly random deals (blue points) or restricted to deals in which both players have an equal share of the face cards (red points.) Apart from very short games, the frequencies in all cases closely resembles an exponential distribution (note the log-scale on the y-axis). Although dealing an equal share of face cards marginally increases expected game length, it does not change the decay rate of the exponential tail of the distribution.
  • Figure 2: Historical records since 1992 for the longest known terminating game of bmn in terms of cards played (A) and tricks (B). A full list of historical records known to the authors is presented in \ref{['Appendix A']}.
  • Figure 3: Illustration of a family of non-terminating games, all converging to the same 62-trick cycle. Starting packs shown in dogeared boxes are those with the topmost player starting; those in plain boxes start with the lower player.