- Membership
- MAA Press
- Meetings
- Competitions
- Community
- Programs
- Students
- High School Teachers
- Faculty and Departments
- Underrepresented Groups
- MAA Awards
- MAA Grants

- News
- About MAA

**December 2005**

No doubt you've heard somebody say, "luckation is everything." No? Don't be so sure, many people can't distinguish it from the more familiar expression you're probably thinking of.

Let's define *luckation* to be the appearance of getting lucky at the way something works out, when, in reality, the observed outcome has been carefully planned and engineered all along. (Some may feel that is an abuse of the English language. Trust me, I'm a mathemagician―so I can define anything I want as long as it doesn't conflict with previously defined terms, or cause the audience to spontaneously combust.)

We consider three examples of luckation below―hence the phrase "luckation, luckation, luckation." Their successful outcomes depend on your surreptitiously glancing at one key card, hopefully when nobody is looking. Knowledge of the bottom card is required for one effect, the middle card for another, and (you guessed it) the top card for the third.

There are many fairly well known effects in which several cards are dealt out side by side, face up, and each is then used as the basis for "completion to 13." By this we mean that on, for instance, 9♦, one counts out loud, "Ten, eleven, twelve, thirteen," dealing out 4 more cards face up onto the base card (thus forming a stack of 5 cards). No cards are dealt on top of K♠, which has value 13 and is considered to be a complete stack already. (Suits plays no role here, only values.) This can be traced back to a trick by Audrey V. Walsh & Hal Haber, which appeared in Jean Hugard's *Card Manipulations 1* (1933), with variations due to Orville Meyer in that author's subsequent *More Card Manipulations 2* (1939).

Here's a treatment of this based on "13" from *Card Tricks Anyone Can Do* (Castle Books, 1968) by Temple Patton. It's really an example of a mathematical *force* (a way to persuade a spectator to select a particular card while maintaining an illusion of free choice), used to enable a foolproof prediction.

Hand a complete deck out to be shuffled, and ask a spectator to hold the deck ready for dealing. Declare that you need a lucky number, adding, "So, first, let's eliminate unlucky numbers." Ask what number is universally regarded as being unlucky. With any luck, several people will suggest 13. Comment, "That's what they all say, but I looked on the internet and found that `lucky 13' gets ten times as many hits as `unlucky 13.' So, I'm sticking with thirteen." Have that many cards dealt out into a scattered face-down arrangement. Take back the rest of the deck.

Add, "I nearly forgot," jotting something down on a piece of paper and folding it over, but leaving it on the table in full view. "I think I know how this is all going to come out. I've just noted a prediction; we'll check later to see if I got lucky."

Have any one of the 13 displayed cards turned face-up, and say, "Let's see what your lucky number is." No matter what value is revealed, look disappointed and comment, "No, I don't think that's it. Let's try another." Have a second card turned face-up, and again express dissatisfaction. Then say, "Maybe it's a case of third time lucky." Have a third card turned face-up, but look exasperated. "None of those was it." Gather up the ten remaining face-down cards and tuck them at the bottom of the deck. Shuffle a little, casually, as you stress how you had no influence on the choice of the cards on the table.

Say, "Let's try dealing out cards on each of these to reach thirteen, my lucky number." Hand out the deck again, and assuming the three displayed cards are, say, 4♣, 8♦ , and Q♥, have cards dealt face-down onto each, to complete each pile to 13, urging the spectator to count out loud. E.g., five cards are dealt onto the 8♦, saying "nine, ten, eleven, twelve, thirteen." Each base card should protrude slightly from the bottom of its pile, so that its value is visible. The spectator holds the remainder of the deck.

Suddenly perk up, "I'm going to take one last stab at your lucky number. Recall, you determined *three* numbers when you chose those three cards at random. Add up those three numbers: *4 + 8 + 12 = 24*. Maybe *that's* your lucky number. Please count down to the 24th card, I bet that's the luckation we need. (Of course, everyone will assume you just said, "location.")

When this card is located and turned over for everyone to see, and your paper is unfolded and checked, they will be found to agree. Gloat discreetly, "I knew thirteen was lucky; it must be true if it's on the internet. You don't believe me? All I can say is: luck on the web."

When the ten unused cards are placed at the bottom of the deck, the original bottom card moves up 10 places in a deck of size *52 - 3 = 49*, in other words *the force/prediction card moves to position 39 from the top*. The casual shuffling suggested must retain the bottom stock; this is not hard to do, and will convince many in the audience that the card is now lost in the deck.

Next, assume the selected cards have values *a*, *b* and *c*. First, *13 - a* cards are dealt on top of the card with value *a*, then *13 - b* cards are dealt on top of the card with value *b*, and finally *13 - c* cards are dealt on top of the card with value *c*. Overall, *(13 - a) + (13 - b) + (13 - c) = 39 - (a + b + c)* cards have been dealt off the top, leaving the force card at position *(39 - (39 - (a + b + c)) = a + b + c* from the top. Since this sum is the value you belatedly declare to be the spectator's lucky number, the final count down leads to the force card as required, nicely matching your prediction.

Thanks to Peter Duffie for the Hugard references, and also for suggesting a justification for the spectator's selection of three from 13.

Hand the deck out, in two parts, to two volunteers to shuffle separately. Take these back, and reassemble the deck, shuffling a little more. Now, invite a tall spectator to cut off and conceal a pile of cards, roughly a third of the deck. Have a second, shorter spectator cut off and hand you about half of what remains, concealing the rest. (You get the top half at this stage, which is of course the middle third of the original deck.)

Glance through these, remarking that your astute mathemagical powers allow you to guess exactly how many cards each spectator has in their pile, even though they themselves do not know.

"I'm going to try to guess how many cards are in each of your piles. Guessing isn't as easy as you might think, it requires the utmost concentration. Silence please, and keep the cards hidden." Feign deep thought for a moment, and then relax and proceed as follows.

"I've got it. We mathemagicians call this process *unaddition*. Let me demonstrate. I have eighteen cards here―when I was young I had the kind of teachers who made sure I could count all the way to twenty without a calculator. My later specialized training in university allows be to deduce that you must have thirty-four cards between you, since *52 - 18 = 34*. That's subtraction by the way; please hold your applause until the end." You are now ready for the surprising finish.

"Here comes the unaddition, which is strictly graduate school stuff. There are many ways to break up thirty-four into two numbers, such as *34 = π + (34 - π) *. Is everybody still with me? However, when I unadded just now, I couldn't help noticing that *34 = 15 + 19*. I wonder if that's it: please count your cards now." It will be discovered that the taller person has fifteen cards, and the shorter nineteen.

Shrug, "It was just a lucky break." (If your audience includes any Hopf algebraists, you can claim that this is an example of coaddition!)

To perform this effect, first split the deck into two exact halves, and hand these out to two volunteers for separate shuffling. One sure-fired way to get this right is to start with the deck casually held in one hand at your side, with the tip of your little finger maintaining an earlier exact split. This is generally known as a *pinky break*, but you should think of it as a lucky break.

Take the halves back, and before reassembly peek at the bottom card of the half which goes on top: that's now the 26th card in the deck. As long as the middle third you retain after the two subsequent cuts include this key card, in the same position, you can deduce the size of each spectator's pile. (You can even do a little casual shuffling at either end, retaining the middle part.) Simply look at the faces of your cards, and note the position of your key card relative to each end of the pile.

For instance, if you find your key card is *12* from the bottom (face) of your pile, and *8* from the top, then *12 - 1 = 11* cards are missing from the bottom half, and *11* from the top half. Hence the shorter person must have *26 - 11 = 15* cards, and the taller person must have *26 - 8 = 18* cards.

Tricks using the 26th card go back a long way of course, and similar effects can be found in Hugard's 1937 *Encyclopedia of Card Tricks* (which was based on a work largely compiled by Glen Gravatt). Special thanks to Robin Robertson for that reference, and for simplifying what we'd originally proposed.

Ask your audience, "Do I look honest to you?" Proceed before people have a chance to come up with any smart answers. "You can trust my look," pause, "I know I do."

Hand a full deck out for shuffling, then take it back and shuffle it some more. Say, "One of the first things you learn in magic is how to memorize the names of all the cards in the deck. It takes some work, but anyone can do it." Wait for that to sink in, continuing, "Let's try an experiment with this shuffled deck."

Give it to a spectator, who is requested to repeatedly deal the cards alternately into two piles, one face down, the other face up. The face up pile is discarded each time, and then more dealing done using the face down pile. Fuss about how you *could* pay attention to the card faces as they are revealed, and check them off against a mental check list. "That's the easy way to do this trick. I'd rather try something else. I'm going to just guess, and trust my luck."

When only one card remains, you name it, right before it is turned over.

"Like I said, trust my look."

You must know the top card of the deck before any dealing is done. One way to do this is to peek at the bottom card when you get the deck back, shuffle it to the top, and do some riffle shuffles without disturbing it. Next, proceed as above, directing the spectator carefully at each of the six deals: for most of them have the first card dealt face down, but for the second deal and the last deal (when only two cards remain), have the first card dealt face up. The last card is the original top one.

This dealing principle is well known. After the first deal, the key card is the last card of a pile of size 26. After the second deal, it's the first card of a pile of size 13. After the third deal, it's the last card of a pile of size 7. After the fourth deal, it's the first card of a pile of size 4. After the fifth deal, it's the last card of a pile of size 2. Then, after the sixth and final deal, it's the last remaining card.

Of course this is related to the fact that *2 ^{5} < 52 < 2^{6}*, or equivalently,