Problem Solving Group

Here’s another one that we never solved.

Does a polyhedron exist with all of its plane sections triangular?

[Engel, Problem Solving Strategies 12.3.2 #60]

“Associated with each face of a solid convex polyhedron is a bug which crawls along the perimeter of the face, at varying speed, but only in the clockwise direction. Prove that no schedule will permit all the bugs to circumnavigate their faces and return to their initial positions without incurring a collision.” [Peter Winkler, Mathematical Puzzles, p. 82.]