A flexi puzzle is a fun little toy, consisting of wood blocks and a string that runs through it.
The pattern of colors is repeated twice; if a color is found at position $N$, it is found a second time at position $N+6$. For example, in the picture above, the color red occurs at positions $1$ and $7$.
You can twist the puzzle along the cutout lines in each square. You can also rotate squares.
Assuming that you can only perform 90° rotations and orthogonal twists, prove whether or not two pieces of the same color can be orthogonally adjacent on a standard flexi puzzle of 12 pieces. A picture or description of such a pattern would suffice as a proof to the positive.