N-dimensional sequential move puzzles

The Rubik's cube is the original and most well-known of the three dimensional sequential move puzzles. There have been many virtual implementations of this puzzle in software. 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. Hence, they can be simulated by software. As with the mechanical sequential move puzzles, there are records for solvers, although not yet the same degree of competitive organisation.

Several of the puzzles described in this article have never been solved (as of May 2008). These include the 6×6×6×6×6 (6⁵) 5-cube, the 7⁵ 5-cube and the 120-cell 4D puzzle.

Terms used in this article

*Vertex. A zero-dimensional point at which higher dimension figures meet.
*Edge. A one-dimensional figure at which higher dimension figures meet.
*Face. A two-dimensional figure at which (for objects of dimension greater than three) higher dimension figures meet.
*Cell. A three-dimensional figure at which (for objects of dimension greater than four) higher dimension figures meet.
*"n"-Polytope. A "n"-dimensional figure continuing as above. A specific geometric shape may replace polytope where this is appropriate, such as 4-cube to mean the tesseract.
*"n"-cell. A higher dimension figure containing "n" cells.
*Piece. A single moveable part of the puzzle having the same dimensionality as the whole puzzle.
*Cubie. In the solving community this is the term generally used for a 'piece'.
*Sticker. The coloured labels on the puzzle which identify the state of the puzzle. For instance, the corner cubies of a Rubik cube are a single piece but each has three stickers. The stickers in higher dimensional puzzles will have a dimensionality greater than two. For instance, in the 4-cube, the stickers are three dimensional solids.

For comparison purposes, the data relating to the standard 3³ Rubik cube is as follows;Achievable combinations:

: $\{\}\; =frac\{15!\}\{2\}cdot\; \{left(\; frac\{4!\}\{2\}\; ight)\}^\{14\}\; cdot\; 4$

: $\{\}\; sim\; 10^\{28\}\; ,!$

4⁴ 4-cube

2⁵ 5-cube

6⁵ 5-cube

Achievable combinations:

: $=\; frac\{600!\}\{2\}\; cdot\; frac\{1200!\}\{2\}\; cdot\; frac\{720!\}\{2\}\; cdot\; frac\{2^\{720\{2\}\; cdot\; frac\{6^\{1200\{2\}\; cdot\; frac\{12^\{600\{3\}$

: $sim\; 10^\{8126\},$

This calculation of achievable combinations has not been mathematically proved and can only be considered an upper bound. Its derivation assumes the existence of the set of algorithms needed to make all the "minimal change" combinations. There is no reason to suppose that these algorithms will not be found since puzzle solvers have succeeded in finding them on all similar puzzles that have so far been solved. Nevertheless, as of May 2008, the puzzle has neither been solved nor all the algorithms found needed for the final proof.

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:

: $=4!,!\; =\; 24$

Note that 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.

It is also worth noting that 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 = 8x6x4 = 192,

for the 4-cube is rotations of a 3-polytope (cube) in 3-space = 6x4 = 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.

1D Projection

Another alternate dimension puzzle is a view achievable in David Vanderschel's 3D Magic Cube. A 4-cube projected on to a 2D computer screen is an example of a general type of an n-dimensional puzzle projected on to a (n-2)-dimensional space. The 3D analogue of this is to project the cube on to a 1-dimensional representation, which is what Vanderschel's programme is capable of doing.

Vanderschel bewails the fact that nobody has claimed to have solved the 1D projection of this puzzle. [ [http://games.groups.yahoo.com/group/4D_Cubing/message/330 Vanderschel posting on the 4D Cubing group at Yahoo] ] However, since records are not being kept for this puzzle it might not actually be the case that it is unsolved.

ee Also

*Combination puzzles
*Rubik's cube
*Megaminx
*Rubik's cube group
*Tesseract
*120-cell
*List of Rubik's cube software

References

*H. J. Kamack and T. R. Keane, "The Rubik Tesseract", available online [http://udel.edu/~tomkeane/RubikTesseract.pdf here] .

External links

* [http://www.superliminal.com/ Superliminal]
* [http://www.gravitation3d.com/magiccube5d/anatomy.html Gravitation3d's Anatomy of a d-dimensional Rubik's Cube]

olving records

* [http://www.superliminal.com/cube/halloffame.htm 4D Hall of Fame]
* [http://www.gravitation3d.com/magiccube5d/hallofinsanity.html 5D Hall of Insanity]

oftware downloads

* [http://www.superliminal.com/cube/cube.htm Superliminal's 4D Magic Cube] The page also has links to other 4D cube implementations.
* [http://david-v.home.texas.net/MC3D/ David Vanderschel's 3D Magic Cube]
* [http://www.superliminal.com/cube/mc2d.html Superliminal's 2D Magic Cube] (Java Applet)
* [http://www.gravitation3d.com/magiccube5d/index.html Roice Nelson and Charlie Nevill's 5D Magic Cube]
* [http://www.gravitation3d.com/magic120cell/index.html Roice Nelson's Magic120Cell]