Seeing Red

University of Waterloo mathematician Ross Honsberger chose this problem for his 2004 collection Mathematical Delights; it’s a generalization of a problem that Robert Gebhardt had offered in the Fall 1999 issue of Pi Mu Epsilon Journal. Paint the outside of an n × n × n cube red, then chop it into n3 unit cubes. Put the unit cubes in a box, mix them up thoroughly, withdraw one at random, and throw it across a table. What’s the probability that it comes to rest with a red face on top?