Wednesday, April 16, 2008

Something Strange Here, Something Almost Magical


Today’s post and tomorrow’s post will make up a kind of free-form, disconnected diptych.

I say disconnected because I’ve been feeling kind of unhinged lately . . .

Today is the last day of my first two years of blogging. Tomorrow will be the first day of my next two years of blogging.

These two days will be long excerpts from writers I like a lot.

In fact, tomorrow will be my favorite passage from my favorite author. People who know me [‘People who know me’ ?! ] probably can guess what author appears tomorrow. But I bet they can’t guess what book my favorite passage comes from.

Today’s post will be an excerpt from an obscure novel by science fiction great Arthur C. Clarke.

Clarke died last month. I wanted to do a post about Clarke last month, but back then I had Other Things on my mind. (I still have those same Other Things on my mind but I’m forcing myself to work around them.) I’m glad I waited because this post fits perfectly as my last post of two years of Impossible Kisses.

When I was very young, like in fifth or sixth grade, for a while I only read three fiction writers, Robert Heinlein, Isaac Asimov and Arthur C. Clarke.

Even back then I realized Heinlein was a little odd and wrote with a kind of restrained madness. Nobody who read a lot of Heinlein was shocked when the rumor got around that Charles Manson was a big fan of “Stranger in a Strange Land.”

Asimov in many ways was the opposite of Heinlein. Asimov wrote with an odd kind of detachment and his characters all lived with a strange kind of detachment. This made for a kind of warm, human sweetness in such novels as “The Robots of Dawn,” but it was still an odd way of writing about odd characters.

Arthur C. Clarke was always the most balanced of the three.

Clarke certainly had his own oddities. I think it was even clear to kids reading him that he wasn’t, so to speak, as fond of women as was, say, Robert Heinlein. Clarke’s...um...proclivities (okay, the rumors always were that he lived in Sri Lanka because over there nobody cared if a grown man lived with young teenage boys) lead to some unusual novels written with an unusual combination of enthusiasm and cynicism.

The excerpt today comes from a science fiction novel that isn’t so much about science as it is about lost friendship, lost love and the bittersweet moments and bittersweet memories that shape a person’s whole life as that life goes on . . .

But today’s quote isn’t about any of that.

Today’s quote is Impossible Kisses material.

One of the young boys in the novel (this is Clarke writing, after all) begins a hobby with pentominoes. These are very real things and all the stuff Clarke writes about them is true.


I’ve never been a pentomino buff myself, but I’ve seen them and once I wrote some programs about them. People who know me [them again!] know I can sometimes be a little obsessive [okay, I’m going to pause for a second to let those people who know me roll their eyes and say ‘sometimes’? ‘a little’?] and pentominoes are things that people can get obsessive about so I was always on my guard.

But the young boy in the novel witnesses a bit of real world magic as a friend of his dives into the very deep waters of the amazing shapes:



For a long time, Duncan stared at the collection of twelve deceptively simply 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, I, L, P, N and the ultimate alphabetical sequence T, U, V, W, X, 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 the magnitude 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, how 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 a 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, although 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 at 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:

      1 004 539 160 000 000

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 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.

One day, Duncan would realize all men had such powers, but might use them only once in a lifetime. In Karl, the gift was exceptionally well developed; from that moment onward, Duncan had learned to take seriously even his most outrageous speculations.





Arthur C. Clarke
















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