The mathemagician tosses a case holding a deck of cards to the first row of his one-thousand-strong audience. The deck is tossed back from person to person till it ends up close to the end of the hall, out of control of the performer. The final deck holder is asked to open the case, and give the cards a straight cut. The deck passes to a second spectator who also is also asked to cut the deck and pass it on. Finally, when the fifth spectator cuts, she is asked to keep the top card, and pass the deck back to the fourth spectator, who removes the current top card and passes it the third spectator and so on backward. When each of the five spectators has a card, the performer says, "Please make a mental picture of your card, and try to send it to me telepathically." As they do this, the performer concentrates, and then appears confused. "You're doing a great job, but there is too much information coming in all at once. Would those of you who have a red card stand up and concentrate?" The first and third spectators stand. The performer appears relieved. "That's perfect. I see a seven of hearts." One of the two spectators indicates that that is indeed his card. "And a jack of diamonds? Yes!" Now, focusing on the other three spectators, the mathemagician names all three black cards!
Mathematician-magicians Ron Graham (left) and Persi Diaconis (right) and their book Unlike some other magic tricks, this one does not rely on sleight-of-hand or accomplices. It is a purely mathematical trick, described by Persi Diaconis and Ron Graham in their new book "Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks." As we have remarked before, there is something about stage magic (and incidentally, skepticism!) that captivates mathematicians who love puzzles. The list of amateur and professional magicians among puzzle masters is legion: we have in our columns featured the puzzles of Martin Gardner, Raymond Smullyan, Persi Diaconis, Fitch Cheney, Norman Gilbreath, and Colm Mulcahy among others. Diaconis and Graham have done a great job putting together the math behind magic tricks in their wonderful 9-by-11-inch 244-page book (which I'm holding between the fingers of my left hand in the above picture).
So how is the card trick done? Not so fast! We first have some puzzles to solve…
1. Consider the binary sequence 11100010. If you take a window of length 3 bits and move it along the sequence one bit at a time looping back to the beginning to complete a cycle, you will get all the possible sets of three bits, without repeats: 111, 110, 100, 000, 001, 010, 101 and 011. This is a truly magical property. Can you construct a different 8-bit sequence that has this property? Sequences that can be made identical to the above sequence by rotation don't count. How many such sequences are there?
2. Can you find a constructive rule to create a window-4 sequence like the above? It should be 16 bits long. Here's something more difficult: how many such window-4 sequences are there? If you can get past that, try making a window-5 sequence. And the Holy Grail is to try and figure out a general formula for the number of window-n sequences with this property.
3. The magic trick that we described involves a window-5 sequence of the above type. Can you come up with an elegant scheme to map playing cards to such a sequence? (Hint: the "deck" has only 32 cards. People don't notice – or complain!) Can you see how the unique property of the sequence allows the magic trick to be performed?
These kind of sequences have a special name which will be revealed later. That is, if one of our smart readers doesn't discover it first! On account of its unique properties, these sequences have applications not just in magic tricks, but also in graph theory, robotic vision, cryptography and DNA analysis. Diaconis and Graham elucidate all this, and then move to magic variations that lead to the Gilbreath Principle, and further afield, to the Mandelbrot Set and Penrose tilings. The book is a virtuoso mathematical performance by two mathematician-performers!
I invite readers to describe any mathematical magic that impressed you. And for our word challenge, how about constructing the longest cyclic window-3 sequence of legal three letter words without repeats? Here's an 8-word example: ALAMANAG, which gives ALA, LAM, AMA, MAN, ANA, NAG, AGA and GAL. (Here's a list of all legal 3-letter words.)
For our word ladder, how about trying to go from UNIQUE to SEQUENCE? That should be challenging as well!
The extended Numberplay word-ladder rules are given below.
Numberplay Word Ladder Rules and Scoring 1. You can change a single letter in place, as in traditional word ladders — for example, SOAP to SOUP. 2. You can add or subtract a single letter in place to increase or decrease the length of a word — for example, MATH to MATCH or vice versa. 3. You can change a single letter and rearrange the letters of the word to give another word of the same length — for example, MUSEUM to SUMMED (with a U changed to D and then rearrangement). 4. All words in the ladder have to be unique.
The aim is to make the shortest possible ladder. If two ladders are the same length, the following familiar Scrabble tie-break scoring will be used. For each step in the ladder, except the original words, add points for each letter as follows: Q,Z: 10; J,X: 8; K: 5; F,H,V,W,Y: 4; B,C,M,P:3; D,G: 2; All others: 1. The ladder with the highest tie-break score wins.
If you want to post a diagram or graphic with your comment, or suggest a new puzzle, please e-mail it to numberplay@nytimes.com.
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