Patience Sorting - The Card Game

The Card Game

The game begins with a shuffled deck of cards, labeled .

The cards are dealt one by one into a sequence of piles on the table, according to the following rules.

1. Initially, there are no piles. The first card dealt forms a new pile consisting of the single card.
2. Each new card may be placed either on an existing pile whose top card has a value higher than the new card's value, thus increasing the number of cards in that pile, or to the right of all of the existing piles, thus forming a new pile.
3. When there are no more cards remaining to deal, the game ends.

The object of the game is to finish with as few piles as possible. D. Aldous and P. Diaconis suggest defining 9 or fewer piles as a winning outcome for, which has approximately 5% chance to happen.