For a quarter of a century, stretching from 1956 to 1983, Martin Gardner wrote an extremely popular and influential monthly column called "Mathematical Games" for Scientific American. Last time here, in the June 2010 Card Colm, we marked the passing of this intellectual giant and prolific writer, who died in May at the age of 95, with a selection of mathematical card tricks—in his own words—from the first four of the resulting 15 books. This time, we sample further expositions on the topic from books eight and ten of the series.
All 15 books can now be enjoyed in full, text searchable form on a wonderful CD-ROM available from the MAA called Martin Gardner's Mathematical Games. Decades after these 300 odd thought provoking columns first appeared, this card column's author is proud to be one of the many people directly inspired by the MG legacy. Some of Martin's more famous (non-card) columns are now available at http://maa.org/pubs/focus/mg.html.
As before, the crystal clear words in italics below are all Martin's, but the section headings are our own.
At the end, we explain a curious trick which was aired in the New York Times in June.
Chapter 7 of Mathematical Magic Show (1977) is simply called "Playing Cards." The first principle discussed there is one we've explored here on previous August occasions, specifically in August 2005 and August 2006, as well as in a "no deal" variation—curiously enough, something related shows up below as Face Up Gamble—in August 2009.
In Chapter 9 of my New Mathematical Diversions from Scientific American, I mentioned briefly a curious principle discovered by a young amateur magician, Norman Gilbreath. Arrange a deck of cards so its red and black cards alternate. Cut the deck to form two piles, breaking the deck so that the top cards of each pile are of different colors. If the two piles are now riffle-shuffled into each other, every pair of cards, from the top down, will consist of one red and one black card. (You can let someone else do the single shuffle and then play him 26 rounds of matching colors. You each take a card from the top of the deck; your opponent wins if the cards match. Of course you win every time.) Gilbreath later discovered that his principle is only a special case of what magicians now call the Gilbreath general principle. It applies to any repeating series of symbols and can best be explained by a few examples.
Arrange a deck so that the suits repeat throughout in the same order, say spades, hearts, clubs, and diamonds. From the top of this deck deal the cards one at a time to the table to form a pile of 20 to 30 cards. (Actually it does not matter in the least how many cards are in this pile.) Riffle-shuffle the two parts of the deck together. Believe it or not, every quartet of cards, from the top down, will now contain a card of each suit. Dozens of subtle card tricks exploiting Gilbreath's general principle have been published in magic periodicals. The simplest trick is to let someone deal and shuffle, take the deck behind your back (or under a table), then pretend to feel the suits with your fingers and bring out the cards in groups of four, each containing all four suits.
It is necessary that one packet be reversed before the shuffle. Dealing cards to the table does this automatically. Another method is to cut off a portion of the deck, turn it over and shuffle this face-up packet into the rest of the deck, which remains face down. A third method is to take cards singly from the top of the deck and push them into the pack, inserting the first card near the bottom, the next anywhere above the previously inserted card (directly above it if you wish), the third above that, and so on until you have gone as high as you can. This is equivalent to cutting off a packet, reversing its order and riffle-shuffling. The deck's original order is destroyed, of course, but the cards remain strongly ordered in the sense that each group of four cards contains all four suits.
A trick applying the Gilbreath principle to a repeating series of length 52 is to arrange one full deck so that its cards are in the same order from top to bottom as the cards in a second deck are from bottom to top. If the two decks are riffle-shuffled into each other and then cut exactly at midpoint, each half will be a complete deck of 52 different cards!
Martin then continues:
Gilbreath's general principle points up how poorly a riffle-shuffle randomizes. This inefficiency of the riffle-shuffle provided another mathemagician, Rev. Joseph K. Siberz of Boston College, with what may well be the first computer program that teaches a computer how to do a mystifying card trick. The trick uses 52 IBM punched cards, each bearing the name of a different playing card. Both program and "deck" are entered in the computer, which then prints the following instructions:
If the card was, say, the five of hearts, the computer quickly prints out: "Your card was the five of hearts. Don't ask me how I do it. Magicians never reveal their secrets. Take another card and I shall do it again." If the person has failed to follow instructions exactly, the computer sometimes finds the card anyway, perhaps after asking the spectator for additional information: "I am having trouble determining the color of your card. Please help me by turning on switch B if it is black or switch C if it is red." This is followed by, "Thank you. Your card is . . ." If the instructions were not followed and the computer cannot find the card, it prints, "You did not follow my directions. Please take another card and try again." If this happens again, the computer politely asks for still another try, but after a third goof it says, "I won't find your $$=) $* * card if you refuse to do it my way. Please try again."
- Give the deck several single cuts and a riffle-shuffle.
- Cut the deck into two piles.
- Look at the top card of one pile and remember it.
- Bury this card in the pile from which it came, then riffle-shuffle the two piles together.
- Cut the deck, complete the cut, and repeat several times if you wish.
- Now give the deck back to me and I shall find your card.
It is not hard to see how the program finds the card. The two riffle-shuffles merely break the deck's original cyclic order into four interlocking sequences. If the instructions are followed correctly, a single card will be missing from where it should be in one of those sequences. While it is identifying the card, the computer memorizes the deck's new order and therefore is all set for an immediate repetition of the trick. The reader can easily perform the trick himself by recording the order of a deck or using an unopened pack, which comes from the manufacturer in a simply ordered sequence that can be memorized while you remove the joker and the extra cards.
After a spectator has followed the instructions given above you can take the twice-shuffled deck to another room; by checking the cards off on your list it is easy to determine the single card that is out of place.
Martin next mentions a curious probabilistic observation which reminds us of the one put to good use here in Better Poker Hands Guaranteed in June 2006, namely that there is a 95% chance of getting at least "one pair" among ten random cards.
On the television show Maverick, popular in the mid-1960's, the gambler Bart Maverick bet someone he could take 25 cards, selected at random, and arrange them into five poker hands each of which would be a straight or better. (The hands higher than a straight are flush, full house, four of a kind, straight flush, and royal flush.) The same bet was made on television in 1967 by Paul Bryan in an episode of Run for Your Life. It is what gamblers call a "proposition"--a bet for which the odds seem against the person making it when actually they are strongly in his favor.
If the reader will experiment with 25 randomly chosen cards, he will be surprised at the ease with which five hands can be arranged. Try the flushes first (there will be at least two), then look for straights and full houses. I have no idea of the actual probability of success, but it is extremely high. Indeed, the question arises: Is success always possible? The answer is no. There are sets of 25 cards that cannot be partitioned into five poker hands of straight or better.
He then provides an example of such a set of 25 cards, before moving on to other topics.
"Mathematical Tricks With Cards" (Chapter 19) of Wheels, Life, and Other Mathematical Amusements (1983) has eight extraordinary pages jam-packed with terrific card material, most of which we now present.
Many excellent card deceptions are based on a parity principle, but the underlying even-odd structure is usually concealed so ingeniously that if you follow the directions with cards in hand you are likely to astonish yourself. Consider the following trick invented about 1946 by the Chicago card expert Ed Marlo. Magicians classify it as an "oil and water" effect, for reasons that will be apparent in a moment. There are many ways of achieving the same effect by secret and difficult "moves," but this version is entirely self-working.
Remove 10 red and 10 black cards from the deck and arrange them in two face-up piles, side by side, with all red cards. on the left and all black cards on the right. First you tell your watchers that you will demonstrate what you intend to do by using only five cards of each color. With both hands simultaneously remove the top card from each pile and place them, still face up, on the table at the bottom of each pile. Do the same with the next two top cards, but this time cross your arms before you place the two cards on the two new piles you are starting. This puts a black card on the red one and a red card on the black one. The next transfer of a pair of cards is made with uncrossed arms, the next with crossed arms, and the fifth and last pair is dealt with arms uncrossed. In other words, five simultaneous deals are made, with arms crossed only on alternate deals. On each side you now have a pile of five face-up cards with their colors alternating. Put either pile on the other one. Spread the 10 cards to show that colors alternate throughout.
Square the cards and turn the packet face down. From its top deal the cards singly and face up to form two piles again, dealing alternately to the left and right. Call attention to the fact that this procedure naturally separates the colors. At the finish you will have five reds on the left and five blacks on the right.
State that you will repeat this simple series of operations with all 20 cards. Begin as before, with 10 face-up reds on the left and 10 face-up blacks on the right. Transfer the cards to form two new piles, just as you did before, crossing your arms on alternate deals so that the colors alternate in each pile. After all 20 cards are dealt put one pile on the other, square the cards, turn the packet over and hold it face down in your left hand.
Deal 10 cards face up to form two piles, dealing from left to right and observing aloud that this brings the reds together on the left and the blacks together on the right. After the 10 cards have been dealt face up do not pause but continue smoothly and deal the remaining 10 cards face down. It is best to put down the cards so that they overlap in two vertical rows.
Pick up the five face-down cards on the left with your left hand and the five face-down cards on the right with your right hand. Cross your arms and put the cards down. You explain that you have transferred half of the cards of each pile to the pile of the opposite color but that like oil and water the colors mysteriously refuse to mix. Turn over the face-down cards. To everyone's surprise (you hope) the reds are back with the reds and the blacks are back with the blacks!
Readers should have little difficulty discovering why it works with any set of cards containing an even number of cards of each color and why it did not work when you demonstrated it with 10 cards.
After you have finished the oil-and-water trick put the two piles together with either color on top. Turn the packet face down and spread it in a fan. You are ready to perform a red/black trick invented by Karl Fulves and published in his magic periodical, The Pallbearers Review, September, 1971.
Ask someone to pull slightly forward any 10 cards he pleases. The fan will resemble the one shown below. With your right hand count the jogged (protruding) cards to make sure there are 10. Do this by removing the cards one at a time from right to left, putting them into a face-down pile as you count from one to 10. Close up the 10 cards remaining in your left hand and place them in a second face-down pile alongside the first.
Tell your audience that an amazing thing has happened. Although 10 cards were selected randomly, the colors in the two piles are so ordered that every nth card in one pile has a color opposite to the color of the nth card in the other pile. To prove this, turn over the top cards of each pile simultaneously. One will be red and the other black. Place the black under the red, turn the pair over and put it aside to form a new face-down pile. Repeat the procedure with the cards now on top of the two original piles. They will be red-black too. Indeed, every pair you turn will be red-black!
As you show the pairs always put the black card under the red before you turn them over and place them on the third pile. When you finish, the cards in this face-down pile will have alternating colors.
But wait, there's more!
Now you are ready to perform a truly mystifying trick in which parity is conserved in spite of repeated shuffling. Known as Color Scheme, it was invented by Oscar Weigle, an amateur magician who is now an editor at Grosset & Dunlap. It sold as a manuscript in magic stores in 1949.
Give the packet of 20 cards to someone and ask him to hold it under the table where neither he nor anyone else can see the cards. Tell him to mix the cards by the following procedure. (It is known as the Hummer shuffle, after Bob Hummer, the magician who first used it in tricks.) Turn over the top two cards (not one at a time but both together as if they were one card), place them on top and cut the packet. Your assistant is to keep repeating this procedure of turn two, cut, turn two, cut for as long as he wishes. The procedure will, of course, result in a packet containing an unknown number of randomly distributed reversed cards.
With the cards still held under the table, tell your assistant to do the following. Shift the top card to the bottom. Then turn over the next card, produce it from under the table and place it on the table. This procedure is repeated--card to bottom, reverse next card and deal-until 10 cards have been dealt to the table. It will be apparent that the cards have become mysteriously ordered. All the face-up cards are the same color and all the face-down ones are of the opposite color.
The second and climactic half of the trick, which Weigle confesses is a "bare-faced swindle," now unfolds. Your assistant is still holding 10 cards under the table. Ask him to shuffle them by separating them into two packets; then, keeping all the cards flat (no card must be allowed to turn over), weave the two packets into each other in a completely random way. You can demonstrate how to do this by using the 10 cards already dealt. After your assistant has executed the shuffle a few times, ask him to turn over the packet and shuffle the same way a few more times. If he likes, he can give the packet a final cut.
Now he continues with the dealing procedure he used before: card to bottom, next card reversed and dealt. (The final card is reversed and dealt.) In spite of the thorough mixing the result is exactly the same as before. All the face-up cards match the former face-up cards in color, and the same is true of all face-down cards.
Martin moves on to present some creations of the prolific—and still active today—Karl Fulves (whose name has three syllables total):
One of the oldest themes in card magic is to produce in some startling fashion a card that has been randomly selected and replaced. Here is a simple method that exploits a binary sorting technique. Fulves published it in his periodical in November, 1970.
Take 16 cards from a shuffled deck and spread them face down on the table without mentioning how many cards you are using. A viewer selects a card, looks at it and places it on top of the deck. The remaining cards in the spread are squared and put on top of the deck above the chosen card. Ask him to cut off about half of the deck, give or take half a dozen cards. Actually he can take between 16 and 32 cards. He hands this packet to you.
Hold the packet in both hands. As your left thumb slides the cards one at a time to the right, move your right hand forward and back so that every other card, starting with the first one, is jogged forward. The resulting fan of cards will resemble the one in the last figure above except that the jogged cards are not randomly distributed.
Strip all the projecting cards from the fan and discard them. Square the remaining cards and repeat the procedure, jogging forward all the cards at odd positions, starting with the first card. Strip them out and discard. Continue in this way until one card is left. Before turning it over ask for the chosen card's name. It will be the card you hold.
A completely different method of locating a selected card can be found in several books on card magic. Turn your back and instruct someone to cut a shuffled deck into three approximately equal piles. He turns over any pile and then reassembles the deck by sandwiching the face-up pile between the other two, which remain face down. He is told to remember the top card of the face-up pile. With your back still turned, ask him to cut the deck several times, then give it one thorough riffle-shuffle. The shuffle will of course distribute the face-up cards randomly throughout the deck.
Turn around, reverse the pack and spread it in a row. Look for a long run of face-up cards, remembering that a cut may have split the run so that part of it is at each end of the spread. The first face-down card above the run is the chosen one. Slide it from the spread, have the card named and then turn it over.
Next is something far from well known that is worthy of further exploration.
Our last trick, based on a curious shuffling principle discovered by Fulves, is presented as a gambling proposition. All cards of one suit (the suit can be chosen by the victim) are removed from the deck. Assume that the discarded suit is diamonds. The remaining cards are arranged so that each triplet has three different suits in the same order. (Card values are ignored.) Again the victim may specify the ordering. Suppose he chooses spades, hearts and clubs. The 39-card deck is arranged from the top down so that the suits follow the sequence spades, hearts, clubs, spades, hearts, clubs and so on.
Place the deck face up in front of the victim. Ask him to cut it in two packets and riffle-shuffle them together. As he makes the cut, note the suit exposed on top of the lower half. We shall call this suit k. After the single shuffle the deck is turned face down. The cards are now taken from the top three cards at a time, and each triplet is checked to see if it contains two cards of the same suit.
It is hard to believe, but:
This assumes, of course, a spades-hearts-clubs ordering. If the ordering is otherwise, the three rules must be modified accordingly; that is, spades must be changed to whatever suit is at the top of each triplet, and so on. Let m stand for the suit that you know cannot show twice in any triplet, and a and b for the suits that can.
- If k is spades, no triplet will contain two spades.
- If k is hearts, no triplet will contain two clubs.
- If k is clubs, no triplet will contain two hearts.
Before dealing through the deck to inspect the triplets, make the following betting proposition. For every triplet containing a pair of m's you will pay the victim $10. In return he must agree to pay you 10 cents for every pair of a's or b's. It seems like a good bet for the victim, but it is impossible for you to lose, and the swindle can be repeated as often as you please. Just arrange the cards again and allow the victim to make the single riffle-shuffle. Naturally you always promise to pay him for doublets of the suit that you know cannot show. The fact that this suit may vary from deal to deal makes the bet particularly mystifying.
As Fulves has observed, the triplets have other unexpected properties. Of the triplets containing pairs the a's and b's will alternate; after a pair of a's the next pair will be b 's and vice versa. Pairs of one suit always include a top card of the triplet. Pairs of the other suit always include a bottom card.
No explanation of these tricks will be given. Readers will find it stimulating, however, to analyze each trick to see if they can comprehend exactly why it operates with such uncanny precision.
The chapter ends with this Addendum:
Peter T. Sarjeant extended Fulves' shuffling trick to the four suits of a full deck. Arrange the cards so that from top down the sequence is a repetition of clubs, diamonds, hearts, spades. As before, the deck is placed face up and cut about in half. Note the suit on the top of the bottom half. Call it k. The halves are then interlaced with a single riffle-shuffle.
When cards are taken four at a time from the top you will find the following true of each quadruplet:
Knowledge of these facts can, of course, be the basis of a variety of betting swindles.
- If k is clubs, there will be no pair of hearts and no pair of clubs.
- If k is diamonds, any suit may be paired.
- If k is hearts, there will be no pair of diamonds and no pair of spades.
- If k is spades, any suit may be paired.
Edward M. Cohen proposed the following variation of Fulves' trick involving a selected card that goes sixteenth from the top of the deck. He likes to begin by forming a square array of 16 cards, face down on the table. A spectator picks a row. Another person picks a column. The card at the intersection is turned face up and remembered. This card goes to the bottom of the deck. The remaining 15 cards are swept into a pile and the deck placed on top of them. The chosen card is now sixteenth from the bottom.
Anyone may now cut the deck about in half (it is only necessary that the lower portion contain more than 16 and less than 32 cards). The top half is discarded. Hand the lower half to someone with the request that he deal it into two face-up piles, alternating piles as he deals. The pile that gets the last card is discarded. The other pile is turned face down, and this procedure is repeated until only one card remains. It will be the chosen card.
Hundreds of more elaborate card tricks have been based on the binary principles that underlie this trick, but the one just described is as simple, effective, and as easy to perform as any.
That concludes this month's sampling of Martin Gardner's Scienfitic American expositions of mathematical card principles.
A curious new "Martin Gardner speller" trick was recently aired in the New York Times puzzle blog, which we now reproduce for your convenience. Over the decades, Martin often reported meeting on regular occasions with famed numerologist Dr. Matrix. It appears that the good Dr. Matrix blogged in the NTY in June (click here), where Numberplay author Pradeep Mutalik was reported to have written:
Dr. Matrix has just shown me a fantastic trick that I must share with you. He took out a deck of cards, and handed it to me, saying, "Please remove all of the red cards, we won't be needing those." I did as he asked, fanning through the card faces. I noticed no particular order, but was surprised to see that each card had a large letter of the alphabet written neatly on it with a heavy black marker. He continued, "What remains are twenty-six black cards. Please hold this half deck face down, and cut the cards several times. Then, start dealing into a pile on the table."
After I had dealt nine or ten cards he instructed me to stop whenever I wished. I did so after dealing a few more. He then had me place the remaining cards in a second pile beside the ones just dealt. "You now have two piles of cards, which I'd like you to push together so as to further randomize the half packet." I complied, noting that the cards now clumped together in unpredictable ways as I combined the piles into a single packet. Finally, he had me deal out exactly thirteen cards into a new pile, and give it to him. "Please pick up the other thirteen cards," he instructed, "and look at the faces. I will do the same with mine. Each of us will see what words we can spell with the letters fate has dealt us. Ignore the card values and suits."
After several minutes, the best I could come up with was DREAM, RANT and GRIN. I glanced over at the neat display Dr. Matrix had arranged, and my jaw dropped. It clearly said MARTIN GARDNER! "He may be gone, but his memory lingers, even in this humble pack of cards," my friendly host said softly. Then he leaned over and gently rearranged my cards until they too spelled out MARTIN GARDNER -- how could I have missed that? I was truly flabbergasted. Had I not, after all, mixed the cards myself? Magic or coincidence? You be the judge.
Can readers figure this out and perform the trick themselves? Perhaps use an old deck which doesn't mind being written on.
It makes use of the illustrated card mixing procedure:
(This is Swedish magic maestro Lennart Green's "Rosette shuffle", which is equivalent to a riffle shuffle of the two piles.)
In June we assured readers that all would be revealed here in August, but Ravi from Hyderabad beat us to the punch in the NYT on 18 July 2010. It is, when all is said and done, a simple application of the Gilbreath General Principle with which we started above.
The black cards should start in some fixed (and non-obvious, as commmented upon in the NYT on 14 August 2010) order of the letters of the words MARTIN GARDNER (cycled twice), and it's a good idea to have totally random letters on the red cards. These latter ones are more likely to be noticed as they are pulled out at the beginning, which adds to the illusion of their being no prior setup. Alternatively, one can simply use half a deck from the outset, and not display too many card faces early on.
Colm Mulcahy (firstname.lastname@example.org) completed his PhD at Cornell in 1985, under Alex 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 http://www5.spelman.edu/~colm/cards.html.