N-dimensional Sequential Move Puzzle - 3x3 2D Square

3x3 2D Square

Geometric shape: square

Interestingly, a 2-D Rubik type puzzle can no more be physically constructed than a 4-D one can. A 3-D puzzle could be constructed with no stickers on the third dimension which would then behave as a 2-D puzzle but the true implementation of the puzzle remains in the virtual world. The implementation shown here is from Superliminal who quite perversely call it the 2D Magic Cube.

The puzzle is not of any great interest to solvers as its solution is quite trivial. In large part this is because it is not possible to put a piece in position with a twist. Some of the most difficult algorithms on the standard Rubik's Cube are to deal with such twists where a piece is in its correct position but not in the correct orientation. With higher-dimension puzzles this twisting can take on the rather disconcerting form of a piece being apparently inside out. One has only to compare the difficulty of the 2×2×2 puzzle with the 3×3 (which has the same number of pieces) to see that this ability to cause twists in higher dimensions has much to do with difficulty, and hence satisfaction with solving, the ever popular Rubik's Cube.

Achievable combinations:

The centre pieces are in a fixed orientation relative to each other (in exactly the same way as the centre pieces on the standard 3×3×3 cube) and hence do not figure in the calculation of combinations.

This puzzle is not really a true 2-dimensional analogue of the Rubik's Cube. If the group of operations on a single polytope of an n-dimensional puzzle is defined as any rotation of an (n – 1)-dimensional polytope in (n – 1)-dimensional space then the size of the group,

• for the 5-cube is rotations of a 4-polytope in 4-space = 8×6×4 = 192,
• for the 4-cube is rotations of a 3-polytope (cube) in 3-space = 6×4 = 24,
• for the 3-cube is rotations of a 2-polytope (square) in 2-space = 4
• for the 2-cube is rotations of a 1-polytope in 1-space = 1

In other words, the 2D puzzle cannot be scrambled at all if the same restrictions are placed on the moves as for the real 3D puzzle. The moves actually given to the 2D Magic Cube are the operations of reflection. This reflection operation can be extended to higher-dimension puzzles. For the 3D cube the analogous operation would be removing a face and replacing it with the stickers facing into the cube. For the 4-cube, the analogous operation is removing a cube and replacing it inside-out.