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 Technology News | April 2006 
Puzzles, Origami and Other Mind-Twisters
 Edward Rothstein - NYTimes
| | Kate Jones with one of her puzzle designs at the Gardner gamesfest. (Erik S. Lesser/NYTimes) |
Two hundred green marbles are in a green jar, and 200 red marbles are in a red jar. Thirty marbles are removed from the green jar and put into the red jar, which is then shaken and stirred. Thirty marbles are then scooped from that mixture and put back into the green jar. Which jar has more of the wrong color marbles?
Simple, right? The scoop removes only a very small proportion of green marbles, so more green remain in the red jar.
Except that's wrong: both jars begin and end with 200 marbles, so any green marble missing from the green jar had to have been replaced by a red one, and vice versa. The two jars end up with exactly the same number of wrongly colored marbles.
That's an old puzzle. I had heard it posed about glasses of water and wine, with a teaspoon of wine being added to the water, and another teaspoon of the mixture spooned back. But even well-known puzzles retain their power, as was made clear again and again last month at the seventh "Gathering for Gardner." These conferences of mathematicians, puzzlers, game-players and magicians at the Ritz-Carlton here began as personal tributes to Martin Gardner, Scientific American's legendary Mathematical Games columnist, and now take place without the master's presence (he is 91). During four days of talks and tricks, the oldest puzzles mixed freely with the newest.
Puzzles are a strong lure. So when the mathematician Solomon W. Golomb discussed the marble problem as one of his "favorite quickies," or when the mathematician Peter Winkler posed puzzles about people with blue and red dots on their foreheads, or about prisoners doomed to die if they don't find their names inside boxes, or when the Google software engineer and puzzle master Wei-Hwa Huang explained how he should have solved the puzzle that cost him the championship in a sudoku competition in Italy last month, one willingly put aside concerns of daily life.
It hardly mattered that there is no such thing as a tribe of truth tellers, or a man with a pet monkey and a pile of coconuts, or any other of those strange inhabitants of puzzle universes. There are no sudoku number-puzzle grids in nature, either. We imagine them, along with marbles in jars, or magic squares, or eggs covered with flexible tiles. We stage esoteric treasure hunts and construct three-dimensional models of four-dimensional objects. The only requirement is that everything be clearly defined with a limited number of laws governing behavior.
Puzzles can seem magical because they really are from a made-up universe that is bounded and simplified. Within that universe, though, free play is allowed.
That is why these conferences, organized with almost theatrical dash by the mathematician Elwyn Berlekamp, the puzzle enthusiast (and generous sponsor) Tom Rodgers and the magician Mark Setteducati also include demonstrations of sleight of hand, feats of memory and exotic juggling. They are also attracting more and more invited attendees (over 270 this time, about 100 more than the last conference).
The mixture becomes surprisingly provocative. When the Swedish physician turned card virtuoso Lennart Green seems to clumsily drop a shuffled deck of cards and then shows how it magically organized itself by suit from high to low, or causes any card a viewer names to fly out from the center of a deck, order is created out of chaos, logic out of confusion. Sleight of hand or sleight of mind? It hardly matters.
When Robert J. Lang, a laser physicist, talks with passion about origami — the Japanese art of paper folding — the line between play and discovery also completely dissolves. Mr. Lang is what might be called an origamist (see www.langorigami.com). He wrote the book "Origami Design Secrets: Mathematical Methods for an Ancient Art" (AK Peters, 2003) and has created tarantulas, delicate herons, 12-spined shells and big-horned elk out of single, uncut, folded sheets of paper.
This is maximal effect with maximal constraints and minimal materials. In that respect, origami is like the puzzle about marbles or a riddle about truth tellers or a game like Go involving black and white stones on a square board: artifice with the ability to amaze. How much can you understand or create when there is so little provided, so many restrictions, and so much possibility?
Mr. Lang points out that while in 1950 there only about 100 standard origami designs, in recent years a "mathematics of origami," studying the theory of folds and constructions, has evolved. As a result, he says, more than 30,000 origami designs now exist, and in September a fourth international conference on origami will be held at the California Institute of Technology.
Once restricted to domestic decoration, origami has also become useful for designing everything from foldable tourist maps to expandable heart stents. Mr. Lang has worked on the folding of air bags in a car, and on a design for a collapsible telescope-mirror more than 300 feet wide that might unfold in space.
This is also a legacy that Mr. Gardner leaves to generations of researchers, teachers and entertainers: don't try to understand the whole world at once. Take only a small part of it. Or better yet: invent your own universe in which there are very few elements and very few rules — a game, a puzzle, a theory. These circumscribed and artificial worlds are like sheets of paper subject to the rules of folding, yet they can yield remarkable results having almost uncanny power. The science fiction writer Arthur C. Clarke once said that any sufficiently advanced technology is indistinguishable from magic, but maybe it's also true that any sufficiently powerful "magic" eventually evolves into essential technology.
That magic is produced when one begins to see a baffling puzzle from a different perspective: what once seemed impenetrable suddenly becomes transparent. The effect resembles the images involving optical illusions or Escher-like transformations of foreground and background that are often displayed at these gatherings. The eye and mind look at one thing and start to see another. Instead of seeing fish in the sea, as in one famous Escher image, one gradually begins to see birds in the sky. Instead of thinking about how many green marbles are in the scoop, one thinks about the unchanging numbers of marbles in the jars. Solving puzzles often means seeing double — an experience singularly magical. And afterwards, one returns to daily life — absurdly confident that some day it too will begin to make sense. |
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