Martin Gardner has had a long career writing about recreational mathematics, which includes games, puzzles, and magic tricks based on mathematical principles (see MAKE, Volume 12, page 80, “Mathemagician”). Gardner generously agreed to share with MAKE readers a few that can be described briefly and performed with no sleight-of-hand skills.
The Gilbreath Principle
A number of card tricks are based on a principle that magician Norman Gilbreath introduced to magic. It’s an application of combinatorial mathematics (which we will spare you here). Gardner discusses it in his books New Mathematical Diversions from Scientific American, Chapter 9, and Mathematical Magic Show, Chapter 7. We’ll describe here the simpler versions.
Prepare the deck ahead of time with the cards in black/red alternation. No other order is necessary. When you start this trick, you can do any false shuffle that doesn’t change the card order. But if you don’t have those skills, don’t bother.
Have a spectator cut the deck and riffle shuffle the two parts together just once. Tell him to fan the cards and look at their faces to confirm that they are well mixed. Say, “Look near the middle of the deck and find two adjacent cards of the same color. Don’t tell me the color, but cut the cards between those two, and complete the cut.”
The dirty work has been done. The deck is now ordered as a sequence of pairs, and each pair has one red and one black card. The spectator doesn’t know this and wouldn’t likely notice even when looking at the card faces.
Some versions of this trick suggest you take the deck under the table and pretend to be searching for red and black cards by touch, then bring them out and show them as pairs. But people become suspicious when the deck is out of their sight.
(I can’t imagine why. Don’t they trust magicians?) The following method keeps the deck in full view.
Take the deck, facedown, and explain that when red and black cards are adjacent to each other, as many must be, their opposite polarities give off radiation that you can sometimes sense, using your enhanced psychic powers.
Peel off cards from the top of the deck, ignoring an even number of cards: two, four, six, etc. Then say, “Aha! Here’s a pair.” Show that the next two are a red/black pair, laying them on the table. Keep doing this, each time laying the ignored cards in a separate facedown stack on the table, or moving them to the bottom of the deck.
Don’t skip the same number of cards each time — make it look as if you are really searching for the radiation from paired cards. Do this until you have ten or more pairs, or enough to convince everyone that you really can do it. It’s good to leave some of the deck in the discard pile.
If someone is still not convinced, do the same procedure on the discard pile, for it still has the same ordering. But don’t run the entire deck, or else your spectators will become bored or realize that the deck already had complete ordering of red/black pairs.
Of course you must be able to reliably count off an even number of cards. If you bungle this, you may still get a red/black pair, but eventually you won’t. Oh well, no one is perfect. When this happens, count off an odd number the next time and you’ll be back on track.
Gardner says, “The point is that one riffle shuffle doesn’t destroy all order in the deck. In this trick it leaves the cards in red-black pairs. At least eight riffle shuffles are necessary and sufficient to destroy all order in a deck. That was first proved by a good friend of mine, Persi Diaconis, now a prominent statistician.”
Two decks together require nine shuffles, and six decks require 12 shuffles. The riffle shuffle looks pretty, but it’s not a good mixer. It merely interleaves two runs of cards. The order within each run is preserved.
Here’s another example. Use a brand-new deck that has its factory ordering of cards. In a new deck, the cards are ordered by suits and numerically within each suit. Don’t shuffle the deck. Have someone select a card. Cut the deck and give it one riffle shuffle. It now has two interlaced ordered sequences. Return the chosen card to the deck. Fan the deck so you can see the faces, and find the selected card. It’s highly unlikely the card will go into the original sequence, so the card out of sequence will be obvious.
Gardner recently told me of an extension of this idea. Take two fresh, sealed decks of the same kind. Remove the jokers and advertising cards. Turn one deck upside down and riffle shuffle the two decks together. Now count out 52 cards from the top, and you will have two decks of 52, and each one will have all the cards a deck should have. But they will have two interleaved orderly sequences of cards. Actually, you don’t need new decks. Any two decks that have the same order will give the same result.
The Permuting Cards
Gardner showed me a version of another trick, using four playing cards bolted together so they are in order, alternating red and black. “The bolt keeps them in order,” he said. Indeed, it is hard to see how they could possibly get out of order, since the bolt has a locking nut. Yet with one flourish, Gardner rotated the rightmost two cards all the way around the bolt, and the cards magically rearranged to red, red, black, black.
It seems to defy physics, and mathematics, too.
When I figured it out, I wondered whether the principle could be made to work with a larger number of cards, since this version doesn’t lend itself to anything but a short magical surprise.
As usual, there’s a mathematical (topological) principle underlying the trick. The cards are cut and interleaved, but that fact is hidden by the bolt. The trick is in how you cut the cards — with razor blade or scissors.
Use a paper punch to make holes in all four cards, carefully positioned so all four are aligned. Only the middle two cards are cut, along the S-shaped solid lines as shown above (ignore the dotted lines for now). Then hold card 2 above and to the right of card 1, and slide tab D under tab A. Now card 2 is both above and beneath card 1, and when card 2 is rotated clockwise, it will slide through and under card 1. Add the uncut cards on top and bottom, then align the holes, and bolt them all together. Use a bolt that fits the holes snugly. A larger-diameter hole and bolt work best, though I’ve gotten away with ¼” bolts when using thin cards. The rotation may be repeated, restoring the cards to their original order.
There are other ways to make the cuts, but after extensive research in the TTT laboratories, we have concluded that this is the most foolproof.
A six-card version requires two pairs of cards to be cut as described above. Slide the right tab of card 2 under the left tab of card 1. Then slide the right tab of card 3 under the left tabs of both 1 and 2. Finally slide the right tab of card 4 under the left tabs of 1, 2, and 3. Add the uncut cards on top and bottom, align the holes, and bolt them all together.
How about going for broke? Prepare three sets of two cut cards, interleave them, and sandwich them between two uncut cards, for a total of 8 cards. Can it possibly work? Yes, quite well, but with so many cards it’s best to use thinner cards, such as 3″×5″ file cards. You can number them, or use different colors, and develop your own presentation routine.
Note, as you play with this, that the “cut” cards undergo a cyclic permutation when you rotate them 180°. The top and bottom cards stay put and serve to hide the surgery you did on the others. You can “cut” the fanned cards at any point before rotating, but cutting next to the end cards does nothing interesting. Of course the action may be repeated, cutting at different points. When you wish to restore the cards to the original order, cut between the cards that were originally next to the end cards.
Yes, the idea can be modified to an odd number of cards. A five-card version with three inner cut cards is a good start, which we leave as an exercise for the reader. Hint: Look at the dotted line in the diagram.
The photos at left show a version I made to tease my physics students. Students should know the order of colors in the spectrum, first studied by Newton. They are: red, orange, yellow, green, blue, and violet. But some can’t remember them. I show them this arrangement that I call “Newton’s crutch.” I tell them that by bolting the colors together, we ensure they can’t get out of order. Then I do “the move” and the colors are disordered.