Seeing Red - Futility Closet

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?

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Credit for the solution goes to Amy Kuiper of Michigan’s Alma College and to the Grand Valley State University Problem Solving Group of Allendale, Mich. The face that receives our final scrutiny might be any one of the faces; all are equally likely. So the probability that it’s red is just $\displaystyle \frac{\textup{the number of red faces}}{\textup{the total number of faces}} = \frac{6n^{2}}{6n^{3}} = \frac{1}{n}.$ (Clayton W. Dodge, “Problem Department,” Pi Mu Epsilon Journal 11:1 [Fall 1999], 47-63. Gebhardt’s is Problem 948, on page 56.)

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