## Return to Answer

3 updated reference and description of results

My paper "Cycles in War" addresses this question, too. I was interested in characterizing the kinds of cycles that can occur. In other words, what does the structure of a cycle in War actually look like? I simplified the problem by assuming that wars are not possible (i.e., the cards have a strict ranking from 1 to n, where n is the number of cards in the deck). Even in this simpler version I found it difficult to characterize all of the cycles. However, I did find some interesting cycle structures. For example, in the following deal of a 52-card deck cycles, assuming case that the winning card goes to the bottom of the winning player's deck before the losing card, I was able to find a way to construct a deal of an $n$-card deck that cycles, for any $n$ that is not a power of 2 or three times a power of 2.

For example, the following deal of a 52-card deck cycles.


26 46  1  7  8 27  9 28 29 47  2 10 11 30 12 31 32 48  3 13 14 33 15 34 35 49

16 36 17 37 38 50  4 18 19 39 20 40 41 51  5 21 22 42 23 43 44 52  6 24 25 45


It takes over 30,000 battles for the deck to return to this ordering. The mathematical argument for why this deal cycles is in the paper, which has been accepted for publication by the journal Integers but has not appeared in print yet. Among other things, the re-loading rules do make a difference, as in this example and as other people have already noted here. Also, given that characterizing cycle structures when wars are not possible turns out to be difficult (or, at least, I found it so), one should expect that characterizing cycles in the standard version of the game in which wars are possible would be even more difficult.

Edit: The paper has now been published on the Integers web site, in the games section, as Vol. 10, Article G2, 2010, pp. 747-764.

2 added 416 characters in body; deleted 1 characters in body; added 68 characters in body; added 28 characters in body; deleted 1 characters in body; deleted 67 characters in body

My paper "Cycles in War" addresses this question, too. I was interested in characterizing the kinds of cycles that can occur. In other words, what does the structure of a cycle in War actually look like? I simplified the problem by assuming that wars are not possible (i.e., the cards have a strict ranking from 1 to n, where n is the number of cards in the deck). Even in this simpler version I found it difficult to characterize all of the cycles. However, I did find some interesting cycle structures. For example, including one the following deal of a 52-card deck cycles, assuming that works the winning card goes to the bottom of the winning player's deck before the losing card.


26 46  1  7  8 27  9 28 29 47  2 10 11 30 12 31 32 48  3 13 14 33 15 34 35 49

16 36 17 37 38 50  4 18 19 39 20 40 41 51  5 21 22 42 23 43 44 52  6 24 25 45


It takes over 30,000 battles for 52-card decksthe deck to return to this ordering. The details are mathematical argument for why this deal cycles is in the paper, which has been accepted for publication by the journal Integers but has not appeared in print yet. Among other things, the re-loading rules do make a difference, as several in this example and as other people have already noted here. Also, given that characterizing cycle structures when wars are not possible turns out to be difficult (or, at least, I found it so), one should expect that characterizing cycles in the standard version of the game in which wars are possible would be even more difficult.

1

My paper "Cycles in War" addresses this question, too. I was interested in characterizing the kinds of cycles that can occur. In other words, what does the structure of a cycle in War actually look like? I simplified the problem by assuming that wars are not possible (i.e., the cards have a strict ranking from 1 to n, where n is the number of cards in the deck). Even in this simpler version I found it difficult to characterize all of the cycles. However, I did find some interesting cycle structures, including one that works for 52-card decks. The details are in the paper, which has been accepted for publication by the journal Integers but has not appeared in print yet. Among other things, the re-loading rules do make a difference, as several people have already noted here. Also, given that characterizing cycle structures when wars are not possible turns out to be difficult (or, at least, I found it so), one should expect that characterizing cycles in the standard version of the game in which wars are possible would be even more difficult.