Simple Arithmetic

The smallest perfect square between 100 and 1,000 inclusive is 100 itself, since 10² = 100. The largest can be found by noting that ${\displaystyle \sqrt{1,000}\approx <!--\; \approx \; -->31.6}$; this makes ${\displaystyle {31}^2=961}$ the greatest perfect square in this range.

Since the squares of the integers from 10 to 31 all fall in this range, this makes 31−10+1=22 perfect squares.