A bridge catalogue has a machine for dealing cards. It can take a randomly arranged deck and divide it into four piles in any way you choose, but the order of the cards in each pile will be in the same order as they were in the deck.

Example: Deck starts as {5H 4C 3S 4S 4H 2D 3D 5S 2S 2C 5D 2H 5C 3C 4D 3H} Sort by suits creates piles: {5H 4H 2H 3H} {4C 2C 5C 3C} {2D 3D 5D 4D} {3S 4S 5S 2S}

Suppose that we run a deck through the machine several times, each time taking the four piles and placing them on top of each other in a fixed order. How many runs does it take to sort a deck into complete rank order, from ace of clubs to king of spades?