A man who sells carpets in the covered market can tell you, without pausing, that a square of side fifty-one and a square of side forty-nine differ in area by exactly two hundred.
He does it in his head, and he does not know the two squares themselves. Ask him what fifty-one times fifty-one is and he will shrug. Ask him the difference and he answers before you have found a pencil.
His method is two steps. Take the gap between them — two. Take the sum of them — one hundred. Multiply: two hundred. That is the answer, and it is exact.
He learned it from his father, who sold cloth, and who used it for the same practical reason: offcuts. When a square of cloth is trimmed down to a smaller square, the strip you lose is the difference of two squares, and a man who can price it instantly does not lose money.
The rule is not a market trick. It is the formula every algebra course teaches — the difference of two squares — and it is the reason the middle terms vanish when you multiply two brackets that differ only in a sign.
He was pleased, when a student explained this to him, that it works every time. Twenty-five and twenty-four differ by forty-nine. A hundred and three and ninety-seven differ by one thousand two hundred. «Yes,» he said. «I know. I have never once had to check.»