So, the octahedron was too difficult. In NRW, a state of Germany, the final math exam for the Abitur can now be repeated. A mathematician from the University at Bonn was cited to have said that this was a problem of someone who has this kind of problems as a hobby.
Let us have a look at that problem.
Given was A, B, C and S1 in the image above in Cartesian coordinates. Then the distance from C to the plane through A, B and S1 was asked, exactly in this wording. The question was also phrased as to compute the „thickness“ of the octahedron, which was defined as the distance of two parallel planes.
From my point of view, this is a standard job. It is so standard, that I would have called it mathematically uninteresting in previous blogs. You compute the so-called Hesse form of the plane, and insert the point C. I admit that would have been a real problem, if it were not phrased in exactly the standard form to compute the distance of a point to a plane.
If, however, the Hesse form is not known in NRW schools, nor the analytic way to compute the distance to a plane, the problem becomes a lot more difficult, if not impossible. One idea: If we know or compute that the octahedron is regular, then we can compute the other corners and the distance of the centers of two parallel triangles. Another idea: compute the volume of the thing, and notice, that the volume is four times a tetrahedron with the desired height, and the side triangles as a base. This, of course, is certainly not an easy solution.
The second and last question was to compute the coordinates of two of the corners of the cube. Since the cube is parallel to the axes this not too difficult, but requires some thinking.
Weather or not this problem justifies the repetition of the exam, depends on the knowledge about the Hesse form. Of course, it also depends on the other problems, which I do not know.