Circle and Echo

Circe drew a perfect nonagon on a fresh sheet of paper. She liked to start with a clean version of the problem. "Okay," she said, her voice clear and sure. "The simplest way to cut a cake is from the center, out to the corners. Like a pizza." She drew faint lines from the middle of her shape to each of the nine corners.

She pointed with the tip of her pencil. "We could give seven of those slices to the seven yetis. Then we'd have two slices left over." She looked at Edie, expecting a nod. It was a logical start. A complete, circular thought.

Edie tilted her head. "So, what you're saying is we should slice it like a pizza and give one slice to each yeti, which leaves two extra slices," she said, tracing Circe's lines in the air with her finger. Her voice was thoughtful, like she was trying the idea on for size.

"Yes, exactly," Circe said.

"Okay," Edie continued. "I hear that. And it's a fast way to get started. But the problem is that the two leftover slices need to be divided up between the seven yetis. And cutting up those two little triangles into seven perfectly equal bits seems... tricky. And messy."

Circe looked at her drawing. Edie was right. Those two leftover slices would have to be crumbled into seven tiny, equal piles. A crumb-based disaster. Not neat at all.

Edie picked up her own pencil, which was not as sharp as Circe's. "So, you showed that slicing it from the center makes leftover pieces that are hard to share," she began, making sure she had it right. Circe nodded.

"Right," Edie said. "So what if we don't start from the center? What if we try to make the cake itself easier to divide? It has nine sides, and we have seven yetis. Nine and seven don't play well together."

She drew a wavy line on her own paper, just for something to do. "What if we just... cut two of the sides off?" she suggested. She drew two heavy lines on the drawing, lopping off two of the nonagon’s corners. "Look. If we cut these bits off, the big piece left in the middle is a heptagon. A seven-sided shape."

She smiled. "Then we could cut that from the center, and it would make seven perfect, equal slices for the seven yetis!"

Circe looked at the new drawing. It was a bold idea. A seven-sided cake was much easier to divide for seven yetis. The logic was strong. But a new problem appeared, sitting right there on Edie’s paper: two lonely, triangle-shaped pieces of cake that had been cut off.

"Okay, I hear you," Circe said, picking up the thread. "You're saying that if we cut off two corners, we're left with a seven-sided cake that's easy to divide. That's really clever, Edie. It turns the hard part of the problem into an easy one."

Edie beamed. "Exactly!"

"But," Circe added gently, tapping the two cutoff pieces on Edie's drawing, "we still have these leftovers. We can't just throw them out. The yetis would know. They can smell wasted cake from a mile away."

Edie's smile faltered. "Oh. Right. So we have the same problem as before. Leftovers."

"So, your idea of changing the shape is smart, but we still have to divide the leftovers," Circe restated. She paused, looking at the perfect nonagon she had drawn earlier. "And my idea of slicing from the middle was simple, but the slices weren't easy to share. What if we combine them?"

She pointed to her own drawing. "What if we don't cut all the way to the center? What if we draw a smaller nonagon inside the big one?" She sketched it quickly. A small, nine-sided shape floated in the middle of the larger one. Now, instead of nine triangles, there were nine trapezoids—long, four-sided pieces—pointing inward. "Now we have nine identical pieces," she said.