We calculated the square root of every four-digit number, and if we did not get an integer, we rounded it to the nearest integer. Did we round up or down more often?
Solution. Rounding is necessary if the square root of an integer $k$ falls strictly between $n$ and $n+1$ (where $n$ is a positive integer), that is,

$$
n<\sqrt{k}<n+1
$$

Squaring both sides, this precisely means that

$$
n^{2}<k<(n+1)^{2}=n^{2}+2 n+1
$$

or equivalently,

$$
n^{2}+1 \leq k \leq(n+1)^{2}-1=n^{2}+2 n
$$

Since

$$
n^{2}+n<\left(n+\frac{1}{2}\right)^{2}=n^{2}+n+\frac{1}{4}<n^{2}+n+1
$$

the value of $\sqrt{k}$ should be rounded down when $n^{2}+1 \leq k \leq n^{2}+n$, and rounded up when $n^{2}+n+1 \leq k \leq n^{2}+2 n$. Therefore, rounding down and rounding up each occur exactly

$$
\left(n^{2}+n\right)-\left(n^{2}+1-1\right)=n=\left(n^{2}+2 n\right)-\left(n^{2}+n+1-1\right)
$$

times.

Thus, for $32^{2}=1024<k<10000=100^{2}$, we will round up exactly as many times as we round down, and no rounding is needed for perfect squares. However, if $31.5^{2}=992.25<1000 \leq k<1024=32^{2}$, then the value of $\sqrt{k}$ should be rounded up. From this, it is clear that we rounded up exactly 24 more times than we rounded down.