The Rubik’s Cube is the original and best-known of the three-dimensional sequential move puzzles. Over the years, many virtual implementations of this puzzle have been created in software. Extending these ideas naturally leads to the development of sequential move puzzles in four or more dimensions. The rules governing their operation are rigorously defined mathematically and are closely related to principles of three-dimensional geometry. As a result, they can be simulated in software, and some have even been realized as physical puzzles. As with their three-dimensional counterparts, there are records for solvers, although there is not yet the same level of competitive organisation.
Glossary
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's 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's cube is as follows; Piece countNumber of vertices (V)8Number of 3-colour pieces8Number of edges (E)12Number of 2-colour pieces12Number of faces (F)6Number of 1-colour pieces6Number of cells (C)1Number of 0-colour pieces1Number of coloured pieces (P)26Number of stickers54
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. 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.
Magic 4D Cube
Geometric shape: Tesseract
The Superliminal MagicCube4D software implements many twisty puzzle versions of 4D polytopes including N4 cubes. The UI allows for 4D twists and rotations plus control of 4D viewing parameters such as the projection into 3D, 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 countRoice Nelson,Anatomy of a d-dimensional Rubik's Cube, available online here and archived 25 December 2008.Number of vertices16Number of 4-colour pieces16Number of edges32Number of 3-colour pieces32Number of faces24Number of 2-colour pieces24Number of cells8Number of 1-colour pieces8Number of 4-cubes1Number of 0-colour pieces1Number of coloured pieces80Number of stickers216 Achievable combinations:Eric Balandraud, Calculating the Permutations of 4D Magic Cubes, available online here and archived 25 December 2008.
24 4-cube Piece countNumber of vertices16Number of 4-colour pieces16Number of edges32Number of 3-colour pieces0Number of faces24Number of 2-colour pieces0Number of cells8Number of 1-colour pieces0Number of 4-cubes1Number of 0-colour pieces0Number of coloured pieces16Number of stickers64 Achievable combinations:
44 4-cube Piece countNumber of vertices16Number of 4-colour pieces16Number of edges32Number of 3-colour pieces64Number of faces24Number of 2-colour pieces96Number of cells8Number of 1-colour pieces64Number of 4-cubes1Number of 0-colour pieces16Number of coloured pieces240Number of stickers512 Achievable combinations:
54 4-cube Piece countNumber of vertices16Number of 4-colour pieces16Number of edges32Number of 3-colour pieces96Number of faces24Number of 2-colour pieces216Number of cells8Number of 1-colour pieces216Number of 4-cubes1Number of 0-colour pieces81Number of coloured pieces544Number of stickers1000 Achievable combinations:
Magic 5D Cube
Geometric shape: penteract
Magic Cube 5D by Roice Nelson is capable of rendering 5-cube puzzles in six sizes from 25 to 75. Allows 5D twists and controls for rotating the cube in multiple dimensions, 4-D and 5-D perspective controls, cubie and sticker spacing and size controls, similar to Magiccube4D.
However, a 5-D puzzle is much more difficult to comprehend than a 4-D puzzle. An essential feature of the Roice's 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 screenshots.
Roice maintains a Hall of Insanity for record breaking solvers of this puzzle. As of 6 January 2011, there have been two successful solutions for the 75 size of 5-cube.Roice Nelson, MagicCube5D unsolved puzzles listed online here and archived 25 December 2008.
35 5-cube Piece countNumber of vertices32Number of 5-colour pieces32Number of edges80Number of 4-colour pieces80Number of faces80Number of 3-colour pieces80Number of cells40Number of 2-colour pieces40Number of 4-cubes10Number of 1-colour pieces10Number of 5-cubes1Number of 0-colour pieces1Number of coloured pieces242Number of stickers810 Achievable combinations:MC5D Permutation Counts
25 5-cube Piece countNumber of vertices32Number of 5-colour pieces32Number of edges80Number of 4-colour pieces0Number of faces80Number of 3-colour pieces0Number of cells40Number of 2-colour pieces0Number of 4-cubes10Number of 1-colour pieces0Number of 5-cubes1Number of 0-colour pieces0Number of coloured pieces32Number of stickers160 Achievable combinations:
45 5-cube Piece countNumber of vertices32Number of 5-colour pieces32Number of edges80Number of 4-colour pieces160Number of faces80Number of 3-colour pieces320Number of cells40Number of 2-colour pieces320Number of 4-cubes10Number of 1-colour pieces160Number of 5-cubes1Number of 0-colour pieces32Number of coloured pieces992Number of stickers2,560 Achievable combinations:
55 5-cube Piece countNumber of vertices32Number of 5-colour pieces32Number of edges80Number of 4-colour pieces240Number of faces80Number of 3-colour pieces720Number of cells40Number of 2-colour pieces1,080Number of 4-cubes10Number of 1-colour pieces810Number of 5-cubes1Number of 0-colour pieces243Number of coloured pieces2,882Number of stickers6,250 Achievable combinations:
65 5-cube Piece countNumber of vertices32Number of 5-colour pieces32Number of edges80Number of 4-colour pieces320Number of faces80Number of 3-colour pieces1,280Number of cells40Number of 2-colour pieces2,560Number of 4-cubes10Number of 1-colour pieces2,560Number of 5-cubes1Number of 0-colour pieces1,024Number of coloured pieces6,752Number of stickers12,960 Achievable combinations:
75 5-cube Piece countNumber of vertices32Number of 5-colour pieces32Number of edges80Number of 4-colour pieces400Number of faces80Number of 3-colour pieces2,000Number of cells40Number of 2-colour pieces5,000Number of 4-cubes10Number of 1-colour pieces6,250Number of 5-cubes1Number of 0-colour pieces3,125Number of coloured pieces13,682Number of stickers24,010 Achievable combinations:
Magic Cube 7D Geometric shape: hexeract (6D) and hepteract (7D)
Andrey Astrelin's Magic Cube 7D software is capable of rendering puzzles of up to 7 dimensions in twelve sizes from 34 to 57.
As of July 2024, in terms of puzzles exclusive to Magic Cube 7D, only the 36, 37, 46, and 56 puzzles have been solved.Magic Cube 7D
Magic 120-cell
Geometric shape: 120-cell (also called the hecatonicosachoron or dodecacontachoron)
The 120-cell is a 4-D geometric figure (4-polytope) composed of 120 dodecahedra, 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, 3-D puzzle, which has the shape of a dodecahedron.
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 has created a "Hall of Fame" for solvers, who must provide a log file for their solution. As of April 2017, the puzzle has been solved twelve times.
Piece countDavid Smith, An Upper Bound for the Number of Different Positions of the Colored Magic120-Cell, available online here and archived 25 December 2008.Number of vertices600Number of 4-colour pieces600Number of edges1,200Number of 3-colour pieces1,200Number of faces720Number of 2-colour pieces720Number of cells120Number of 1-colour pieces120Number of 4-cells1Number of 0-colour pieces1Number of coloured pieces2,640Number of stickers7,560 Achievable combinations:
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.
3x3 2D square
Geometric shape: square
A 2-D Rubik type puzzle can no more be physically constructed than a 4-D one can.David Vanderschel, , 21 August 2006. "MC2D's (reflecting) moves would require a 3rd dimension to implement them physically". Retrieved 4 April 2009, 9 July 2012. 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 call it the 2D Magic Cube.