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