In a spatial rectangular coordinate system, a sphere centered at the origin has a radius of 3 units. How many lattice points lie on the surface of the sphere?
Let's take a point $P$ on the surface of the sphere, with coordinates $x, y, z$. We know that the radius of the sphere is 3 units, so using the Pythagorean theorem, we can write the following equation:

$$
x^{2}+y^{2}+z^{2}=9
$$

We need to find the integer solutions to this equation. The equation holds if one of the coordinates is +3 or -3, and the other two are 0, that is,

$$
x= \pm 3, y=0, z=0, y= \pm 3, z=0, x=0, z= \pm 3, x=0, y=0
$$

This gives us a total of 6 possibilities.

If none of the coordinates is 3, then the absolute value of two of them must be 2, and the third coordinate must be +1 or -1. The two coordinates with an absolute value of 2 can be chosen in 3 different ways. For example, if $x=y= \pm 2$ (this gives 4 possibilities) and $z= \pm 1$, this results in 8 solutions. Similarly, it is possible that $y=z= \pm 2$ and $x= \pm 1$ or $x=z= \pm 2$ and $y= \pm 1$, which gives a total of 24 cases.

Thus, a total of 30 lattice points fall on the surface of the sphere.

Mult Attila (Békéscsaba, Rózsa F. Gimn., 9. o.t.)