G2 shall be my favorite group! Well, for now. For a while. Or maybe for longer. It's the smallest of the exceptional Lie groups. Found 1893 by Élie Cartan. It was he who suggested to think about it in terms of rolling balls. https://en.wikipedia.org/wiki/G2_%28mathematics%29
Picture two spheres, one three times larger than the other. Imagine them rolling on another without slipping nor twisting. Rolling surfaces have their own branch of mathematics: contact geometry!
So, let's solve the puzzle: Start at the center s of our, say, white hemiball. After a quarter turn we get to -s the center of the black hemiball. Now all directions point back at s! Any other quarter turn leads back!