Let the face of the $2 \times2$ Rubik's cube be defined as the ordered quadruple of colors of it's pieces in a fixed direction (let's say counterclockwise). Does the set of $6$ quadruples define a unique cube state? Of course the state has to be solvable and color scheme of the solved state is fixed.
At first I thought this is trivially true, but I failed to prove it and found that it's far from trivial.