A Tiling Problem

A while ago, Andy and I were at a maths challenge. There were a few problems with the answers we'd been given to the questions. Andy has already documented one of them, I'm going to comment on the other.

Here's a tiling pattern (imagine, if you will, that it was drawn by someone more competent than me, so all three polygons are regular every time they occur):

Now imagine that we use this pattern to tile the entire plane - so that we can ignore what happens at the boundary. The question is: what is the ratio of hexagons to squares to triangles in the final tiling (actually, the question was "what is the ratio of squares to triangles?", but you might as well do all three). The answer is probably not what you think immediately.

The answers we were given to the problem said that the ratio was 1:1. This is quite a tempting answer the first time you see the pattern, as it looks like theres one triangle per "corner" of a hexagon and one square per "edge". It doesn't take long to see that this approach is wrong.

Here's the first method I tried to work out the right answer - look at what happens at the corners.

At each corner, there is 1 hexagon, 1 triangle and two squares. So the ratio of "hexagon vertices to square vertices to triangle vertices" must be 1:2:1. Fairly straighforward. Now, each hexagon has 6 vertices, each triangle has 3 vertices, and each square has 4, so the ratio of hexagons:squares:triangles must be 1/6:2/4:1/3, or 1:3:2, if you prefer integers.

It's relatively easy, but a bit more boring to perform a simliar analysis using the edges (a bit more boring because there are two "types" of edge, and you have to be a bit careful to figure out what ratio the types of edge occur in. You get the same answer.

I also had several other methods, all of which gave the same (I think you will now agree) correct answer.

When I pointed out that the answer to this question was wrong to the relevant person in a position of authority, she informed that it wasn't, because you can make a shape with squares, triangles and hexagons in the ratio 3:3:1 and use this to tile the plane. This is an interesting approach, and one that I wholeheartedly endorse - the only problem is that the only shape I can make with squares, triangle and hexagons in that arrangement which tiles the plane contains 1 hexagon, 2 triangles and 3 squares... I'll leave it as an exercise to check this pattern actually does tile the plane.

I'm not quite sure what the point of writing this post was - I think I just quite enjoyed the problem and wanted to share it. If there is one, it's probably something to do with using multiple approaches to increase your certainty in your answer, but I really quite like the vertex-based argument, so I don't think that can be it.