"What's happening over there?" he asked.
At first, he had thought it was one of the fire fountains that sometimes erupted in unstable areas; but in that case the city would have been in danger, and he would have heard of it long ago. Then he realized that the brilliant yet smoky column of light burning steadily on the hill crest three or four kilometers away could only be man-made.
"There's a fusor running over at Huygens. I don't know what they're doing, but that's the oxygen burn-off."
"Oh, one of Armand's projects. Doesn't it annoy you?"
"No — I think it's beautiful. Besides, we need the water. Look at those rain clouds... real rain. And I think there's something growing over there. I've noticed a change in color on the rocks since that flame started burning."
"That's quite possible — the bioengineering people will know all about it. One day you may have a forest to look at, instead of all this bare rock."
He was joking, of course, and she knew it. Except in very restricted areas, no vegetation could grow here in the open. But experiments like this were a beginning, and one day...
Over there in the mountain, a hydrogen fusion plant was at work, melting down the crust of Titan to release all the elements needed for the industries of the little world. And as half that crust consisted of oxygen, now needed only in very small quantities in the closed-cycle economies of the cities, it was simply allowed to burn off.
"Do you realize, Duncan," said Grandma suddenly, "how neatly that flame symbolizes the difference between Titan and Earth?"
"Well, they don’t have to melt rocks there to get everything they need."
"I was thinking of something much more fundamental. If a Terran wants a fire, he ignites a jet of hydrocarbons and lets it burn. We do exactly the opposite. We set fire to a jet of oxygen, and let it burn in our hydromethane atmosphere.
This was such an elementary fact of life — indeed an ecological platitude — that Duncan felt disappointed; he had hoped for some more startling revelation. His face must have reflected his thoughts, for Grandma gave him no chance to comment.
"What I'm trying to tell you," she said, "is that it may not be as easy for you to adjust to Earth as you imagine. You may know — or think you know — what conditions are like there, but that knowledge isn't based on experience. When you need it in a hurry, it won't be there. Your Titan instincts may give the wrong answers. So act slowly, and always think twice before you move."
"I've no choice about acting slowly — my Titan muscles will see to that."
"How long will you be gone?"
"About a year. My official invitation is for two months, but now the trip's being paid for, I'll have funds for a much longer stay. And it seems a pity to waste the opportunity, since it's my only one."
He tried to keep his voice as cheerfully optimistic as he could, though he knew perfectly well the thoughts that must be passing through Grandma's mind. They were both aware that this might be their last meeting. One hundred fourteen was not an excessive age for a woman — but, truly, what did Grandma have to live for? The hope of seeing him again, when he returned from Earth? He liked to think so...
And there was another matter, never to be referred to, yet hovering in the background. Grandma knew perfectly well the main purpose of his visit to Earth, and the knowledge must, even after all these years, be like a dagger in her heart. She had never forgive Malcolm; she had never accepted Colin; would she continue to accept him when he returned with little Malcolm?
Now she was hunting around, with a clumsiness quite unlike her normal precise movements, in one of the cubbyholes of her work desk.
"Here's a souvenir to take with you."
"What — oh, it's beautiful!" He was not being excessively polite; sheer surprise had forced the reaction from him. The flat, crystal-lidded box he was now holding in his hands was, indeed, one of the most exquisite works of geometrical art he had ever seen. And Grandma could not have chosen any single object more evocative of his youth and of the world that, though he was now about to leave it, must always be his home.
As he stared at the mosaic of colored stones that exactly filled the little box, greeting each of the familiar shapes like an old friend, his eyes misted and the years seemed to roll away. Grandma had not changed; but he was only ten...
7
A Cross of Titanite
"You're old enough now, Duncan, to understand this game... though it's very much more than a game."
Whatever it is, thought Duncan, it doesn't look very exciting. What can you do with five identical squares of white plastic, a couple of centimeters on a side?"
"Now the first problem," continued Grandma, "is to see how many different patterns you can make, by putting all these squares together."
"While they lie flat on the table?"
"Yes, with the edges matching exactly — overlapping isn't allowed."
Duncan started to shuffle the squares.
"Well," he began, "I can put them all in a straight line like this... then I can switch the end one to make an L... and the one at the other end to make a U..."
He quickly produced half a dozen different assemblies of the five squares, then found that he was repeating himself.
"I think that's all — oh, stupid of me."
He had missed the most obvious figure of all — the cross, or X, formed by putting one square in the middle and the other four surrounding it.
"Most people," said Grandma, "find that one first. I don’t know what this proves about your mental processes. Do you think you've found them all?"
Duncan continued to slide the squares around, and eventually discovered three more figures. Then he gave up.
"That's the lot," he announced confidently.
"The what about this one? Said Grandma, moving the squares swiftly to make a figure that looked like a humpbacked F.
"Oh!"
"And this..."
Duncan began to feel very foolish, and was much relieved when Grandma continued: "You did fairly well — you only missed these two. Altogether, there are exactly twelve of these patterns — no more and no less. Here they are. You could hunt forever — you won't find another one."
She brushed aside the five little squares, and laid on the table a dozen brightly colored pieces of plastic. Each was different in shape, and together they formed the complete set of twelve figures that, Duncan was now quite prepared to admit, were all that could be made from five equal squares.
But surely there must be more to it than this. The game couldn't have finished already. No, Grandma still had something up her sleeve.
"Now listen carefully, Duncan. Each of these figures — they're called pentominoes, by the way — is obviously the same size, since they're all made from five identical squares. And there are twelve of them, so the total area is sixty squares. Right?"
"Um... yes."
"Now sixty is a nice round number, which you can split up in lots of ways. Let's start with ten multiplied by six, the easiest one. That's the area of this little box — ten units by six units. So the twelve pieces should fit exactly into it, like a simple jigsaw puzzle."
Duncan looked for traps — Grandma had a fondness for verbal and mathematical paradoxes, not all of them comprehensible to a ten-year-old victim — but he could find none. If the box was indeed the size Grandma said, then the twelve pieces should just fit into it. After all, both were sixty units in area.
Wait a minute... the area might be the same, but the shape could be wrong. There might be no way of making the twelve pieces fit this rectangular box, even though it was the right size.
"I'll leave it to you," said Grandma, after he had shuffled pieces around for a few minutes. "But I promise you this — it can be done."
Ten minutes later, Duncan was beginning to doubt it. It was easy enough to fit ten of the pieces into the frame — and once he had managed eleven. Unfortunately, the hole then left in the jigsaw was not the same shape as the piece that remained in his hand — even
though, of course, it was of exactly the same area. The hole was an X, the piece was a Z...
Thirty minutes later, he was fairly bursting with frustration. Grandma had left him completely alone, while she conducted an earnest dialogue with her computer; but from time to time she gave him an amused glance, as if to say "See — it isn't as easy as you thought..."
Duncan was stubborn for his age. Most boys of ten would have given up long ago. (It never occurred to him, until years later, that Grandma was also doing a neat job of psychological testing.) He did not appeal for help for almost forty minutes...
Grandma's fingers flickered over the mosaic. The U and the X and L slid around inside their restraining frame — and suddenly the little box was exactly full. The twelve pieces had been perfectly fitted into the jigsaw.
"Well, you knew the answer!" said Duncan, rather lamely.
"The answer?" retorted Grandma. "Would you care to guess how many different ways these pieces can be fitted into their box?"
There was a catch here — Duncan was sure of it. He hadn't found a single solution in almost an hour of effort —and he must have tried at least a hundred arrangements. But it was possible that there might be — oh —a dozen different answers.
"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 the danger signal. Obviously, there was 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."
"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 al 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 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 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 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.
The 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 unchanged.
"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 as 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 glimmer 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 that all men had such powers, but might use them only once in a lifetime. In Karl, the gift was exceptionally well developed; form that moment onward, Duncan had learned to take seriously even his most outrageous speculations.
That was twenty years ago; whatever had happened to that little set of plastic figures? He could not remember when he had last seen it.
But here it was again, reincarnated in colored minerals — the peculiar rose-tinted granite from the Galileo Hills, the obsidian of the Huygens Plateau, the pseudomarble of the Herschel Escarpment. And there — it was unbelievable, but doubt was impossible in such a matter — was the rarest and most mysterious of all the gemstones found on this world. The X of the puzzle was made of Titanite itself; no one could ever mistake that blue-black sheen with its fugitive flecks of gold. It was the largest piece that Duncan had ever seen, and he could not even guess at its value.
"I don't know what to say," he stammered. "It's beautiful — I've never seen anything like it."
He put his arms around Grandma's thin shoulders — and found, to his distress, that they were quivering uncontrollably. He held her gently until the shaking stopped, knowing there were no words for such moments, and realizing as never before that he was the last love of her empty life, and he was leaving her to her memories.
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