I saw this trick at Futility Closet and didn't believe it. Tried it a couple times and it worked. I couldn't understand how it worked, so I requested the book from the library.
Impossible? Surprising Solutions to Counterintuitive Conundrums, by Julian Havil is not really a book of magic tricks. It's about mathematics - very advanced math. You can browse it at Google Books; I consider it too complex to add to my list of recommended books. But it was worth struggling through just to retrieve this card trick.
The trick itself is impressive. The deck has to be fixed as shown in the image above. It can then be cut as many times as desired without changing the result. After cutting, the deck is shuffled - but not shuffled in the typical sense of cut and then shuffle. Before shuffling, cards are dealt from the top to form the second pile, and that second pile is then shuffled into the first one. Basically one reverses the order of half the pile, then shuffles the piles together.
After the cutting and the shuffling, the deck will still be arranged in 13 consecutive sets of four cards of four different suits. Also, alternatively, it will also be arranged in four sets of 13 cards deuce through ace (mixed-suit straights). I've tried this several times, and it has always worked. The explanation starts out something like this...
My understanding is that what it boils down to is that a single shuffle of a deck of cards (in-shuffle, out-shuffle, perfect or imperfect) is not sufficient to randomize the order of the cards. Whether the shuffle is clumsily done or professionally done, the deck will still be "ordered" to the extent that ordered groups can be dealt from it.
The nice thing about the trick is that it doesn't require any sleight-of-hand. That means you can hand the deck to someone else, and they can do the trick. This will be entertaining some of my family and friends in the winter months to come...