# Rainbow Icosahedra

This puzzle was inspired by playing with a Magz magnetic ball-and-stick toy: how to color (in the usual mathematical sense) the edges of an icosahedron with 6 colors?

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.

Problem. 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.

Observation 2. 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.

#### Some properties of the tilings

No tiling has a symmetry group of even order, despite the abundance of even-order subgroups in the icosahedral group. In particular, no tiling is invariant under reflection, 180-degree rotation, or central inversion coupled with renaming of colors.

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.

#### Related 5-colorings

Branko Grunbaum suggested looking for 5-colorings in which each combination of colors occurs on exactly two faces with the colors in opposite sequence on the two faces. All 13 such colorings have even-order symmetry groups. One example has the full symmetry of the icosahedron: PostScriptPDF.

#### Dual problems

By duality, one can color the edges of a dodecahedron with 6 colors so that every combination of 3 colors meets at exactly one vertex, or with 5 colors so that every sequence of three colors occurs clockwise about exactly one vertex.

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.

Modified February 11, 2006; March 24, 2008.