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“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.

“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 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 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.

“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