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Aha! Insight

by Martin Gardner

"The creative act owes little to logic or reason. In their accounts of the circumstances under which big ideas occurred to them, mathematicians have often mentioned that the inspiration had no relation to the work they happened to be doing. Sometimes it came while they were traveling, shaving or thinking about other matters. The creative process cannot be summoned at will or even cajoled by sacrificial offering. Indeed, it seems to occur most readily when the mind is relaxed and the imagination roaming freely."

-- Morris Kline, SCIENTIFIC AMERICAN, March 1955.




xperimental psychologists like to tell a story about a professor who investigated the ability of chimpanzees to solve problems. A banana was suspended from the center of the ceiling, at a height that the chimp could not reach by jumping. The room was bare of all objects except several packing crates placed around the room at random. The test was to see whether a lady chimp would think of first stacking the crates in the center of the room, and then of climbing on top of the crates to get the banana. 

The chimp sat quietly in a corner, watching the psychologist arrange the crates. She waited patiently until the professor crossed the middle of the room. When he was directly below the fruit, the chimp suddenly jumped on his shoulder, then leaped into the air and grabbed the banana. 

The moral of this anecdote is: A problem that seems difficult may have a simple, unexpected solution. In this case the chimp may have been doing no more than following her instincts or past experience, but the point is that the chimp solved the problem in a direct way that the professor had failed to anticipate. 

At the heart of mathematics is a constant search for simpler and simpler ways to prove theorems and solve problems. It is often the case that a first proof of a theorem is a paper of more than fifty pages of dense, technical reasoning. A few years later another mathematician, perhaps less famous, will have a flash of insight that leads to a proof so simple that it can be expressed in just a few lines. 

Sudden hunches of this sort-hunches that lead to short, elegant solutions of problems-are now called by psychologists "aha! Reactions." They seem to come suddenly out of the blue. There is a famous story about how William Rowan Hamilton, a famous Irish mathematician, invented quatemions while walking across a stone bridge. His aha! Insight was a realization that an arithmetic system did not have to obey the commutative law. He was so staggered by this insight that he stopped and carved the basic formulas on the bridge, and it is said that they remain there in the stone to this day. 

Exactly what goes on in a creative person's mind when he or she has a valuable hunch? The truth is that nobody knows. It is some kind of mysterious process that no one has so far been able to teach to, or store in, a computer. Computers solve problems by mechanically going step-by-step through a program that tells them exactly what to do. It is only because computers can perform these steps at such incredible speeds that computers can solve certain problems that a human mathematician cannot solve because it might take him or her several thousand years of nonstop calculation. 

The sudden hunch, the creative leap of the mind that "sees" in a flash how to solve a problem in a simple way, is something quite different from general intelligence. Recent studies show that persons who possess a high aha! Ability are all intelligent to a moderate level, but beyond that level there seems to be no correlation between high intelligence and aha! Thinking. A person may have an extremely high I.Q., as measured by standardized tests, yet rate low in aha! Ability. On the other hand, people who are not particularly brilliant in other ways may possess great aha! Ability. Einstein, for instance, was not particularly skillful in traditional mathematics, and his records in school and college were mediocre. Yet the insights that produced his general theory of relativity were so profound that they completely revolutionized physics. 

This book is a careful selection of problems that seem difficult, and indeed are difficult if you go about trying to solve them in traditional ways. But if you can free your mind from standard problem solving techniques, you may be receptive to an aha! Reaction that leads immediately to a solution. Don't be discouraged if, at first, you have difficulty with these problems. Try your best to solve each one before you read the answer. After a while you will begin to catch the spirit of offbeat, nonlinear thinking, and you may be surprised to find you aha! Ability improving. If so, you will discover that this ability is useful in solving many other kinds of problems that you encounter in your daily life. Suppose, for instance, you need to tighten a screw. Is it necessary to go in search of a screwdriver? Will a dime in your pocket do the job just as well? 

The puzzles in this collection are great fun to try on friends. In many cases, they will think for a long time about a problem, and finally give it up as too difficult. When you tell them the simple answer, they will usually laugh. Why do they laugh? Psychologists are not sure, but studies of creative thinking suggest some sort of relationship between creative ability and humor. Perhaps there is a connection between hunches and delight in play. The creative problem solver seems to be a type of person who enjoys a puzzling challenge in much the same way that a person enjoys a game of baseball or chess. The spirit of play seems to make him or her more receptive for that flash of insight that solves a problem. 

Aha! Power is not necessarily correlated with quickness of thought. A slow thinker can enjoy a problem just as much, if not more, than a fast thinker, and he or she may be even better at solving it in an unexpected way. The pleasure in solving a problem by a shortcut method may even motivate one to learn more about traditional solving techniques. This book is intended for any reader, with a sense of humor, capable of understanding the puzzles. 

There certainly is a close connection, however, between aha! Insights and creativity in science, in the arts, business, politics, or any other human endeavor. The great revolutions in science are almost always the result of unexpected intuitive leaps. After all, what is science if not the posting of difficult puzzles by the universe? Mother Nature does something interesting, and challenges the scientist to figure out how she does it. In many cases the solution is not found by exhaustive trial and error, the way Thomas Edison found the right filament for his electric light, or even by a deduction based on the relevant knowledge. In many cases the solution is a Eureka insight. Indeed, the exclamation "Eureka!" comes from the ancient story of how Archimedes suddenly solved an hydraulic problem while he was taking a bath. According to the legend, he was so overjoyed that he leaped out of the tub and ran naked down the street shouting "Eureka! Eureka!" (I have found it!)

We have classified the puzzles of this book into six categories: combinatorial, geometric, number, logic, procedural and verbal. These are such broad areas that there is a certain amount of unavoidable overlap, and a problem in one category could just as well be regarded as in one of the others. We have tried to surround each puzzle with a pleasant, amusing story line intended to put you in a playful mood. Our hope is that this mood will help you break away from standard problem solving routines. We urge you, each time you consider a new puzzle, to think about it from all angles, no matter how bizarre, before you spend unnecessary time trying to solve it the long way. 

After each problem, with its delightful illustrations by the Canadian graphic artist Jim Glen, we have added some notes. These comments discuss related problems, and indicate how, in many cases, the puzzles lead into significant aspects of modern mathematics. In some cases, they introduce problems that are not yet solved. 

We have also tried to give some broad guidelines for the channels along which aha! Thinking sometimes moves: 

1. Can the problem be reduced to a simpler case?
2. Can the problem be transformed to an isomorphic one that is easier to solve?
3. Can you invent a simple algorithm for solving the problem?
4. Can you apply a theorem from another branch of mathematics?
5. Can you check the result with good examples and counterexamples?
6. Are aspects of the problem give that are actually irrelevant for the solution, and whose presence in the story serves to misdirect you? 

We are rapidly entering an age in which there will be increasing temptation to solve all mathematical problems by writing computer programs. The computer, making an exhaustive trial-and-error search, may solve a problem in just a few seconds, but sometimes it takes a person hours, even days, to write a good program and remove all its bugs. Even the writing of such a program often calls for aha! Insights. But with the proper aha! Thinking, it may be possible to solve the same problem without writing a program at all. 

It would be a sad day if human beings, adjusting to the Computer Revolution, became so intellectually lazy that they lost their power of creative thinking. The central purpose of this collection of puzzles is to exercise and improve your ability in this technique of problem solving.

(Introduction, pages vi-viii)
 
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Last Update
August 28, 2003

 
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