Crisca

"Twenty-three times forty-one," Crisca said, drawing the numbers with a gap. "Step one, the ones: three times one, three — that goes in the ones place. Step two, cross the middle: two times one plus three times four — that's two plus twelve, fourteen — that goes in the tens place. Step three, the tens: two times four, eight — that goes in the hundreds place. So I have eight hundred, fourteen tens, and three ones. Fourteen tens is a hundred and forty, so it carries: eight hundred plus one forty is nine forty, plus three — nine hundred forty-three." Crisca tapped it. "Ones times ones, cross-add the middle, tens times tens. Three placed pieces."

"The criss-cross works," Crisca said, "because when you multiply two two-digit numbers, the answer only ever gets contributions in three spots: the very back from ones-times-ones, the front from tens-times-tens, and the middle from the two ways a ten can meet a one. That middle 'cross' is the step people forget — you have to do it both ways and add them. Once you see it as three fixed positions instead of a stack to build, the paper disappears. You just say three small products in order and let each land in its place, carrying tens forward like always."

The student tried 32 × 21. "Ones: two times one, two. Cross: three times one plus two times two — three plus four, seven. Tens: three times two, six. Six hundred, seven tens, two ones — six hundred seventy-two." They looked up, cautious. "Thirty-two times twenty-one is six hundred seventy-two?" Crisca nodded. "Check it however you like — it holds. The only trap is forgetting the middle needs both crosses. Do both, add them, and place your three pieces back to front." The student did another, murmuring "ones, cross, tens," and it clicked out clean.