Magic Cube 2D Applet

by Melinda Green

A Rubik's cube in 2D is a square with sides divided into thirds. Twisting with rotations in 2D is not possible but to make the puzzle interesting we can allow reflections. Although this probably doesn't make it a proper analog to the higher dimensional versions, it is still the same type of puzzle and is close enough to make it interesting to study since its complete state graph is small enough to be easily drawn.

The interactive puzzle is the large square on the left. [Now obsolete] The puzzle is in a scrambled state. Your job is to make each edge a single color. Click an edge to twist that face. The puzzle will beep when you solve it. Refresh the page to scramble it again.

Complete state graph for MC2D

The diagram displays the complete graph of all the patterns that are possible with the puzzle. It is not the particular colors or orientations of the states that is important but the pattern of their colors. 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 it 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 2x2. Who says no one can know the mind of God? It is interesting to wonder what features of this graph will also be found in the much larger state graphs of the familiar higher dimensional twisty puzzles.

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