Squaring Numbers in Your Head

Squaring a number like 47 by brute-force multiplication ($47 \times 47$) means carrying digits through a two-line calculation — slow and error-prone to do silently in your head. But 47 is close to the friendly round number 50, and there's a shortcut that turns any squaring problem into "distance from a round number" arithmetic, which is almost always faster.

The identity behind the trick

The shortcut is just the algebraic identity for the difference of squares, rearranged:

$$n^2 = (n+d)(n-d) + d^2 ,$$

where $d$ is any convenient distance you choose. In plain English: pick $d$ so that $n+d$ is a round, easy-to-multiply number (usually a multiple of 10), multiply $(n+d)$ by $(n-d)$ — which is easy because one factor is round — then add back $d^2$ to correct for the fact that $(n+d)(n-d)$ isn't quite $n^2$.

Worked example 1: 47²

Let $n = 47$. Choose $d = 3$ so that $n + d = 50$, a round number. Then $n - d = 44$. Compute:

$$47^2 = (50)(44) + 3^2 = 2{,}200 + 9 = 2{,}209 .$$

$50 \times 44$ is easy (half of 4,400), and adding 9 is trivial — the whole calculation takes seconds instead of a full long multiplication. Compare that to multiplying 47 by 47 digit-by-digit, carrying partial products: the round-base version has no carrying at all.

Worked example 2: 96² using a base of 100

Let $n = 96$, choose $d = 4$ so $n + d = 100$. Then $n - d = 92$.

$$96^2 = (100)(92) + 4^2 = 9{,}200 + 16 = 9{,}216 .$$

Multiplying by 100 is instant, so the entire square reduces to one subtraction (100 − 96 = 4, giving 92) and one small addition. This version of the trick — basing off 100 — is the single most useful case to have automatic, since so many mental-math questions involve numbers in the 80s and 90s.

What this means in practice

This trick is the workhorse behind faster mental math generally — variance calculations, quick standard-deviation estimates, and option-Greek sanity checks all involve squaring numbers, and doing it via a round base is dramatically faster than direct multiplication once it's automatic. The skill compounds: once squaring near 50 or 100 is instant, estimating square roots (the inverse operation) and multiplying two nearby numbers become easy extensions of the same identity. It's worth drilling until it's reflexive — the whole point is to free up mental bandwidth for the actual interview question, rather than burning it on arithmetic that a well-practiced candidate does without conscious effort.