In "On random sections of the cube" (https://arxiv.org/pdf/math/9812123.pdf), Yossi Lonke considers the intersection of an n-cube with a randomly-chosen k-plane that goes through the origin (chosen from the unique rotationally-invariant probability measure on k-planes). Lonke shows (among other things) that the average number of vertices of a random 2-dimensional central section of the 3-cube is exactly (24/Pi) arctan (1/Sqrt(2)), or about 4.7. Our "Wall of Fire problem" is different: we average over all nonempty sections, not just the central ones. It can be shown that when k is n-1, the expected number of j-dimensional faces of a random cross-section is the same as the number of j-dimensional faces of the k-cube; moreover, we can restrict to just the k-planes of any particular orientation, and the result still holds. To what extent does this generalize to the case where the codimension is bigger than 1? Here's the first interesting case (I think): Fix a 2-plane in 4-space. Look at all the translates of that 2-plane that intersect a fixed 4-cube. The intersections are polygons. What can be said about the average number of sides? It is not obvious to me that the answer is independent of the original choice of a 2-plane. (Thanks to Neil Sloane for bringing the Lonke article to my attention.) Jim Propp