Floppy Cube

This Demonstration introduces the floppy cube as a simplification of the 6-color Rubik's cube; it was first built by Katsuhiko Okamoto. Like other such challenges, the goal is to scramble the puzzle and then return it to a state where all squares are the same color.
A more coded name for the floppy cube is 2-color 3-square. A 2-color 1-square is always solvable and a 2-color 2-square is either solvable or just one twist away from a solution. A 2-color 3-square can be up to six twists away from a solution, and the number of wrongly colored squares does not always decrease with the number of twists. That highlights why puzzles of this kind need to be solved holistically instead of with a step-by-step approach.
In this simplification of the 3×3×1 floppy cube, there are nine 2-sided squares. The "rotate clockwise" control does exactly that to all squares. The "twist top row" control changes XYZ in the top row to Z’Y’X’, where the primes are the reverse opposite-colored sides.

SNAPSHOTS

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DETAILS

A twist of the top row and a quarter-rotation clockwise are the only allowed transformations here. Configurations are labeled by a quaternary code: starting from an all-white square, for each digit from left to right, count the number of quarter-rotations after a single twist. For example, "02" is a top twist followed by a half-rotation and another top twist, effectively realizing a middle twist and a flip. All distinct configurations (rotation and flip invariant) can be selected and then manually transformed back to an all-white square within six twists. There are distinct configurations for a twist number from one to six. Snapshots 1–6 show the transformation from an all-white square to a corner-black square.
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