Magic Cube 2D

A Rubik's cube in 2D is a square with sides divided into thirds. Twisting a 1D face simply flips its direction along with the attached stickers of it's neighboring faces. The resulting puzzle has only 24 states, and no state is further than 4 twists from solved. What makes it interesting is that it is in the same class as it's larger siblings yet small enough to study fully.

The interactive puzzle is the large square on the left. The puzzle is in a scrambled state. Your job is to make each edge a single color. Click an edge to twist that face.

Scramble Reset

Unfortunately, even with just 24 states, the complete state graph is still too tightly interconnected to display well on a flat diagram. However the diagram on the right displays a map of all the patterns that are possible with the puzzle. In other words, it is not the particular colors or orientations of the shown states that is important but the pattern of their colors. In other words, this puzzle has only 8 unique patterns. State 0 is the solved state and state 2 is the checkerboard. The solid lines are transitions that are possible with this puzzle. The dashed lines are twists that would be possible if the puzzle allowed twisting middle slices as a single move. As you can see, you can always perform a middle slice twist move with two regular twists. You can think of this map as the puzzle's solution. If you can find a particular scrambled state on the graph you should be able to see which twists will allow you to return to the solved state.

With this graph it is easy to see that four twists is the greatest number you will ever need to solve it from any position, or three if middle slice twists are allowed. This is nothing less than God's algorithm for the 3x3. Who says no one can know the mind of God? It is interesting to wonder what features of this map will also be found in the much larger state graphs of the familiar higher dimensional twisty puzzles.