Take a regular hexagon, and mark points on each edge. Connect these points with the center of the hexagon to obtain six quadrilaterals. If we choose the marked points so that they divide an edge in one of the proportions 1:3, 2:2, or 3:1, there are exactly six such quadrilaterals (not counting mirror symmetric copies), and as you can see, they can tile a hexagon. I will call these hexons, in analogy of a puzzle by Alan Schoen that I will discuss in the near future. Above are mirror images of this solution, print & cut them out, then glue them together back to back.
Here is a first puzzle: Suppose you flip one of the three pieces over (like here the blue one) that is not mirror symmetric, can you still tile the hexagon?
More complicated is the challenge to use seven hexons of each type to tile the seven hexagons above so that corners only meet corners, not just edges of neighboring hexons.
Likewise, can you properly tile the above shape, so that the vertices of the tiles only meet at other vertices?
After playing with these hexons for a while, you will find it tempting and useful to group three of them together along so that they meet at their 120º-vertex. This can be done in 11 possible ways:
You will get inflated equilateral triangles, where an edge is either straight, or has a kink inside or outside. I have discussed some of these triangles before, and the square shaped analogies (which I called pillows) in a long sequence of blog posts. Understanding how these 11 triangles can fit together will help to create and solve many more puzzles for the hexons. We will begin this next week.
The Inner Frame 