Crafty Counting

To count a set means to put it in one-to-one correspondence with a set of integers {1, 2, 3,...,n}. Direct counting is nice, but in complicated situations it pays to be more crafty. A problem with patterns of colored tiles gives us a chance to illustrate a popular counting principle known by various names. We'll call it the Burnside-Cauchy-Frobenius formula. It is also popularly called the Burnside Orbit-Counting Lemma, though wags refer to it as "not Burnside," because it was known long before Burnside was born. Later Pólya generalized the formula, so some readers may recognize this as Pólya Enumeration.