## Tri-colouring a triangular grid

### Problem 189

Published on Friday, 11th April 2008, 05:00 pm; Solved by 1197Consider the following configuration of 64 triangles:

We wish to colour the interior of each triangle with one of three colours: red, green or blue, so that no two neighbouring triangles have the same colour. Such a colouring shall be called valid. Here, two triangles are said to be neighbouring if they share an edge.

Note: if they only share a vertex, then they are not neighbours.

For example, here is a valid colouring of the above grid:

A colouring C' which is obtained from a colouring C by rotation or reflection is considered *distinct* from C unless the two are identical.

How many distinct valid colourings are there for the above configuration?