## Hexagons with Directed Colored Paths

Take a hexagon and connect pairs of their edges by three paths. If the paths are directed and
colored with three different colors you get 120 distinct pieces. Jared McComb suggested this set and wanted
to construct some figures with it.

Instead of hexagons with directed paths you can also use hexagons with six different colored edges. In this case
the colors of adjacent edges in a construction are determined by three pairs of colors chosen
out of the six given ones. Unfortunately a construction with same colors at adjacent edges can't be transformed
into a construction with matching color pairs. Therefore new solutions are needed.

I cut hexagons from hard foam board, printed pictures of the pieces on self-adhesive foil, and stuck them
on the hexagons. The SVG-file with the pieces is here and the physical pieces
are shown below.

The star shown above is rather easy to solve, because there are many edges at the border with no color constraints.
For some other figures, symmetric under 180 degree rotation I used a different appraoch. Two pieces are dual if only
the direction of the red paths are different. Using only one piece of each pair for half the figure the other half is given
by the remaining pieces if the common edges of both halves match.

Two congruent figures can also be received by using the concept of dual pieces.

For figures with no symmetry under 180 degree rotation this method failed.
Nevertheless I got a solution for a propeller shown below.
A solution for a triangle of size 15 is here,
but for some rings with triangular symmetry a solution seems to be difficult.

Index