This trick demonstrates how a deck of cards can act as a computer, tracking your input and finding your selected card through elimination.
How the Trick Works
Make up a small deck of twelve cards as follows: Use the ace through six of spades and the ace through six of clubs. Arrange these as if they were a new deck: The ace of spades on top followed by the two of spades, and so on, ending with the six of spades. Then the clubs, in reverse order—the six of clubs, followed by the five of clubs, and so on, with the ace of clubs on the bottom.
The packet is going to act as a computer. It needs some input from you. Think of a small number, say, two, three, or four; deal the packet of twelve cards into that number of piles, face-down on the table, from right to left. Reassemble the piles into a single packet by putting one on the next (left to right, or right to left). Repeat once more—think of a new small number, deal that many piles, and reassemble.
The next step of mixing involves a reverse perfect shuffle. Hold the packet face-down as if you were about to deal them in a card game. Spread through the packet, pushing out alternate cards. When you are through, remove the forward cards as a group and place them on top.
The packet of cards is tracking your input. When you are done shuffling, let the computer know by performing a Monge shuffle. To do this, hold the packet in dealing position. Deal the cards from one hand to the other as follows: Push the top card off and take it in the opposite hand. Push the next card off on top of the first card. Push the next card off and put it underneath the first two. The next card is placed on top, the next underneath, and so on until you are done.
To select a card, hold the cards face-down in dealing position. Start moving cards one at a time from the top of the packet to the bottom. Stop whenever you want. The current top card is your choice. Deal it, face-down, off to one side.
The computer will now find out which card you picked by a process of elimination. This final elimination phase uses the down-and-under shuffle. Hold the eleven-card packet face-down in dealing position. Deal the top card down onto the table. Put the next card under the packet (from top to bottom). Deal the next card onto the table, put the next under the packet and so on. This eliminates cards one at a time until one card is left in your hand.
You have selected a card freely from a mixed deck and the cards have determined a card from those remaining. The two cards should be a match in value—like the two of spades and the two of clubs.
Session notes
Chapter 6 is about perfect shuffles and systematic rearrangements. Today we traced the chapter’s main trick, a twelve-card packet that the book uses as a kind of computing device. Tomorrow we cover the mathematics.
0:30 The plan: today the effect end-to-end; tomorrow definitions and theory. I understood the explanation about halfway before the session, visually rather than formally; the last step we puzzled out live.
1:36 A preview of tomorrow: the in-shuffle and out-shuffle interleave two equal halves exactly, differing in which half goes on top. Eight perfect shuffles of a 52-card deck restore the original order. The book’s gambler example: four aces on top, four perfect shuffles, and the aces reach positions 1, 5, 9, and 13, dealing them all to the first player.
4:42 The packet is a stay stack: ace through six of one suit on top, then six down to ace of a second suit below, so the -th card from the top and the -th from the bottom share a value. Everyone built one and followed along. The trick shuffles with moves that preserve this structure; after the spectator sets one card aside and the remainder is processed, the partner card surfaces. Twelve cards are needed only for the final step.
7:51 Phase one: deal into two, three, four, or six piles, reassemble in any order, repeat at will; then a reverse perfect shuffle, pushing out alternate cards and lifting the out-jogged six onto the top. Which half ends on top does not matter.
11:46 Question from the group: is the reverse perfect shuffle the same as dealing into two piles? As permutations, no – dealing reverses the order – but both preserve the invariant. Pratik noted that dealing into six piles of two gives the same result as the reverse perfect shuffle. For pile counts that do not divide twelve, the book specifies a particular pick-up order.
13:51 Phase two: the Monge shuffle, placing each card alternately on top of and beneath the growing pile.
15:12 The selection: the spectator moves cards one at a time from top to bottom, then deals the new top card face down. Nobody knows what it is.
16:15 Phase three: down-and-under elimination on the remaining eleven cards until one remains. It matched the chosen card in value. This step requires exactly twelve cards.
20:01 Dealing preserves the stay stack, shown with six differently-backed decks. Two piles of six are reverses of each other; stacking in either order restores the symmetry. Three piles: the middle pile is itself symmetric and the two outer piles are reverses of each other.
25:01 The book’s proof: deal face up and watch. Sunita’s cleaner formulation: number positions 1 to 12; partners occupy positions summing to 13, and that sum characterises symmetric positions; so it suffices to show that dealing into piles sends partners to positions that still sum to 13. Vinay: track a single card and its partner. Worked example: the two sixes at positions 6 and 7 land at 5 and 8 after dealing into four piles; a partner with three cards above it has its twin with three cards below. We checked cases by hand; a general formula for where position lands is left open.
38:59 The reverse perfect shuffle preserves the stay stack: partners have opposite parity (since 13 is odd), so exactly one from each pair is out-jogged; the out-jogged six appear in the reverse relative order of their twins, by the original symmetry; the result is again a stay stack. The same argument should extend to a true perfect shuffle.
44:26 The Monge shuffle destroys the stay stack but installs a new invariant: the two middle cards become separated by exactly five cards and, checking physically, every partner pair ends up with exactly five cards between them.
46:39 Two questions left open. Sunita: is a reverse perfect shuffle followed by a perfect shuffle the identity? Not resolved; to check tomorrow. Vinay: a packet of cards has stay-stack arrangements; do dealing and the reverse perfect shuffle reach all of them from the initial configuration? The same reachability question we raised for Hummer and Gilbreath. The book, as far as I know, does not address it.
49:18 Moving cards top to bottom preserves “five between partners” cyclically; with twelve cards, five between on one side means five on the other. After the spectator removes a card, its partner is exactly the sixth card of the remaining eleven. Counting down would already work; the elimination is more theatrical.
51:24 Eleven cards, target at position 6: the first pass removes positions 1, 3, 5, 7, 9, 11; the target survives. Five cards remain with the target in the middle; parity flips because the eleventh card went down, so the second pass removes the even positions; the target survives again. It matters that the first card goes to the table, not under.
54:26 From the book: everything before the elimination works for any even packet, but the elimination surfaces the middle card only for packets of size 4, 12, 44, 172, …, given by a recursive rule; four is the base case. Why exactly these sizes, and whether a closed form exists, I do not know.
58:56 A variation: twelve cards in a king-queen-jack repeat (suits irrelevant, I think). Free cut, moves top to bottom, take one card; down-and-under on the rest; then the performer moves five cards top to bottom, removes the new top card, and eliminates again with the first card going under. The four cards set aside are all one value – we got the kings. Why it works is homework.
1:03:59 The book also covers dealing into other pile counts with a careful pick-up order, restoring the stay stack quickly when asked to repeat, and the trick’s history as a sleight-of-hand effect refined into a self-working one by several people.
1:06:06 Tomorrow: the mathematics of perfect shuffles, possibly a way to always win at faro if I can parse the rules, and Sunita’s composition question.
To think about
- Write the map from position to its new position when cards are dealt into piles and picked up left to right, and verify that partner sums are preserved.
- Prove that the Monge shuffle on a stay stack of cards leaves exactly cards between every pair of partners.
- Find all odd for which down-and-under on cards ends on the middle card, and explain the sequence 4, 12, 44, 172, …
- Decide whether a reverse perfect shuffle followed by a perfect shuffle is the identity.
- Explain the king-queen-jack variation.