It is known that the planes of a tetrahedron divide space into 15 parts. Of these parts, what is the maximum number that a line can intersect?
Solution. A line passes from one spatial region to another precisely when it penetrates at least one plane. This can happen at most four times, because a tetrahedron has four faces. The four can indeed be achieved if we choose a line that is not parallel to any face and does not intersect any edge of the tetrahedron. Every such line intersects five spatial regions, so the answer to the problem is: five.

We can easily find a suitable line in the following way. Let $P$ be an interior point of the $BCD$ face of the tetrahedron $ABCD$. The lines $AB$ and $AC$ divide the plane $ABC$ into four regions. In the region opposite to the one containing triangle $ABC$, choose a point $Q$ that does not lie on the plane $APD$. It can be seen that the line $PQ$ intersects all four planes but does not intersect any of the lines $AB, AC, AD, BC, BD, CD$.