Here are the triangles mod n for n = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.
Look at the triangles for n = 2 and n = 4. See how similar they are? If you were to recolor the yellow circles as royal blue and the light blue circles as black, the n = 4 triangle would be the same as the n = 2 triangle. Why do you think that might be?
Now look at the n = 6 triangle. Can you see a color identification that would transform it to the n = 2 triangle? How about to the n = 3 triangle? Answers
Next look at the n = 8 triangle. Can you make an identification that would transform it to the n = 4 triangle? You could then make another identification to transform it to the n = 2 triangle. Answers
Can you see identifications between other triangles? What is the pattern here? Answers