Circle and Echo

"Yes."

"And the two wedges we have left over each need to be divided into seven exactly equal portions. So that nobody — not the yeti who got their wedge first, not the yeti who only gets a little crumb from the leftover wedges — receives even a crumb more than anyone else."

"Yes."

Edie looked at the drawing for a long time. "I want to say something about that," she said, "but I want to say it carefully, because I don't want you to think I am dismissing your idea. Your idea is the right place to start. I am restating it so I am sure I understand what you're proposing before I try to add to it."

"I am paying attention," Circe said.

"What I want to say is: the moment we start dividing those two leftover wedges into seven equal pieces, the geometry stops being friendly. A wedge is a triangle. Cutting a triangle into seven exactly equal pieces is — geometrically possible, but the kind of possible that takes a ruler and a protractor and a lot of trust."

"That is true."

"So I'm not sure my objection is that your idea is wrong," Edie said. "My objection is that your idea solves the easy part of the problem and pushes the hard part of the problem one step further down the road. Which is a normal thing for an idea to do. It just means we need to ask whether there is a different first move that doesn't push the hard part down the road."

Circe was quiet for several seconds. She was not offended. She was, in a way she had been learning to recognize over the past two years of these afternoons, grateful. There was a thing Edie did, that Circe's own mind did not naturally do, which was to receive an idea fully — completely — and then ask, is this the version of the problem we want to keep solving? Circe's instinct was to refine an idea until it worked. Edie's instinct was to ask whether the idea had pointed them at the right shape of question.

"All right," Circe said. "What if we don't start from the center?"

Edie picked up her own pencil, which was a perfectly ordinary, somewhat blunt pencil that had a small bite mark on the eraser end.

"So what you're saying is," she began — and Circe smiled, because she had begun to find this small ritual genuinely useful — "we should look at a different first move. Something that doesn't leave us with leftover wedges that nobody knows how to share."

"Yes."

"All right." Edie tapped her pencil against the drawing. "We have a nonagon. Nine sides. We have seven yetis. Nine and seven are coprime, which is a polite way of saying they have nothing in common. There's no factor we can pull out to make them play nicely. So instead of starting with the nine, what if we start by giving ourselves a seven?"

She drew, on the same paper, a small but firm pair of lines. Two of the nonagon's nine vertices were now sliced off. The remaining shape — Circe traced it with her eyes — was, in fact, a heptagon. Seven-sided. Seven was the number of yetis.

"If we trim two corners off," Edie said, "the inner shape is a heptagon. And a heptagon, sliced from its center, gives us seven equal wedges. Seven yetis, seven wedges, one each. Clean."

Circe looked at the new diagram. It was a beautiful idea. Beautiful in the way Edie's ideas often were — not surgical, but generous. Reshape the problem before you solve it. Circe had once read, in a footnote to a translation of Euler, that the great mathematicians distinguish themselves not by solving the hardest problems but by selecting the right form of those problems.

She was about to say so.

Then she saw the two trimmed corners, sitting in the corner of the drawing like discarded triangle-shaped offcuts, and she felt a small, familiar dissonance.

"Echo," she said quietly. "I want to restate what you just proposed before I add to it. Because you've done me the courtesy of restating my ideas, and I want to do the same."

"All right."

"What you're proposing," Circe said, "is that we change the geometry of the cake itself. We turn it from a nonagon into a heptagon by cutting off two of its corners. Inside the heptagon we make seven clean wedges, one for each yeti, and that solves the hard part of the original problem — the part where nine wedges can't be divided cleanly among seven yetis."

"Yes."

"The thing your idea does not yet solve," Circe said, gently, "is what happens to the two corners we trimmed off."

Edie put her pencil down. "Ah."

"The yetis will know."

"The yetis can smell wasted cake from a mile away."

"They can. So we are now in the same kind of situation as before. Your first move solved the easy part of the problem and pushed a different hard part one step down the road. The hard part is now: how do we divide two triangle-shaped offcuts equally among seven yetis."

"Which is just as hard as dividing two wedge-shaped offcuts among seven yetis."

"Yes."

The two friends sat with this for a moment, looking at each other across the table with the particular companionable resignation of people who have spent enough afternoons together to know that being stuck is a normal stage of the work and not a moral failure.

"All right," Edie said. "I think you're going to tell me that what we actually want is a different kind of move."

"I think what we want," Circe said slowly, "is a move that doesn't generate leftovers. A move where the very act of cutting the cake produces exactly the right number of equal pieces, with no offcuts."

"What you're saying is — instead of starting from the center, or trimming the outside, we look for a third way."

"Yes."

Circe stared at the nonagon for a long time. Edie watched her stare. Circe had a way of looking at things that was not quite the same as looking. She was, Edie had decided some months ago, a person whose face went very still when her mind was working hardest, which was disconcerting if you did not know her and informative if you did.

"What if we draw a smaller nonagon inside the big one," Circe said.

"Restate that for me."

"What if I take the nonagon," Circe said, "and inside it — concentric, with the same nine vertices radiating out to the same nine corners — I draw a smaller nonagon. So the cake is now made of two pieces: a small central nonagon, and around it, a ring made of nine identical trapezoids."

She drew it as she said it. The nine trapezoids ringed the inner nonagon like the petals of a stiff, geometric flower.

Edie hummed. "So what you're saying is — nine identical pieces around the outside, and one small whole nonagon in the middle."

"Yes."

"And we have seven yetis."

"So each yeti receives one trapezoid. That uses seven of the nine outer pieces. We are left with two outer trapezoids and one inner nonagon."

"Which is still leftovers."