**February 2011**

This column's romp *deals with* additive and multiplicative inverses in Z* _{13}*, and can be adapted to work for other values of 13. As such, it might provide an entertaining way to help to cement in the minds of beginning students these key algebraic concepts. The underlying dealing principle is revealed in due course, and once understood, it can be employed in entirely different magical effects for audiences unfamiliar with and uninterested in the mathematical considerations below. It can also be modified to yield further algebraic curiosities, e.g., concerning primitive roots or quadratic residues.

Address a group of mathematically minded people, announcing, "You're about to have some fun with the help of some of the greatest names in mathematics." Run through a deck of cards face up, tossing out all of the Diamonds. Set the rest aside. Arrange the Diamonds in a face-up packet in order, with the King on top and the Ace at the bottom.

Fan the cards for all to see, reminding your your audience of the usual numerical correspondences (Ace = 1, Jack = 11, etc). Say, "Please think of this packet as representing the cyclic group of order thirteen, which is a very lucky group, as we will soon see." Now turn this packet of cards face down, so that the Ace is on top.

Ask for a volunteer who fancies a little low-down double dealing. Hand him or her a piece of paper with the word Hilbert on it, another with Lagrange, and still more with Ramanujan, Pythagoras, Riemann, Descartes and Archimedes. Say, "In a moment, while I turn away, I'm going to ask you to do some dealing and dropping with this packet of cards, using any of these names you wish, in any order you want, as often as you feel like. In a sence, you'll be double dealing with the big boys."

Hold the packet face down as you demonstrate, "For instance, if you were to start with the name Fermat—and by the way, he's not an option right now as he knows too much about where we're going with this—you would count out one card for each letter, F-E-R-M-A-T, into a pile on the table, dropping the rest on top, then do all of that a second time. Keep the cards held close to the table, so that it really is a case of low-down double dealing." Spell out the letters of Fermat slowly as you count six cards into a pile, dropping the rest on top, and then go much faster the second time around, quickly handing the packet to the volunteer. Turn away and reiterate the "double-deal as often as you wish" instructions. The volunteer might, for instance, start with Archimedes (used twice), then move on to Lagrange (twice), and finish with Ramanujan (twice). As advertised, names may be repeated, and it doesn't matter how many are used.

When the volunteer is finished, turn back, and reclaim the packet, truthfully commenting, "These cards are totally randomized now, based on your actions. I honestly have no idea where any particular card is at this stage." Fan the cards face down. Have the volunteer slip out one card, after which you place the others behind your back. Say, "Are you familiar with the concept of the additive inverse of a number? Here, it means what you would need to add to the chosen number to get zero, or equivalently, thirteen. What card do you have?" Suppose it is revealed to be a 4. Without missing a beat, you casually say, "Since 4 + 9 = 13, the additive inverse of four is nine," as you produce the nine from behind your back. You can joke, "You see, in a sense nine is minus four, and I could just *feel* the minus sign implicit on this card. Did I mention that I have very sensitive fingers?"

You can bring forward the remaining eleven cards, and repeat this stunt several more times. If, at any stage, the chosen card is the middle one, you know that it's the King. Don't even ask what it is: simply reach behind your back and dramatically produce nothing.

Declare the experiment over, and gather up all thirteen cards, reassembling the packet into a fan showing the original ascending order, Ace to King, from left to right. "Let's try something different," you say as you flash the card faces.

"Are you familiar with the concept of *multiplicative* inverses? Each number between 1 and 12 can be multiplied by something to yield 1, modulo 13. We exclude 13 itself of course," you say as you discard the King. "It's not too exciting for the Ace, which is its own inverse, or for the Queen either, bearing in mind that 12 times 12 is 1 1 modulo 13." Set those cards aside too.

Now fan the remaining ten cards, and pull out the 8. "What is the multiplicative inverse of this one?" Pause to see if anyone knows. "Since 8 times 5 is 40, which is 1 modulo 13, it's 5!" Replace the 8, and close up the fan, turning it face down. "Once again, I'm going to give you free reign to do as much double dealing as you wish. We have fewer cards this time, so let's exclude Archimedes and Pythagoras. But you can use Gauss, Newton or Euclid instead, as well as the other names I gave you earlier. What a deal!"

"For instance," you say impulsively, "If I were to use Gauss, I'd do this twice," as you demonstrate the dealing out of five cards and dropping the rest on top. Only do it once, however.

Now turn away again and have lots of double dealing done. When the volunteer is finished, turn back, and reclaim the packet, again commenting, "These cards are really mixed up now, based on your choices and actions. Again, I honestly have no idea where any particular card is at this point." Fan the cards face down. Have the volunteer slip out one card, after which you place the others behind your back. Say, "What card do you have?" Suppose it is revealed to be a 4. Without hesitation, casually say, "Since 4 x 10 = 40, which is 1 modulo 13, the multiplicative inverse of 4 is 10," as you produce the ten from behind your back."

You can of course bring forward the remaining eight cards, and repeat the stunt once or twice more.

The observation underlying this month's effects borrows ideas from both the very first *Card Colm*, Low-Down Triple Dealing (October 2004), and the more recent (A) Pi Evolved Set - Harmonic Split Drill (August 2008).

As may easily be verified, the following **Double Dealing Palindrome Principle** holds:

*A palindromic packet of cards remains palindromic when a fixed number
of cards*—

*two rounds of being dealt off into a pile with the rest dropped on top. *

Suppose that the packet contains an even number of cards *2n*, and that the last *n* cards in some sense match the first *n* cards, only in reverse order. For instance, A♣, 2♣, 3♣, 4♣, 5♣, 6♣, 6♥, 5♥, 4♥, 3♥, 2♥, A♥. One can also consider a similar odd-sized packet with a Joker inserted in the middle. See what happens when at least half of the cards are low-down double dealt as explained above.

[Of course, as discussed in Low-Down Triple Dealing, four such rounds of dealing and dropping restores any packet—palindromic or not—to its original order.]

By the time the volunteer is let loose on the low-down double dealing for our two effects this month, the packets are palindromic in the following senses. In the first case each card is paired with its additive inverse, and in the second case each card is paired with its multiplicative inverse.

Many presentational alternatives suggest themselves. For instance, one could be upfront about the final palindromic state of the packets, "milking" the top and bottom cards repreatedly—i.e., sliding them off in pairs—to show the desired "matching" properties (additive or multiplicative inverses respectively). While this gives away part of the secret, it also makes the transition from additive to multiplicative focus more mysterious, as the conditions under which the effects are carried out seem to be the same.

The principle above is almost all that is needed to pull off the two Z_{13} effects just as described earlier, but there are two secret ingredients without which observers may find it difficult to duplicate the production of the matching inverses. These are discussed in the sections that follow.

The Fermat spelling demonstration suggested above is not 100% honest: on the second deal you actually count out seven cards before dropping the rest on top, so that the cards end up in order 6, 5, 4, 3, 2, Ace, King, Queen, Jack, 10, 9, 8, 7:

[Alternatively, that result can be achieved by splitting a King to Ace fan between the 6 and 7, and reassembling in reverse order—just try not to be at 6s and 7s.]

The upshot is that the King is now in the middle, where it remains throughout all subsequent double deals, and most importantly the cards equidistant from it sum to 13 (Ace and Queen, 2 and Jack, 3 and 10, and so on, ending with 6 and 7).

By the Double Dealing Palindrome Principle, this property is retained no matter how many double deals *of seven or more cards* are done. The names which may be used must therefore be restricted; hence the joke about excluding Fermat. When the volunteer slips out a card, you merely note how far it is from the closest end of the fan. Then, behind your back, pull out and bring forward the card in the same position relative to the other end.

There is also a secret for pulling off the multiplicative inverse effect. When removing and replacing of the 8 before the (single) Gauss dealing and dropping demonstration earlier, the 8 is actually put back between the 10 and Jack. As a result, after the Gaussian spelling, the cards are now in order 7, 9, 10, 8, Jack, 6, 5, 4, 3, 2. Don't let anyone see you doing this (hence our heading here).

This time the cards equidistant from the centre multiply to 1 (modulo 13), since 7 x 2 = 14, 9 x 3 = 27, 10 x 4 = 40, 8 x 5 = 40, and 11 x 6 = 65. So the stack is "multiplicative inverse palindromic." This property is invariant under any number of double dealings involving five to nine cards (or indeed ten, if you want to go Greek).