Halvor

"Sixteen times five," Halvor said. "Sixteen is even, so I halve it — eight. And whatever I take from one number, I hand to the other, so I double the five — ten. Now I've got eight times ten." Halvor spread their hands as if it were obvious. "Eighty. And eight times ten is a sum a little kid can do. I turned an ugly sixteen-times-five into a friendly eight-times-ten, and the answer didn't budge, because halving one side and doubling the other keep the product exactly the same."

"The reason it works," Halvor explained, "is that multiplying is really about how much stuff you have in a rectangle — so many rows of so many. If I make half as many rows but twice as long, the rectangle holds the same amount. I've just reshaped it. So whenever one number is even and awkward, I halve it toward something friendly and double the other to pay for it. Fourteen times five? Halve to seven, double to ten — seventy. Twelve times fifteen? Halve to six, double to thirty — one hundred eighty. I keep trading until the numbers are kind, and the answer is safe the whole time because the trade is always fair."

The student tried 18 × 5. "Eighteen is even — halve it, nine. Double the five — ten. Nine times ten... ninety." They stared. "Eighteen times five is ninety? That was nothing." Halvor grinned. "Now push it further when you can. Twenty-four times five — halve, twelve, double, ten — a hundred twenty. Or halve twice: twenty-four to twelve to six, and double five twice, five to ten to twenty — six times twenty, still a hundred twenty. Trade as many times as you like. It never costs anything."