### 45th Putnam 1984

**Problem A1**

S is an a x b x c brick. T is the set of points a distance 1 or less from S. Find the volume of T.

**Solution**

Answer: abc + 2(ab + bc + ca) + π(a + b + c + 4/3).

*Easy.*

T comprises: S, volume abc; a slab on each face, two volume ab, two volume bc, two volume ca; a quarter cylinder on each edge, four with length a, total volume πa, four length b, total volume πb, and four length c, total volume πc; and an eighth sphere radius 1 at each vertex, total volume 4/3 π.

45th Putnam 1984

© John Scholes

jscholes@kalva.demon.co.uk

3 Nov 1999

Last corrected/updated 25 Nov 03