Today we will use the special trillows above to tile the region(or similar ones) below:

Let’s call for the moment the grayish and greenish trillows *simple*, and the brown and purple ones *special*. Here are two puzzles as starters: Can you tile the same region just with the simple trillows? Can you do it using just one special trillow?

As an outline what is involved in creating and solving puzzles, let’s overlay the region we want to tile with a graph as above. The purple tiles will later correspond to special trillows, the brown ones to simple trillows. Let’s get rid of most of the purple tiles. We do so by deleting a yellow bridge between two adjacent purple tiles, and replacing them with brown tiles, keeping the remaining yellow edges. This amounts to partially tiling the purple region with diamonds. By now you should have solved the two puzzles at the beginning.

Now we orient the edges of the yellow graph. This is a bit deliberate. The outer loop we orient counterclockwise, and for the inner graph we make sure that at the two trivalent vertices the not all three edges point in the same direction.

Now we inflate/deflate all triangles according to the direction of the arrows, and obtain a tiling with just two special trillows.

We have used three more gray trillows than green trillows. Can you do it with just one more gray trillow? Or with no green trillows a all? More about this next time.