n-dimensional sequential move puzzle
The Rubik's Cube is the original and best known of the three-dimensional sequential move puzzles.
It is a natural extension to create sequential move puzzles in more than three dimensions.
Although no such puzzle could ever be physically constructed, the rules of how they operate are quite rigorously defined mathematically and are analogous to the rules found in three-dimensional geometry.
As with the mechanical sequential move puzzles, there are records for solvers, although not yet the same degree of competitive organisation.
For comparison purposes, the data relating to the standard 33 Rubik's cube is as follows; Number of achievable combinations
There is some debate over whether the face-centre cubies should be counted as separate pieces as they cannot be moved relative to each other.
In this article the face-centre cubies are counted as this makes the arithmetical sequences more consistent and they can certainly be rotated, a solution of which requires algorithms.
However, the cubie right in the middle is not counted because it has no visible stickers and hence requires no solution.
Arithmetically we should have But P is always one short of this (or the n-dimensional extension of this formula) in the figures given in this article because C (or the corresponding highest-dimension polytope, for higher dimensions) is not being counted.
The Superliminal MagicCube4D software implements many twisty puzzle versions of 4D polytopes including N4 cubes.
Superliminal Software maintains a Hall of Fame for record breaking solvers of this puzzle.
An essential feature of the Roice's implementation is the ability to turn off or highlight chosen cubies and stickers.
Roice maintains a Hall of Insanity for record breaking solvers of this puzzle.
Andrey Astrelin's Magic Cube 7D software is capable of rendering puzzles of up to 7 dimensions in twelve sizes from 34 to 57.
The puzzle is rendered in only one size, that is three cubies on a side, but in six colouring schemes of varying difficulty.
This large number of colours adds to the difficulty of the puzzle in that some shades are quite difficult to tell apart.
The full list of colouring schemes is as follows; The controls are very similar to the 4-D Magic Cube with controls for 4-D perspective, cell size, sticker size and distance and the usual zoom and rotation.
Gravitation3d has created a "Hall of Fame" for solvers, who must provide a log file for their solution.
Its derivation assumes the existence of the set of algorithms needed to make all the "minimal change" combinations.
The implementation shown here is from Superliminal who 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.
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 reflection operation can be extended to higher-dimension puzzles.
For the 4-cube, the analogous operation is removing a cube and replacing it inside-out.
Another alternate-dimension puzzle is a view achievable in David Vanderschel's Magic Cube 3D.
The 3D analogue of this is to project the cube on to a 1-dimensional representation, which is what Vanderschel's program is capable of doing.
Vanderschel bewails that nobody has claimed to have solved the 1D projection of this puzzle.
