Find the biggest 4-digit number which is a perfect square.

The largest $4$ -digit number is $9999$. Let's estimate its square root:
$$\sqrt{10000}=100$$
This means the largest $4$ -digit perfect square must be the square of a number just below $100$, which is $99$.Let's calculate $99^2$:
$$99^2=(100-1{)}^2=10000-200+1=9801$$
The square of the next integer, $100^2=10000$, is a $5$ -digit number. Therefore, $9801$ is the largest $4$ -digit perfect square.