Squill

"Here's the move," Squill said. "Forty-eight is two below fifty. So I spread it two each way — down to forty-six and up to fifty. Now I multiply the friendly pair: forty-six times fifty. Forty-six times fifty is twenty-three hundred — easy, it's just half of forty-six hundred. Then, because I stepped two away in each direction, I add back the square of that step: two times two, four. Twenty-three hundred plus four — two thousand three hundred and four. That's forty-eight squared." Squill smiled. "Step down and up the same amount to reach an easy multiply, then add the little square of the step."

"The reason the little square comes back," Squill said, "is that when you spread a number out equally — one down, one up — the product dips by exactly the step, squared. So you multiply the easy spread pair, then repay that tiny dip. It sounds fussy, but the beauty is the hard part vanishes: you never actually multiply forty-eight by forty-eight. You multiply fifty by something, which your mind can do half-asleep, and then you add a number as small as four or nine or twenty-five. Squaring near a base means the scary square becomes an easy multiply with a pebble on top."

The student tried squaring 53. "Fifty-three is three above fifty. So I spread it three each way — down to fifty and up to fifty-six. Fifty times fifty-six is... twenty-eight hundred. Then add three squared, nine. Two thousand eight hundred nine." They double-checked and their eyes went wide. "Fifty-three squared is two thousand eight hundred and nine." Squill leaned in. "Exactly — fifty times fifty-six is twenty-eight hundred, then plus nine, two thousand eight hundred nine. Trust the base." The student redid it and grinned. "Two thousand eight hundred nine. I never squared fifty-three. I just spread it around fifty and added nine."