The problem is as follows: Four dwarfs had taken a friend, let's call him Joe, to the fair. After some time they finally found the roller coaster. Everyone was already sitting in their own wagon - except Joe. For some time the four of them drove over the rails and they didn't even notice that they were blocking the only four wagons of this train. There were even more people who wanted to ride along. But the wagons drove and drove up and down the most twisting tracks, always dividing the rails into four sections of equal length. It was only after half an hour and a half that they stopped driving. Enthusiastically they told Joe how great it had been. But he only said to them: "Did you actually notice that despite the hilly terrain you were again and again on a (affine) plane in between? They hadn't noticed. But why is Joe right?
Hint: You may use without proof that the determinant is a continuous map.
How can this be done using the determinant (function). Thanks in advance!
Here’s a corresponding German original. It doesn’t seem to be exactly the one you translated (unless you took a lot of artistic license), but it does show the crucial error in your translation: Where you wrote “you were always on a (affine) plane in between”, the German version says “dass Ihr [...] immer wieder zwischendurch auf einer Ebene wart”, and “immer wieder” means “again and again”, “repeatedly”, not “always”.
Since the dwarves were always at equal distances along the rails, they cyclically permuted their positions every quarter of the ride. Since the permutation $(1234)$ is odd, this changed the sign of the oriented volume of the tetrahedron formed by their positions. Thus, by continuity every once in a while in between that oriented volume must have been zero, which means that the four dwarves must have been on a common plane.