User blog:Edwin Shade/The Nano-Graham Problem

Given a 3-D cube, how many unique ways are there to color with two colors all 12 edges, such that no coloring is a mirror image or rotated identical to another coloring ?

Suppose you have selected blue and red as your two colors. Logically, since there are 12 edges to color either blue and red on the cube there is an upper bound of $$2^{12}$$, or 4,096 possible ways to color the cube.

However, many of these coloring's are identical. For instance, there are 12 coloring patterns with 11 red edges and 1 blue edge, but since these colored cubes can be rotated in such a way that they appear all the same from the same reference point, they are not truly original coloring's, but merely the same coloring.

The question above asks how many unique coloring's there are. I have found breaking this problem up into a series of smaller ones is helpful. Let us first consider how many unique coloring's there are if we only allow 11 red edges and 1 blue edge to be colored. The answer is one, that is, just a single edge colored in with blue.

With 10 red edges and 2 blue edges there are exactly 9 unique coloring's.

This is as far as I have gotten at the present time, and I feel fairly confident that there are a low enough number of total unique coloring's that they can be drawn out on a single sheet of paper, (using both sides of course).

If you have additional knowledge of this problem or else-thought please leave a comment below.