cheers, Chelyabinsk.

Lieven le Bruyn is discussing Surreal numbers and chess from a a combinatorial game theory perspective but with a reference to Conway ONAG.

In this setting there would seem to be common ground between chess and cards?

The solitaire version when the player represents 7-10 different players and see what happens is quite exciting as well. There is a Wikipedia article describing the game and a reference to a site with many variations.

Probably, as simple the game is, all the natural probabilistic question you can ask about this game are rather hopeless. Amir Dembo and I considered once (in 1990 or so) a simplified version where you have a pack of n cards with distinct values. In this version there are no “wars” and while the game is still exciting the outcome is clear: the player with the highest card wins. I think we were able to show (for the version where the new piles are shuffled, (or perhaps even a further simplification where each time you pick a random card),) that the number of moves until victory
is in the order of .

Another nice variation of war is for three (or odd number of) players. In each round the middle card player wins and take the other cards. After each player played all his card (just once) the player who collected the medium amount of cards wins.

While I’m at it, here’s another, perhaps better, idea for a probabilistic model of the simplified version of the game. Instead of randomly inserting or removing cards, you randomly change the value of a random card (to jack if it’s a non-jack, and to non-jack if it’s a jack), according to some simple rule, such as each go making a change with probability p, with all decisions independent. The hope would be that for some fairly small p (if that last rule was adopted) the game could be shown to terminate in reasonable time with high probability.

And here are a few more thoughts on the problem (added a bit later). The first is that there is a drastically simplified version of the problem that might actually give decent predictions: it would be interesting to find out. It can be defined for a standard pack but for simplicity let me just define it for the black/red = jack/non-jack game. You just randomize completely. In other words, each time you turn a card over, it’s a jack with probability 1/2 and a non-jack with probability 1/2. (One could actually play this game by tossing coins and never looking at a card.) I haven’t done the exercise, but this should give a pretty simple random walk on a very simple configuration space (the number of cards in the hand of one of the two players and whose turn it is). It would be interesting to know how the length of the game is distributed in this case, and also in the case where each card that appears is a jack with probability 1/13, a queen with probability 1/13, a king with probability 1/13, an ace with probability 1/13, and chicken-feed with probability 9/13.

The second thought (which is bound to be standard) is that the game is reversible, in the sense that if you see the order of the cards at the end of the game, then you can work out how the entire game went up to that point. (The one thing you can’t tell is when the game started, but you sometimes reach a point where you cannot continue backwards, so you know that it started after that point.) It’s an easy exercise to check these facts.

The third thought, which I’m surprised I didn’t have earlier, is that if one wants to analyse the deterministic version of the game, then dynamical systems could possibly help, since it is full of concepts that do justice to our intutions that deterministic processes can “behave randomly”. However, the fact that games terminate presents an initial technical problem that I don’t immediately see how to solve. (Perhaps one could let the number of cards tend to infinity and use the fact that the proportion of configurations that are terminal tends to zero, and thereby obtain a game that one could think of as Beggar-My-Neighbour as played in Asymptopia, or ultra-Beggar-My-Neighbour, or something.)

As for the main point, you say that there’s a cycle for a certain 50-card deck? Can’t this be bootstrapped into a cycle for the full deck? Or did you mean that there’s a known cycle for a certain 24-card deck?

Michael Kleber’s site ( http://people.brandeis.edu/~kleber/ ) lists a new record (as of July 2007) of 975 moves, by Richard Mann and Nick Wu at Oxford.