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).
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).
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 (1,1), (1,4), (4,1), (4,4) are flipped over, and then the cards in positions (1,4), (1,5), (3,4), (3,5) are flipped over, we'll see an arrangement like the following (for some face-up cards):
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.
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 remains.
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).
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.
A big thank you to magic dealer and historian Richard Hatch for pinning down the elusive Bob Hummer dates.