Why use six colors, when it's easy with five? [MathPages.com] Notice that C(6,3), the number of combinations of 6 colors taken 3 at a time, is 20--the same as the number of faces. Now the problem gets interesting! You may enjoy trying it before looking at my solutions.
In how many ways, distinct under symmetries of the icosahedron
and renaming of the colors, can the edges of
a regular icosahedron be colored with 6 colors so that
all 5 edges meeting at any vertex have different colors, and
each combination of 3 distinct colors occurs on the boundary of some face?
Observation 1. In any such coloring each color must appear on exactly 5 of the 30 edges of the icosahedron. For consider one color, say red. C(5,2)=10 is the number of pairs of non-red colors, and hence the number of distinctly colored triangles with one red edge. Since each red edge borders two triangles, there must be 5 red edges.
Think of the 5 edges of one color as a "tile",
albeit a completely disconnected one. Then we can address the problem
without attention to colors:
find all the distinct shapes of tiles;
find all the ways of tiling the icosahedron with these shapes; and
discard tilings in which 2 faces touch the same 3 tiles.
A Haskell program to carry out this plan revealed just 18 tile shapes and 12 tilings.
One tiling has 3-fold rotational symmetry: PostScript or PDF or HTML (lower quality).
Four tilings have 5-fold rotational symmetry: PostScript or PDF.
The other seven tilings lack symmetry: PostScript or PDF.
Every tiling uses tiles of more than one shape. However the 3-fold symmetric tiling can be seen as 3 congruent bicolor supertiles.
Each of the 18 tile shapes occurs in some tiling.
John Conway found there are three 5-colorings of the edges of a dodecahedron such that each of the twelve 5-color necklaces, distinct under rotation and reversal, occurs around some face. These colorings are dual to 5-colorings of the edges of an icosahedron in which every sequence of 5 colors occurs clockwise or else counterclockwise about exactly one vertex.