Premo Card Trick

Following is an example of the Premo card trick described in my blog post. Start with the deck in a known order; following is the "new deck" order, at least for the Bicycle packs that I have experimented with:

After the series of shuffles, cuts, card selection, and subsequent shuffles and cuts in the trick, following is an example of the final arrangement of cards in the deck:

In this example, the selected card was the five of spades. How can we determine this? Each card c has a predecessor p(c) and successor s(c) in the initial new deck ordering. For example, the five of spades is preceded by the six of spades, and followed by the four of spades. (The deck is viewed cyclically, so that the ace of spades is preceded by the two of spades and followed by the ace of hearts.)

For a given candidate card c, consider scanning the cards in the shuffled arrangement from left to right, starting with the predecessor p(c), continuing to c, and ending with the successor s(c), "wrapping around" if necessary from the right (bottom) to the left (top) of the pack. Count the number of cards traversed along this cyclic sequence. For example, for the five of spades, we start with p(c), the six of spades, scan to the right 25 cards (wrapping around in the process) to the five of spades, and continue 34 cards from there to s(c), the four of spades, for a total length of 59 cards.

The selected card will (hopefully) be the unique card with the longest such "path" p(c)->c->s(c). In most cases, as in this example, this longest path is the only path longer than a full cycle of 52 cards, which is simpler to scan quickly than actually counting individual cards. Of course, sometimes the selected card is not the unique card with this longest path property, in which case the trick will fail. But the probability of success is about 0.84, as described in more detail in the blog post.