(by Colm Mulcahy)
Many Fold Synergies
It's been 50 years now since the appearance of Martin Gardner's seminal
Mathematics, Magic and Mystery (Dover, 1956). It was an instant classic,
and it remains as elegant, economical and entertaining today as it was then.
Included within its 175 pages are several brilliant creations by Bob Hummer
(1905--1981), who apparently had little difficulty living up to his reputation
as "The World's Most Eccentric Magician." We review two of those here, from
the 1940s, based on parity principles, along with some two-dimensional
generalizations which also have Hummer roots.
The originals which inspired these can be found in Bob Hummer's Collected
Secrets (1980) by Karl Fulves, a treasure trove of mathemagical wizardry from
one (make that two) of the most creative minds of 20th century mathemagic.
Martin Gardner observes in the introduction, "I believe that if Hummer had
obtained an education in mathematics he might have become a great mathematician
or physicist," and goes on to explain why he believes that.
A spectator is handed about a half of the deck, and invited to shuffle.
Next, the spectator holds the cards out of sight (under the table or behind the
back), while some actions (cutting, turning over pairs of cards) are performed as
often as is wished. You then take the cards back, and do your own manipulations,
again out of sight. Finally, you announce that the number of face-up cards now in
the packet is written on a piece of paper which has been in full view all along.
When checked, you are found to be correct.
To perform this trick, first write the number "13" on a piece of paper and fold
that over. Leave it on the table without drawing attention to it. Then hand the
spectator a packet of size 24 (any even number works, if an appropriate adjustment
is made to the prediction). With the cards hidden from view, ask the spectator
to cut any number of cards to the bottom, and then turn over (as a unit) the top
two cards; these two basic steps being repeated arbitrarily often.
Take the packet back, and hold it out of sight. Reverse alternate cards. You
actually end up with exactly 12 cards facing each way. Here comes the sneaky
bit: flip over the top card, and peek at this as you bring the cards back into
view. If that card is face-up, announce that the number of face-up cards now
in the packet is written on the piece of paper on the table. If it's face-down,
announce instead that the number of face-down cards now in the packet is written
on the piece of paper. You can't lose!
There are many ways to look at the parity principle which is at work here.
Note that each cut or flip brings to the top a pair of cards which is in
one of three basic configurations: both face-down (like at the outset), both
face-up, or one face-up and one face-down. Turning over the top cards merely
interchanges the "both face-up" and "both face-down" situations, while not
altering the third, mixed possibility at all. No matter how many cuts and
flips are performed, this remains true of each of the s pairs which
make up the packet. Also, the total number of face-up cards is always even.
Suppose the given packet of s pairs breaks down into u face-up pairs,
v face-down pairs, and a necessarily even number s-u-v of mixed
pairs, half of which consist of a face-up card followed by a face-down card, the
other half a face-down card followed by a face-up card. After you have reversed
alternate cards, you have u + v + (s-u-v)/2 + (s-u-v)/2 = s face-up cards.
This is "Hummer's 18 Card Mystery" from Bob Hummer's Collected Secrets, by
Karl Fulves, which we first read in Gardner's Mathematics Magic and Mystery.
The final flip was added to avoid the too predictable "half the cards are facing
one way" scenario---just in case anyone is counting.
Charles Hudson later coined the term (Hummer's) CATO Principle (Cut And
Turn Over) for the cutting and flipping under discussion. There is nothing
sacred about turning over the top two cards each time; any even number
works just as well.
Gardner has the next trick as a followup to the last one, under the combined
title "Hummer's Reversal Mystery" (and yes, it's also in the Fulves compendium).
Dead Parity Sketch
Ask to be handed about a third of the deck. Concealing them from view (perhaps
claim that you are counting them?), pass them from hand to hand, flipping those
in the even positions. If you've been given an odd number of cards, the secretly
drop the last one on the floor! Give the packet to a spectator, and request that
they once again be held out of view, while being subjected repeatedly to CATO
actions: cutting and flipping over (as a unit) the top two cards. Then have the
top card brought forward, shown around and noted. Have it replaced flipped
over on the packet. Have any number of additional CATO actions applied to
the cards. Take the packet back, for one final "undercover operation" of your
own, then bring them forward to reveal that they all face one way---except for
the spectator's card of course!
The secret lies in the fact that the packet first handed to the spectator
consists of alternating face-down and face-up cards. This condition is not
altered by subsequent CATO actions. Even better, thanks to the cards being
hidden for most of the action, it's not noticed either. When one card is
noted and flipped over, it is thus "out of sequence" relative to the others.
When you get the packet back, you simply perform the same flipping of alternate
cards, which will expose the noted card.
It should come as no surprise that many magicians have come up with applications
of this principle to red and black card separations. (Simon Aronson took things
even further in the late 1970s with his "Shuffle-Bored" effect.) Peter Duffie &
Robin Robinson have recently taken a fresh look at this area with a whole
chapter in their book Card Conspiracy (2003).
This Is an X Parity
Bob Hummer also extended the parity principle to two-dimensional arrays, and
over the years, this generalization has been explored by many others. (English
computer scientist and magician Alex Elmsley had some ingenious ideas in this
area, which can be found in his collected works. Sadly, this giant of
mathematical magic passed away on 8 January 2006.)
The basic idea is that if a rectangular array of cards, in which all cards start
out face-down, is subjected to repeated flipping over of the four corners of
sub-rectangles of any size, then each row and each column of the array is
sure to contain an even number of face-up cards.
For instance, for a four by five face-down array, if
the cards in positions
(4,4) are flipped over, and then
the cards in positions
(3,5) are flipped over, we'll see an arrangement like the following (for some
Observe that the number of face-up cards in each row or column does not change,
modulo 2, under this kind of corner flipping. That conclusion remains
valid regardless of the starting configuration of the array.
Here's one way to exploit this. Hand out the deck, and request that a jumbled
rectangular array be dealt out. "It doesn't matter how many cards are dealt
out, or which cards are face-up and which are face-down," you claim truthfully.
"A good mix of at least fifteen or twenty cards is recommended."
Demonstrate the flipping over of rectangle corners, perhaps using two hands to
turn over "parallel pairs" of cards in sequence. Emphasize that some cards may
be flipped a second (or even third) time. Secretly note which rows and columns
have odd parity (namely an odd number of face-up cards in them). For instance,
if the first, second and fourth rows, and the third and fifth columns, are "odd"
in this sense, just remember "1, 2, 4 and 3, 5; how odd."
Now have a spectator do more rectangle corner flipping as you turn away. Have
one card selected, shown around and noted, and then returned flipped over.
Encourage more rectangle corner flipping. Next, turn around and survey the scene:
relative to what you noted and remembered, exactly one row and one column will
have changed parity, and like a giant cross, these lines mark the selected card.
There are many ways to proceed from here. You could, for example, ask for a
number (or word) to be called out as you pick up the cards in an apparently
random fashion, while arranging it so that counting out (or spelling) will lead
to the selected card.
Here's Steve of Estonia
In July 1971, Martin Gardner published a paper folding trick called "Paradox
Papers" in the Pallbearers Review. Editor Karl Fulves added some card
effects which this new idea suggested. (Robin Robertson's recent "Paradox Squares
Force," from Puzzlers' Tribute: A Feast for the Mind, A.K. Peters, 2001,
sticks with paper and uses mathematics to force a certain total.)
One nice application is to repeatedly folding a rectangular array of (piles
of) cards until only one pile remained followed.
The basic folding works like this. Start with an array of sixteen face-down
cards, in a regular four by four grid. One of the outside edge rows (or columns)
is folded over "as a unit" (in practice, this requires careful moves with two hands),
so that those four cards are now face up on top of their four previous neighbours.
Another edge is then folded over on top of the rest, and so on, until only one pile
For instance, if the right-most column is folded first, and then the bottom row,
we'd obtain arrangements like this:
Here are the basic facts regarding parity invariance under folds, assuming that
the original array has an even number of rows and an even number of columns:
There have been numerous clever applications of this principle, e.g., to forcing
particular cards (four Aces from a four by four array being a popular option).
If we start with all cards face-down, we end up with a packet of alternating
face-up, face-down cards.
If we start with a chessboard pattern of cards, say with a face-up card in the
upper left position, we end up with a packet of cards all facing the same way.
If we start with a chessboard pattern of cards, say with a face-up card in the
upper left position, in which additional cards are also turned face-down, then
we end up with a packet in which all cards face the same way except for those
extra "shy cards."
Here's a prime application that can be varied. (With a little planning it can
even follow from a five by five version of the last trick). Produce a packet of
twenty-four cards and show them to be jumbled, some face up and some face down.
Remark, "Here's a trick I learned from Steve of Estonia." Shuffle a little and
then deal out into four rows of six cards.
Invite a spectator to direct a
series of folding operations, which you carry out exactly as requested. Start
by asking which edge should be folded first, leading to either eighteen or twenty
piles (some are single cards, some are back-to-back pairs). Fold again and again,
as requested, so that the number of piles shrinks while growing in height.
Stop when there is a single pile of twenty-four cards.
Pick up this packet, and casually spread it to show the face up cards, commenting
on some values. Turn the packet over and repeat, then look surprised and say,
"Wait, it looks as if you succeeded in separating the even values from the odd
ones." Hand the cards out for inspection. With any luck, somebody will point
out that your claim is not quite correct: there is a lone 2 spot among the odds
and there are some 9 spots among the evens. This is where a little mathematics
comes to the rescue.
"So the cards facing one way are a 2, 3, 5 and 7 spot, and the ones facing the
other way are all 4, 6, 8, 9 or 10 spots? That's even more amazing---you have
the primes separated from the composites! Congratulations! By the way,
mathematicians refer to this as the Sieve of Eratosthenes."
There is a secret set-up, which is done ahead of time. First assemble two
packets as follows: one consists of any twelve 4, 6, 8, 9 and 10 spots, and the
other consists of the other eight 4, 6, 8, 9 and 10 spots, and the four Kings.
The packets are mixed separately, and then one is turned over and interwoven
perfectly with the other, so that face-up and face-down cards alternate. Fan the
twenty-four cards so that you can see the Kings face-up: replace them one by
one, with a face-down 2, 3, 5 and 7 spot. (Suits are irrelevant throughout.)
The shuffling referred to should be restricted to casual cutting.
The "shy primes" will stand out after the folding, provided that the array is
laid out properly. In order to achieve the desired chessboard effect, the first
and third rows must be dealt from left to right, and the second and fourth rows
from right to left; in order words weave from side to side as you deal out.
The rest is automatic.
You may opt to have more than four primes emerge, perhaps including Jacks (value
11) and Kings (13) too; or you could force a royal flush instead.
Many more wonderful Bob Hummer (and Alex Elmsley) tricks were featured in various
Martin Gardner's Scientific American columns between 1956 and 1986. All
4500 pages of those columns were recently gathered on a single CD by the MAA as
A big thank you to magic dealer and historian Richard Hatch for pinning down the
elusive Bob Hummer dates.
Colm Mulcahy (firstname.lastname@example.org) has taught
at Spelman College since 1988, where he is wrapping up a three year term as chair
of the department of mathematics. He wants readers to know that Fulves (like
Gardner, Hummer and Elmsley) has two syllables, and that "Here's Steve of Estonia"
is an anagram of "Sieve of Eratosthenes." You can find more of his writing
on mathematical card tricks at