A student keeps subtracting consecutive odd numbers from 121: 121 − 1 = 120, then 120 − 3 = 117, then 117 − 5 = 112, and so on until 0 is reached. How many subtractions does it take?

Useful patterns to spot a perfect square (and reject non-squares):

- **Last digit**: a perfect square ends in 0, 1, 4, 5, 6, or 9 — NEVER in 2, 3, 7, or 8. So 142 cannot be a perfect square (ends in 2).
- **Number of zeros at the end**: must be even. 100 is a perfect square $(10^2)$; 1000 is not.
- **Squares of even numbers are even; squares of odd numbers are odd.**
- **Sum of consecutive odd numbers**: 1 + 3 + 5 + … + $(2n-1)=n^2$. Try it: $1+3+5=9=3^2$.

These let you reject many candidates without doing any heavy work.