# A mathematical card trick

If you search hard enough on the internet you’ll discover a pamphlet from the 1898 by Si Stebbins entitled “Card tricks and the way they are performed” (which I’ll denote by [S98] for simplicity). In it you’ll find the “Si Stebbins system” which he claims is entirely his own invention. I’m no magician, by from what I can dig up on Magicpedia, Si Stebbins’ real name is William Henry Coffrin (May 4 1867 — October 12 1950), born in Claremont New Hampshire. The system presented below was taught to Si by a Syrian magician named Selim Cid that Si sometimes worked with. However, this system below seems to have been known by Italian card magicians in the late 1500’s. In any case, this blog post is devoted to discussing parts of the pamphlet [S98] from the mathematical perspective.

In stacking the cards (face down) put the 6 of Hearts $6 \heartsuit$ first, the 9 of Spades $9 \spadesuit$ next (so it is below the $6 \heartsuit$ in the deck), and so on to the end, reading across left to right as indicted in the table below (BTW, the pamphlet [S98] uses the reversed ordering.) My guess is that with this ordering of the deck — spacing the cards 3 apart — it still looks random at first glance.

Next, I’ll present a more mathematical version of this system to illustrate it’s connections with group theory.

We follow the ordering suggested by the mnemonic CHaSeD, we identify the suits with numbers as follows: Clubs is 0, Hearts is 1, Spades is 2 and Diamonds is 3. Therefore, the suits may be identified with the additive group of integers (mod 4), namely: ${\bf{Z}}/4{\bf{Z}} = \{ 0,1,2,3 \}$.

For the ranks, identify King with 0, Ace with 1, 2 with 2, $\dots$, 10 with 10, Jack with 11, Queen with 12. Therefore, the ranks may be identified with the additive group of integers (mod 13), namely: ${\bf{Z}}/13{\bf{Z}}=\{ 0,1,2,\dots 12\}$.

Rearranging the columns slightly, we have the following table, equivalent to the one above.

In this way, we identify the card deck with the abelian group

$G = {\bf{Z}}/4{\bf{Z}} \times {\bf{Z}}/13{\bf{Z}}$.

For example, if you spot the $2 \clubsuit$ then you know that 13 cards later (and If you reach the end of the deck, continue counting with the top card) is the $2 \heartsuit$, 13 cards after that is the $2 \spadesuit$, and 13 cards after that is the $2 \diamondsuit$.

Here are some rules this system satisfies:

Rule 1 “Shuffling”: Never riff shuffle or mix the cards. Instead, split the deck in two, the “bottom half” as the left stack and the “top half” as the right stack. Take the left stack and place it on the right one. This has the effect of rotating the deck but preserving the ordering. Do this as many times as you like. Mathematically, each such cut adds an element of the group $G$ to each element of the deck. Some people call this a “false shuffle” of “false cut.”

Rule 2 “Rank position”: The corresponding ranks of successive cards in the deck differs by 3.

Rule 3 “Suit position”: Every card of the same denomination is 13 cards apart and runs in the same order of suits as in the CHaSeD mnemonic, namely, Clubs $\clubsuit$, Hearts $\heartsuit$, Spades $\spadesuit$, Diamonds $\diamondsuit$.

At least, we can give a few simple card tricks based on this system.

Trick 1: A player picks a card from the deck, keeps it hidden from you. You name that card.

This trick can be set up in more than one way. For example, you can either

(a) spread the cards out behind the back in such a manner that when the card is drawn you can separate the deck at that point bringing the two parts in front of you, say a “top” and a “bottom” stack, or

(b) give the deck to the player, let them pick a card at random, which separates the deck into two stacks, say a “top” and a “bottom” stack, and have the player return the stacks separately.

You know that the card the player has drawn is the card following the bottom card of the top stack. If the card on the bottom of the top stack is denoted $(a,\alpha) \in G$ and the card drawn is $(b,\beta)$ then

$b \equiv a+3 \pmod{13}, \ \ \ \ \ \ \beta \equiv \alpha +1 \pmod{4}.$

For example, a player draws a card and you find that the bottom card is the $9 \diamondsuit$. What is the card the player picked?

solution: Use the first congruence listed: add 3 to 9, which is 12 or the Queen. Use the second congruence listed: add one to Diamond $\diamondsuit$ (which is 3) to get $0 \pmod 4$ (which is Clubs $\clubsuit$). The card drawn is the $Q \clubsuit$.

Trick 2: Run through the deck of cards (face down) one at a time until the player tells you to stop. Name the card you were asked to stop on.

Place cards behind the back first taking notice what the bottom card is. To get the top card, add 3 to the rank of the bottom card, add 1 to the suit of the bottom card. As you run through the deck you silently say the name of the next card (adding 3 to the rank and 1 to the suit each time). Therefore, you know the card you are asked to stop on, as you are naming them to yourself as you go along.