Problem of the Week 1156

A 3-Dimensional Chessboard

The normal chessboard coloring of the entire plane has the property that every square has four white and four black neighbors (neighbors include diagonal ones).

Consider a 3-dimensional board made from cubes and colored in the usual alternating black-white pattern; each cube has 12 same-colored and 14 opposite-colored neighbors.

Is it possible to assign white or black to each cube so that each has 13 white and 13 black neighbors?

Source: From the nice new problem book, A Mathematical Orchard, Problems and Solutions, by Mark Krusemeyer, George Gilbert, and Loren Larson, MAA Problem Book Series, 2012.

© Copyright 2012 Stan Wagon. Reproduced with permission.

8 October 2012