How many perfect squares lie strictly between 100 and 1000?

The smallest square above 100 is $11^2=121$ and the largest below 1000 is $31^2=961$, because $32^2=1024$. The roots therefore run 11, 12, …, 31, and that is 31 − 11 + 1 = 21 numbers. Subtracting without adding one back gives the off-by-one answer 20; counting $10^2$ and $32^2$ as well would give 23.