I've come across an odd way of estimating the square root of a number, going like this:
- Given a number n,
- Subtract the odd numbers from n in a rising order (1, 3, 5...), until $n \leq 0$
- Count how many numbers you subtracted from n. This is the approximation of the square root.
- $n = 16$
- $n = (16 - 1) = 15$
- $n = (15 - 3) = 12$
- $n = (12 - 5) = 7$
- $n = (7 - 7) = 0$
- We have subtracted 4 odd numbers. Since $\sqrt16 = 4$, the approximation works.
(This script is an implementation in Python.)
However, try as I might, I can't understand why this works. Is there a good explanation available?