Driving the Old Woman to Bed

In the Preface to the Dover reprint of The Mathematics of Games, I said, quoting the second edition of Winning Ways for your Mathematical Plays, that a non-terminating position in the game of "Driving the Old Woman to Bed" (also known as Beggar My Neighbour and Strip Jack Naked) had been found by Marc Paulhus. I have now been told that this was due to a misunderstanding, and that no such position had been found by him. The question was therefore still open.

However, such a position has now been found by Brayden Casella, and he has kindly given us permission to quote it here. It is

     Player 1: ---K---Q-KQAJ-----AAJ--J--
     Player 2: ----------Q----KQ-J-----KA

which eventually condenses into a 62-position loop.

For a demonstration that a loop does eventually emerge, click here. It will be noticed that no fewer than twelve times during the cycle, one side is reduced to a single jack and one other card.