A slab is the set of points strictly between two parallel planes. Prove that a countable sequence of slabs, the sum of whose thicknesses converges, cannot fill space.
A slab thickness d intersects a sphere radius R in a volume less than πR2d. So the entire set of slabs fill less than πR2D of the sphere, where D is the sum of their thicknesses. If we take R > D this is less than the volume of the sphere, so the slabs cannot even fill the sphere.
36th Putnam 1975
© John Scholes
27 Jan 2001