Let's find a point that is at distances $m, n, p$ from three planes. How many solutions does this problem have?
We construct parallel planes at distances $m, n, p$ from the three given planes. The intersection point of these planes will be the desired point.

For each plane, we can construct the corresponding parallel planes on both sides, and if we denote these by $a_{1} a_{2}, b_{1} b_{2}, c_{1} c_{2}$, then we can look for the intersection points of the planes $\left(a_{1} b_{1} c_{1}\right),\left(a_{2} b_{2} c_{2}\right),\left(a_{1} b_{2} c_{2}\right),\left(a_{2} b_{1} c_{1}\right),\left(b_{1} a_{2} c_{2}\right),\left(b_{2} a_{1} c_{1}\right)\left(c_{1} a_{2} b_{2}\right)$ and $\left(c_{2} a_{1} b_{1}\right)$. Therefore, the problem has eight solutions.

(Béla Enyedi, Budapest.)

The problem was also solved by: Bartók I., Deutsch E., Deutsch I., Haar A., Hirschfeld Gy., Kertész G., Liebner A., Pazsiczky G., Riesz K., Sonnenfeld J.