Imperial Earth

“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.” 91 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 hydro methane atmosphere.”

This was such an elementary fact of life-indeed an ecological platitude-that Dqmcan 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 pi 7, 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 forgiven

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 31 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….

A CROSS OF TITANITE

I I You’re old enough now, Duncan, to understand this game … though it’s very much more than a game.1p

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 a

“Then 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?”

GUm … 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 33 making the twelve pieces fit this rectangular box, even though it was the right size.

“I’ll leave you to it,” 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 begi

“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 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 answerl” said Duncan, rather lamely.

“The answer?” retorted Grandma. “Would you care to guess how many diflerent ways those 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.