Finiteness of the playing time in 'Beggar-my-neighbour' card game

TL;DR

This paper proves that in certain card games like 'Beggar-my-neighbour', the expected duration of a game is finite under specific random conditions, extending the result to a broad class of similar games.

Contribution

It establishes the finiteness of the expected playing time for a general class of 'Beggar-my-neighbour'-type card games under randomized starting conditions.

Findings

Expected game duration is finite under specified conditions.

The result applies to a broad class of similar card games.

The proof relies on probabilistic and combinatorial analysis.

Abstract

It is proved that in card games similar to 'Beggar-my-neighbour' the mathematical expectation of the playing time is finite, provided that the player who starts the round is determined randomly and the deck is shuffled when the trick is added. The result holds for the generic setting of the game.