Wednesday, June 13, 2012
An equilateral with side $n$ is divided into smaller equilaterals with size 1 so that they form a triangular lattice. Each segment of length 1 is colored either red, blue, or green. A coloring is called "good" if every equilateral has sides of three different colors. How many good colorings are there? Two colorings are called distinct if at least one segment is colored differently. For example, with $n=1$, there are 6 good colorings. With $n=2$, there are 48 colorings.