Contributed by Jim Propp, University of Massachusetts Lowell

Prof. Jim Propp of the University of Massachusetts Lowell has been corresponding with the Prisoner’s Dilemma about planar cross sections of three-dimensional solids. For example, suppose you have a regular tetrahedron (a triangular pyramid on an equilateral triangle base with the proper height so that all of the faces are equilateral triangles). If you pass it through a cutting plane so that one face is parallel to the plane, all of the cross sections are triangles, which does not seem surprising. You can see such a cross section, and Prof. Propp himself, in the image above. However, if you rotate the tetrahedron so that both of a pair of opposite edges are parallel to the cutting plane and then pass it through the plane, all of the cross sections are rectangles, as shown in figure 1.

Can you find an orientation of the tetrahedron that ‘s exactly in-between? In other words, an orientation with respect to a cutting plane so that when you move the tetrahedron at a constant rate perpendicular to that plane, the average number of sides of the cross-section is three and a half?

Prof Propp would contrast these possibilities for a tetrahedron with the situation for another Platonic solid: he has proven that no matter how you orient a cube, when you pass it through a cutting plane in this way the average number of sides of a cross section is always exactly four (even though any number of sides from three to six can occur for a single specific cross section). An octahedron (the solid with eight equilateral triangular faces made by gluing two square pyramids by their square faces) is more like a tetrahedron in this way: if you slice it perpendicular to a line joining two opposite vertices, all of the cross sections are squares, but if you slice it parallel to one of the faces, all of the cross sections (except the lone initial and final ones) are hexagons. What about the other two Platonic solids, the dodecahedron with twelve pentagonal faces and the icosahedron made up of twenty equilateral triangles?

This problem originally appeared in the Prisoner’s Dilemma in the 2022 Fall issue of the PMP Newsletter. Solutions are no longer being accepted for this Dilemma.