This summer I developed a minicourse for high school students in group theory. We discussed the game of 15, SET, walpaper patterns, and Rubik's cube. Maybe this could be helpful:
www-math.mit.edu/~etingof/groups.pdf
in particular, we used this diagram for recognition of the 17 plane patterns:
www-math.mit.edu/~etingof/Patternsdiagram.pdf