Overlap and the Corner That Was Taken Twice

Overlap is an octopus, and having eight arms has taught her something most creatures never learn: it is very, very easy to grab the same thing twice without noticing. Two arms reach for the same shell from two directions, and suddenly she is holding it once but counting it twice. So Overlap watches corners. Corners are where mistakes hide.

Her puzzle is the opposite of Mayor Tila’s. Tila watches a square grow by b on two sides. Overlap watches a square shrink. You start with a big square, a on every side, and you want the smaller square that is left when each side is shortened by b - a square of side (a - b).

The way you shrink it is to remove a strip. You slice a strip of width b off the right side of the square. Then you slice another strip of width b off the bottom. Two strips gone, two sides shortened. Simple - until Overlap taps the far corner with one careful arm.

“Look here,” she says. “This little square in the corner. The b-by-b one. When I took the strip off the right side - did I take this corner?”

You look. The corner was part of the right strip. “Yes.”

“And when I took the strip off the bottom - did I take this corner again?”

You look again, and your stomach does a little flip, because the corner was part of the bottom strip too. “…Yes.”

“So I removed it twice,” Overlap says, “but it was only ever there once.”

This is the whole trap, and Overlap has seen a hundred people fall into it. To find the shrunken square, they take the big a² and subtract a strip and subtract another strip - that is a² minus ab minus ab, or a² minus 2ab. And they stop there, feeling finished. But they have taken the little corner away two times when it should have gone only once. They have removed too much. The square they have described is smaller than the square that is actually left.

The fix is not to panic and start over. The fix is small and exact: add the corner back one time. That corner is b by b, which is b². Put it back, and the count is honest again. a² minus 2ab - the two strips - plus b², the one corner you overcharged yourself for.

That is (a - b)². It is a² - 2ab + b². The minus in the middle comes from removing two strips; the plus at the end comes from giving back the corner that got removed twice.

Overlap does not feel clever when she finds the double-count. She feels relieved - the specific, loosening relief of catching an error before it costs you, of realizing “wait, I paid for that twice” and getting the extra back. Anyone who has ever untangled a bill, or freed an arm from grabbing the same shell twice, knows the feeling. It is not triumph. It is the quiet exhale of now the books are square.

“People think the plus b² is strange,” she says, coiling comfortably. “Why add, when you’re making the square smaller? But it isn’t strange at all. You over-removed. Adding it back is just honesty. You only ever get to take a thing away as many times as it was really there - and my corner was only ever there once.”

And Overlap settles into the small b-by-b corner she rescued, the one that belongs to two strips but counts for one, exactly where a careful octopus knows it should be.