Anathem

CALCA 1: Cutting the Cake

A supplement to Anathem by Neal Stephenson

“LET’S SAY THAT EACH serving will be a square, the same width as the spatula. Go ahead and cut in one corner of the pan.”

Dath cut the cake thus:

and then made more cuts thus, to produce the four servings I’d asked for:

“I can’t believe you’re doing this!” Arsibalt muttered.

“If it worked for Thelenes…” I muttered back. “Now shut up,” and I turned my attention back to Dath who was awaiting further instructions. “How many servings do we have there?” I asked him.

“Four,” he said, slightly unnerved by my ridiculously easy question.

“Now, what if you cut a similar figure but with sides twice as long? So instead of each side being two units-two spatula-widths-it would be-?”

“Four units?”

“Yes. We have four servings here already-if you doubled the size of the figure, how many people could we serve then?”

“Well, two times four would be eight.”

“I agree that two times four is eight. Go ahead and try it,” I said. Dath made more cuts thus:

Halfway through, he saw his error and made a wry face, but I encouraged him to keep going until he was finished. “Sixteen,” he said. “We actually have sixteen servings. Not eight.”

“So, just to review: when we cut a square grid that is two units on a side, we get how many servings?”

“Four.”

“And you just told me that a four-unit grid gives us sixteen. But what if we only wanted eight servings? How many units would our grid have to be?”

“Three?” Dath said, cautiously. Then his eyes dropped to the cake and he counted it out. “No, that gives nine servings.”

“But we’re getting warmer. And now an important thing has changed, which is that you know you don’t know.”

Dath’s eyebrows went up. “That’s important?”

“It’s important to us in here,” I said.

I couldn’t remember what Thelenes had done next when he had done this with a slave-boy on the Plane six millennia ago, and had to ask Orolo.

I spun the cake around, presenting Dath with an unmarked corner. “Go ahead and cut one square big enough for four servings. You don’t have to cut the individual servings out of it.”

“Can I make lines on the frosting?” he asked.

“If it helps.”

With some hints and nudges from Cord, Dath produced a square like this:

“Good,” I said, “now add three more squares just like it.”

Extending lines he’d already made and adding some new ones, Dath enlarged it to this:

“Now, remind me, how many servings can we get out of that whole area?”

“Sixteen.”

“All right. Now look only at the square in the lower right-hand corner.”

“Is there a way you can divide it exactly in half with only one cut?”

He got ready to slice along one of the dotted lines, but I shook my head. “Arsibalt here is very particular about his cake and he wants to be sure no one gets a larger slice than him.”

“Thank you very much, wise Thelenes,” Arsibalt put in.

I ignored him. “Can you make one cut that’s guaranteed to satisfy him? The pieces don’t have to be square. Other shapes are okay-like triangles.”

With that hint, Dath made a cut like this:

“Now, do the others the same,” I said. He made it like this:

“When you made the first diagonal cut, you cut a square exactly in half, right?”

“Right.”

“And is the same true of the other three diagonal cuts and the other three squares?”

“Of course.”

“So, let’s say I rotate the pan and you look at it this way”:

“What shape do you see in the middle there?”

“A square.”

“And how many servings worth of cake are contained in that square?”

“I don’t know.”

“Well, it’s made up of four triangles, right?”

“Yeah.”

“Each of those triangles is half of a small square, right?”

“Right.”

“And how many servings in a small square?”

“Four.”

“So each triangle has enough cake for how many servings?”

“Two.”

“And the square that’s made up from four such triangles has enough cake for-”

“Eight servings,” he said, and then realized: “which is the problem we were trying to solve before!”

“We’ve been trying to solve it the whole time,” I corrected him, “it just takes a minute or two. So, can you cut us eight servings then, please?”

“That’s it,” I said.

“We can eat now?”

“Yes. Do you see what just happened?”

“Uh…I cut eight equal servings of cake?”

“You make it sound easy…but it was hard, in a way,” I said. “Remember, a few minutes ago, you knew how to cut four servings. That was easy. You knew how to cut sixteen. That was easy too. Nine, no problem. But you didn’t know how to cut eight. It seemed impossible. But by thinking it through, we were able to come up with an answer. And not just an approximate answer, but one that is perfectly correct.”