### 36th Putnam 1975

**Problem B2**

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.

**Solution**

A slab thickness d intersects a sphere radius R in a volume less than πR^{2}d. So the entire set of slabs fill less than πR^{2}D 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

jscholes@kalva.demon.co.uk

27 Jan 2001