Cutting up space with spherical or cube-shaped cookie-cutters.

[Neil Sloane]

Dear SeqFans, Last year I spent a lot of time working on "Dissecting a Pancake with an Exotic Knife" (arXiv:2511.15864) but that was just for 2-D. Does anyone know anything about 3-D? OEIS A046127 claims to give the max number of regions you can divide space into by drawing n spheres (they can have different radii), but does not give a proof or a construction, nor a reference. What about the analogous questions for a cube or a tetrahedron? Surely these problems must have been studied. Draw n overlapping (hollow) cubes, of different sizes if you want; what is the max number of regions this divides space into? (A000125 does the cutting up a cake with a planar knife problem.)

[Neil Sloane]

PS I was wrong when I said that A046127 does not give any details about the maximum number of regions that 3-D space can be divided into using spherical cookie-cutters. The entry gives a reference (not a link) to Yaglom and Yaglom, and when I dug up my copy, I found that it gives a fairly  full discussion of the solution.

What had misled me was this: The most symmetric solution to the 2-D problem is to arrange n circles in a ring, with each circle cutting all the others. Of course you can't do that in 3-D (you can't arrange n points uniformly around a sphere unless n is small).  But as Yaglom and Yaglom point out, in 2-D you can also arrange n circles on a straight line so that each one cuts all the others, and that you can do in any dimension!

I have updated A046127 with annotated scans of a few pages from Yaglom and Yaglom.

The cube- and regular tetrahedron-shaped cookie cutters are still unsolved.