If you make blue, yellow, and hot pink all the same color and black, light blue, and violet all the same color, then the pattern in the mod 6 triangle becomes that of the mod 2 triangle. Watch:
Notice that this identification colors 0, 2, and 4 alike and 1, 3, and 5 alike. Why would this lead to the same triangle as the mod 2 addition?
If you change light blue to royal blue, hot pink to black, and violet to yellow, then the triangle mod 6 becomes the triangle mod 3.
For a larger view of the identifications that transform mod 6 to mod 2 and mod 3, click here.