(by Colm Mulcahy)
Low-Down Double Dealing With the Big Boys
This column's romp deals with additive and multiplicative inverses in
Z13, 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
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.
Begin Dual Lode
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
A palindromic packet of cards remains palindromic when a fixed number
of cards -- representing at least half of the packet -- is subjected to
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♥,
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
[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 Z13
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,
[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.
Limit Optical Input
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
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).
Colm Mulcahy (firstname.lastname@example.org)
completed his PhD at Cornell in 1985, under
F.T.W. Rosenberg. He has been in the department of mathematics at Spelman
College since 1988, and writing Card Colms---the only MAA columns to
actively encourage lying on a regular basis---bi-monthly since October 2004. For
more on mathematical card tricks, including a guide to topics explored in previous
Card Colms, see
"Begin dual lode " is an anagram of "double dealing,"
"act disoriented" is an anagram of "addition secret," and finally,
"limit optical input" is an anagram of "multiplication tip.
Follow Colm on Twitter at @CardColm