Rive and Yoke

Rive went first. "Finding a number's factors is finding every way it rives into equal rows. Thirty beasts — pens of two? Fifteen pens, clean: two is a factor. Pens of three? Ten pens, clean: three is a factor. Pens of five? Six pens: five is a factor." He was thorough and slow, testing each pen-size in turn, and the children began to see that a factor was simply a pen-size that used up the whole herd with nothing standing loose. But it was slow work, trying every possible pen-size one after another, and a child near the back asked, plaintively, whether there was a way to know where to start instead of testing every number blindly. Rive paused — because he had spent years doing exactly that blind testing, and had never found a better first move.

Yoke, at the fence beside him, laughed. "There is, and it is my whole trade. Before you try to rive a number at all, ask the easiest question in the world: is it even? Pair the herd off — this beast with that, this with that. If every beast finds a partner, the number is even, and that means—" she paused, and looked at Rive, and something dawned on both at once — "that means two rives it cleanly. An even number can always be split into pens of two. So you never have to test two on an even number. You already know." She showed them with the herd: thirty paired off with none left standing. "Even. So two is a factor — I didn't have to try it, I could see it. And an odd number leaves one beast standing, so pens of two never come out clean — two is not a factor, and you skip it without testing at all."

And then, at the fence, the two of them built the thing that was better than either craft alone — and had the children do it, hands on the herd, to feel it work. "Take thirty," Rive said, "and find its factors the fast way. Yoke — the easy question first." Yoke had them pair the herd: even. "So two is in," said Rive, "and I can halve it — thirty splits into two fifteens, so now I only have to understand fifteen." He set fifteen apart. "Yoke — is fifteen even?" They paired and found one left standing. "Odd," said Yoke. "So skip two entirely — don't waste a moment." "Then I try three," said Rive, "and fifteen rives into three fives — and five and three are stubborn, un-riveable primes, so we're done." The children looked at what they had built: thirty, taken all the way apart into two, three, and five, and they had never once tested a pen-size blindly. Every hard split had been aimed by an easy question. Ask if it's even. If yes, two's a factor — halve it. If no, skip two and move on. The pairing-glance had become the first, fastest move of the factor-hunt — the parity test standing at the front door of factorization.