Let T be a equilateral triangle with each side having length 1. Imagine T in a fixed position in the plane, say with the bottom side on the x-axis and the opposite angle above it. Let S be the set of coloured triangles obtainable from T by painting each side with one of the colours red and blue. Any combination of colours is allowed, for example all sides could have the same colour. Note that S has 8 elements: for example the bottom side being red and all other sides being blue is a different painting than the leftmost side being red and all other sides being blue. Define a relation R on S by s1 R s2 if and only if si can be rotated so that the rotated coloured triangle is identical to s2. Prove that R is an equivalence relation and find the equivalence classes. (The elements of your sets can be pictures of the coloured triangles.)

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