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N-dimensional sequential move puzzlesFrom Wikipedia, the free encyclopedia
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5D 25 puzzle partial cutaway demonstrating that even with the minimum size in 5-D the puzzle is far from trivial. The 4-D nature of the stickers is clearly visible in this screen shot.


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[1] and the 120-cell 4D puzzle.[2]

Contents[hide] 1 Terms used in this article 2 Magic 4D Cube 2.1 34 4-cube 2.2 24 4-cube 2.3 44 4-cube 2.4 54 4-cube 3 Magic 5D Cube 3.1 35 5-cube 3.2 25 5-cube 3.3 45 5-cube 3.4 55 5-cube 3.5 65 5-cube 3.6 75 5-cube 4 Magic 120-cell 5 3x3 2D square 6 1D Projection 7 See also 8 References 9 External links 9.1 Solving records 9.2 Software downloads

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;

Piece Count
Number of vertices (V)	8	Number of 3-colour pieces	8
Number of edges (E)	12	Number of 2-colour pieces	12
Number of faces (F)	6	Number of 1-colour pieces	6
Number of cells (C)	1	Number of 0-colour pieces	1
Number of coloured pieces (P)	26
Number of stickers	54

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. A different number of pieces may be given in different sources. 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.

4-cube 34 virtual puzzle, solved. In this projection one cell is not shown. The position of this cell is the extreme foreground of the 4th dimension beyond the position of the viewers screen.



4-cube 34 virtual puzzle, rotated in the 4th dimension to show the colour of the hidden cell.



4-cube 34 virtual puzzle, rotated in normal 3D space.



4-cube 34 virtual puzzle, scrambled.



4-cube 24 virtual puzzle, one cubie is highlighted to show how the stickers are distributed across the cube. Note that there are four stickers on each of the cubies of the 24 puzzle but only three are highlighted, the missing one is on the hidden cell.



4-cube 54 virtual puzzle with stickers of the same cubie made to exactly touch each other.


Geometric shape: tesseract The Superliminal MagicCube4D software is capable of rendering 4-cube puzzles in four sizes, namely 24, the standard 34, 44 and 54. As well as the ability to make moves on the cube there are controls to change the view. These include controls for rotating the whole cube in 3-space and 4-space, 4-D perspective, cubie size and spacing, and sticker size.
Superliminal Software maintains a Hall of Fame for record breaking solvers of this puzzle.

34 4-cube

Piece Count[3]
Number of vertices	16	Number of 4-colour pieces	16
Number of edges	32	Number of 3-colour pieces	32
Number of faces	24	Number of 2-colour pieces	24
Number of cells	8	Number of 1-colour pieces	8
Number of 4-cubes	1	Number of 0-colour pieces	1
Number of coloured pieces	80
Number of stickers	216

Achievable combinations:[4]

24 4-cube

Piece Count[3]
Number of vertices	16	Number of 4-colour pieces	16
Number of edges	32	Number of 3-colour pieces	0
Number of faces	24	Number of 2-colour pieces	0
Number of cells	8	Number of 1-colour pieces	0
Number of 4-cubes	1	Number of 0-colour pieces	0
Number of coloured pieces	16
Number of stickers	64

Achievable combinations:[4]

44 4-cube

Piece Count[3]
Number of vertices	16	Number of 4-colour pieces	16
Number of edges	32	Number of 3-colour pieces	64
Number of faces	24	Number of 2-colour pieces	96
Number of cells	8	Number of 1-colour pieces	64
Number of 4-cubes	1	Number of 0-colour pieces	16
Number of coloured pieces	240
Number of stickers	512

Achievable combinations:[4]

54 4-cube

Piece Count[3]
Number of vertices	16	Number of 4-colour pieces	16
Number of edges	32	Number of 3-colour pieces	96
Number of faces	24	Number of 2-colour pieces	216
Number of cells	8	Number of 1-colour pieces	216
Number of 4-cubes	1	Number of 0-colour pieces	81
Number of coloured pieces	544
Number of stickers	1000

Achievable combinations:[4]


Magic 5D Cube
5-cube 35 virtual puzzle, close in view in solved state.



5-cube 35 virtual puzzle, scrambled.



5-cube 75 virtual puzzle, with certain pieces highlighted. The rest are shaded out to aid the solver's comprehension of the puzzle.



5-cube 75 virtual puzzle, solved.



Software control panel for rotating the 5-cube, illustrating the increased number of planes of rotation possible in 5 dimensions.


Geometric shape: penteract The Gravitation3d Magic 5D Cube software is capable of rendering 5-cube puzzles in six sizes from 25 to 75. As well as the ability to make moves on the cube there are controls to change the view. These include controls for rotating the cube in 3-space, 4-space and 5-space, 4-D and 5-D perspective controls, cubie and sticker spacing and size controls. If this all sounds very similar to Subliminal's 4D cube, that would be because it is. Gravitation3d acknowledge on their website the debt that they owe to Subliminal for many of the concepts.
However, a 5-D puzzle is incredibly more difficult to comprehend on a 2-D screen than a 4-D puzzle is. An essential feature of the Gravitation3d implementation is the ability to turn off or highlight chosen cubies and stickers. Even so, the complexities of the images produced are still quite severe as can be seen from the screen shots.
Gravitation3d maintains a Hall of Insanity for record breaking solvers of this puzzle. As of May 2008 there have been no claimed solvers listed for either the 65 size or the 75 size of 5-cube.[1]
[url=][/url]
35 5-cube

Piece Count[3]
Number of vertices	32	Number of 5-colour pieces	32
Number of edges	80	Number of 4-colour pieces	80
Number of faces	80	Number of 3-colour pieces	80
Number of cells	40	Number of 2-colour pieces	40
Number of 4-cubes	10	Number of 1-colour pieces	10
Number of 5-cubes	1	Number of 0-colour pieces	1
Number of coloured pieces	242
Number of stickers	810

25 5-cube

Piece Count[3]
Number of vertices	32	Number of 5-colour pieces	32
Number of edges	80	Number of 4-colour pieces	0
Number of faces	80	Number of 3-colour pieces	0
Number of cells	40	Number of 2-colour pieces	0
Number of 4-cubes	10	Number of 1-colour pieces	0
Number of 5-cubes	1	Number of 0-colour pieces	0
Number of coloured pieces	32
Number of stickers	160

45 5-cube

Piece Count[3]
Number of vertices	32	Number of 5-colour pieces	32
Number of edges	80	Number of 4-colour pieces	160
Number of faces	80	Number of 3-colour pieces	320
Number of cells	40	Number of 2-colour pieces	320
Number of 4-cubes	10	Number of 1-colour pieces	160
Number of 5-cubes	1	Number of 0-colour pieces	32
Number of coloured pieces	992
Number of stickers	2,560

55 5-cube

Piece Count[3]
Number of vertices	32	Number of 5-colour pieces	32
Number of edges	80	Number of 4-colour pieces	240
Number of faces	80	Number of 3-colour pieces	720
Number of cells	40	Number of 2-colour pieces	1,080
Number of 4-cubes	10	Number of 1-colour pieces	810
Number of 5-cubes	1	Number of 0-colour pieces	243
Number of coloured pieces	2,882
Number of stickers	6,250

65 5-cube

Piece Count[3]
Number of vertices	32	Number of 5-colour pieces	32
Number of edges	80	Number of 4-colour pieces	320
Number of faces	80	Number of 3-colour pieces	1,280
Number of cells	40	Number of 2-colour pieces	2,560
Number of 4-cubes	10	Number of 1-colour pieces	2,560
Number of 5-cubes	1	Number of 0-colour pieces	1,024
Number of coloured pieces	6,752
Number of stickers	12,960

75 5-cube

Piece Count[3]
Number of vertices	32	Number of 5-colour pieces	32
Number of edges	80	Number of 4-colour pieces	400
Number of faces	80	Number of 3-colour pieces	2,000
Number of cells	40	Number of 2-colour pieces	5,000
Number of 4-cubes	10	Number of 1-colour pieces	6,250
Number of 5-cubes	1	Number of 0-colour pieces	3,125
Number of coloured pieces	13,682
Number of stickers	24,010

Magic 120-cell
120-cell virtual puzzle, close in view in solved state



120-cell virtual puzzle, solved


Geometric shape: 120-cell or hecatonicosachoron The 120-cell is a 4-D geometric figure (4-polytope) composed of 120 dodecahedrons, which in turn is a 3-D figure composed of 12 pentagons. The 120-cell is the 4-D analogue of the dodecahedron in the same way that the tesseract (4-cube) is the 4-D analogue of the cube. The 4-D 120-cell software sequential move puzzle from Gravitation3d is therefore the 4-D analogue of the Megaminx dodecahedral 3-D puzzle.
The puzzle is rendered in only one size, that is three cubies on a side, but in six colouring schemes of varying difficulty. The full puzzle requires a different colour for each cell, that is 120 colours. This large number of colours adds to the difficulty of the puzzle in that some shades are quite difficult to tell apart. The easiest form is two interlocking tori, each torus forming a ring of cubies in different dimensions. The full list of colouring schemes is as follows;

• 2-colour tori.
• 9-colour 4-cube cells. That is, the same colouring scheme as the 4-cube.
• 9-colour layers.
• 12-colour rings.
• 60-colour antipodal. Each pair of diametrically opposed dodecahedron cells is the same colour.
• 120-colour full puzzle.

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. Additionally, there is the ability to completely turn off groups of cells based on selection of tori, 4-cube cells, layers or rings.
Gravitation3d state on their website that they will create a "Hall of Fame" for solvers when the first solver provides them with their solution. As of May 2008 there have been no claims to have solved this puzzle.[2]

Piece Count[5]
Number of vertices	600	Number of 4-colour pieces	600
Number of edges	1,200	Number of 3-colour pieces	1,200
Number of faces	720	Number of 2-colour pieces	720
Number of cells	120	Number of 1-colour pieces	120
Number of 4-cells	1	Number of 0-colour pieces	1
Number of coloured pieces	2,640
Number of stickers	7,560

Achievable combinations:[5]
This calculation of achievable combinations has not been mathematically proven 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
2-cube 3×3 virtual puzzle


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.

Piece Count[3]
Number of vertices	4	Number of 2-colour pieces	4
Number of edges	4	Number of 1-colour pieces	4
Number of faces	1	Number of 0-colour pieces	1
Number of coloured pieces	8
Number of stickers	12

Achievable combinations:
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 ProjectionAnother 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.[6] However, since records are not being kept for this puzzle it might not actually be the case that it is unsolved.

[ 本帖最后由 大头 于 2009-1-1 00:42 编辑 ]

bhw19930503

看不懂啊~~~郁闷  

射石饮羽

难得熬一次夜，坐到了沙发，不容易啊，英文看不懂，图也很深奥的样子

汪小光

很爽啊，不过，这是真正的N维嘛？N维又是怎么定义的呢？在三维的空间里，有可能做出多维的魔方吗？

carloshn123

看的晕了……本来就够深奥……还是英文…………

Cielo

上午发现在这版回不了帖现在可以回了！
这网页上说的都比较笼统，没说具体的一步是怎么转的，我再看看吧！

谢谢大头！新年快乐！

ZJY

貌似比相对论还要复杂啊

kexin_xiao

大头最近频发原著啊，学习！可惜给你加不了分了，抱歉！

Xwam

晕了,看不太明白,好难得样子

nileibin

咱看不懂....