# Yin Yang Solitaire

Here is John Finn's description of Peter's game of Yin Yang solitaire.

We start with a brand new deck of cards, which in America are ordered so that if we put the deck face-down on the table, we have

Ace through King of Hearts,
Ace through King of Clubs,
King through Ace of Diamonds,
King through Ace of Spades.

Hearts and Clubs are thus the High suits, and Diamonds and Spades the Low. (Some would term these Yin and Yang, but not according to any scheme that I believe would satisfy Georges Osawa, who says that tomatoes and eggplants are both extremely yin because of their purple color.)

We shuffle the deck of cards 7 times, then cut it, and then start removing and revealing each card from the top of the deck, making a new pile of them face-up (so if this were all we did, we'd just have the deck unchanged after going through it once, except that the deck would be lying face-up on the table).

We start the pile for each suit when we discover its ace, and add cards of the same suit to each of these 4 piles, according to the rule that we must add the cards of each suit in order.

Thus a single pass through the deck is not going to accomplish much in the way of completing the 4 piles, so having made this pass, we turn the remaining deck back over, and make another pass.

We continue this until we complete either the two high piles (hearts & clubs), or the two low piles (diamonds & spades). If the high piles get completed first, we call the game a win; it's a loss if the low piles get completed first.

If the deck has been thoroughly permuted (by having put the cards through a clothes dryer, say), then the lows and highs will be equally likely to be first to get completed. Thus our expected proportion of wins will be 1/2.

But it turns out that after 7 shuffles and a cut, we are significantly more likely to complete the highs before the lows, so our proportion of wins will be greater than 1/2.