“I’d guess there might be twenty ways of putting those pieces into the box,” he replied, determined to be on the safe side.

“Try again.” That was a danger signal. Obviously, there was 34 much more to this business than met the eye, and it would be safer not to commit himself.

Duncan shook his head.

“I can’t imagine.~P

“Sensible boy. Intuition is a dangerous guide though sometimes it’s the only one we have. Nobody could ever guess the right answer. There are more than two thousand distinct ways of putting these twelve pieces back into their box. To be precise, 2,339. What do you think of that?”

It was not likely that Grandma was lying to him, yet Duncan felt so humiliated by his total failure to find even one solution that he blurted out: “I don’t believe it!”

Grandma seldom showed annoyance, though she could become cold and withdrawn when he had offended her. This time, however, she merely laughed and punched out some instructions to the computer.

“Look at that,” she said.

A pattern of bright lines had appeared on the screen, showing the set of all twelve pentominoes fitted into the six-by-ten frame. It held for a few seconds, then was replaced by another obviously different, though Duncan could not possibly remember the arrangement briefly presented to him. Then came another… and another, until Grandma canceled the program.

“Even at this fast rate,” she said, “it takes five hours to run through them all. And take my word for it-though no human being has ever checked each one, or ever could-they’re all different.”

For a long time, Duncan stared at the collection of twelve deceptively simple figures. As he slowly assimilated what Grandma had told him, he had the first genuine mathematical revelation of his life. What had at first seemed merely a childish game had opened endless vistas and horizons-though even the brightest of ten-year-olds could not begin to guess the full extent of the universe now opening up before him.

This moment of dawning wonder and awe was purely passive; a far more intense explosion of intellectual delight occurred when he found his

first very own solution to the problem. For weeks he carried around with him the set of twelve pentominoes in their plastic box, playing with them at every odd moment. He got to know each of the dozen shapes as personal friends, calling them by the letters which they most resembled, though in some cases with a good deal of imaginative distortion: the odd group, F, 1, L, P, N and-the ultimate alphabetical sequence T, U, V, W,

Y…” Y, Z.

And once in a sort of geometrical trance or ecstasy which he was never able to repeat, he discovered five solutions in less than an hour. Newton and

Einstein and Chen-tsu could have felt no greater kinship with the gods of mathematics in their own moments of truth…. It did not take him long to realize, without any prompting from Grandma, that it might. also be possible to arrange the pieces in other shapes besides the six-by-ten rectangle. In theory, at least, the twelve pentominoes could exactly cover rectangles with sides of five-by-twelve units, four-by-fifteen units, and even the narrow strip only three units wide and twenty long.

Without too much effort, he found several examples of the five-by twelve and four-by-fifteen rectangles. Then he spent a frustrating week, trying to align the dozen pieces into a perfect three-by-twenty strip. Again and again he produced shorter rectangles, but always there were a few pieces left over, and at last he decided that this shape was impossible.

Defeated, he went back to Grandma-and received another surprise.

“I’m glad you made the effort,” she said. “Generalizing–exploring every possibility-is what mathematics is all about. But you’re wrong. It can be done. There are just two solutions; and if you find one, you’ll also have the other.”

Encouraged, Duncan continued the hunt with renewed vigor. After another week, he began to realize them.agnitude of the problem. The number of distinct ways in which a mere twelve objects could be laid out essentially in a straight line, when one also allowed for the fact that most of them could assume at least four different orientations,

was staggering. Once again, he appealed to Grandma, pointing out the unfairness of the odds. If there were only two solutions, bow long would it take to find them?

“I’ll tell you,” she said. “If you were a brainless computer, and put down the pieces at the rate of one a second in every possible, way, you could run through the whole set in”—she paused for effect–-~‘rather more than six million, million years.”

Earth years or Titan years? thought the appalled Duncan. Not that it really mattered … “But you aren’t a brainless computer,” continued Grandma. “You can see at a glance whole categories that won’t fit into the pattern, so you don’t have to bother about them. Try again….”

Duncan obeyed, though without much enthusiasm or success. And then he had a brilliant idea.

Karl was interested, and accepted the challenge at once. He took the set of pentominoes, and that was the last Duncan heard of him for several hours.

Then he called back, looking a little flustered.

“Are you sure it can be done?” he demanded.

“Absolutely. In fact, there are two solutions. Haven’t you found even one?

I thought you were good in mathematics.”

“So I am. That’s why I know how tough the job is. There are over a quadrillion possible arrangements to be checked.”

“How do you work that out?” asked Duncan, delighted to discover something that had baffled his friend.

Karl looked at a piece of paper covered with sketches and numbers.

“Well, excluding forbidden positions, and allowing for symmetry and rotation, it comes to factorial twelve times two to the twenty-first-you wouldn’t understand why! That’s quite a number; here it is.”

He held up a sheet on which he had written, in large figures, the imposing array of digits:

1 004 539 160 000000

Duncan looked at the number with satisfaction; he did not doubt Karl’s

arithmetic. “So you’ve given up.”

“NO! I’m just telling you how hard it is.” And Karl, looking grimly determined, switched off.

The next day, Duncan had one of the biggest surprises of his young life. A bleary-eyed Karl, who had obviously not slept since their last conversation, appeared on his screen.

“Here it is,” he said, exhaustion and triumph competing in his voice.

Duncan could hardly believe his eyes; he had been convinced that the odds against success were impossibly great. But there was the narrow rectangular strip, only three squares wide and twenty long, formed from the complete set of twelve pieces…. With fingers that trembled slightly from fatigue, Karl took the two end sections and switched them around, leaving the center portion of the puzzle untouched.

“And here’s the second solution,” he said. “Now I’m going to bed. Good night-or good morning, if that’s what it is.”

For a long time, a very chastened Duncan sat staring at the blank screen.

He did not yet understand what had happened. He only knew that Karl had won against all reasonable expectations.

It was not that Duncan really minded; he loved Karl too much to resent his little victory, and indeed was capable of rejoicing in his friend’s triumphs even when they were at his own expense. But there was something strange here, something almost magical.

It was Duncan’s first glimpse of the power of intuition, and the mind’s mysterious ability to go beyond the available facts and to short-circuit the process of logic. In a few hours, Karl had completed a search that should have required trillions of operations and would have tied up the fastest computer in existence for an appreciable number of seconds.