Rubik's Revenge

From Wikipedia, the free encyclopedia

Rubik's Revenge in solved state

The Rubik's Revenge is the 4×4×4 version of Rubik's Cube. Invented by Péter Sebestény, the Rubik's Revenge was nearly called the Sebestény Cube until a somewhat last-minute decision changed the puzzle's name to attract fans of the original Rubik's Cube. Unlike the original puzzle, it has no fixed facets: the centre facets (four per face) are free to move to different positions. The internal mechanics are rather different: the centre cubelets slide in grooves on an internal ball, which cannot be seen unless the puzzle is disassembled. The edge and corner cubelets glide on tracks formed by the edges of the centre cubelets in much the same way as in the 3×3×3 version. A new mechanism was introduced by the East Sheen company, where the mechanism is an expansion of that of the 3×3×3.

Methods for solving the 3×3×3 cube work for the edges and corners of the 4×4×4 cube, as long as one has correctly identified the relative positions of the colours — since the centre facets can no longer be used for identification.

1 Workings
- 1.1 Permutations
2 Solutions
- 2.1 Records
3 See also
4 External links

[edit] Workings

Rubik's Revenge in scrambled state

The puzzle consists of the fifty-six unique miniature cubes ("cubies") on the surface. However, the center four cubes of each face are merely single square facade hooked in to a grooved ball. These provide structure for the other pieces to fit into and rotate around. The core piece is a ball with a groove on each of the three intersecting orbits that holds the four pieces in each of the six centres while letting them rotate. This is the largest change to the 3×3×3 cube, because the centre pieces can move in relative to each other, unlike the fixed centres on the original. The Cube can be taken apart without much difficulty, typically by turning one side through a 45° angle and prying an "edge cubie" away from the "centre cubies" until it dislodges. It is a simple process to solve a Cube by taking it apart and reassembling it in a solved state; however, this is not the challenge.

There are twenty-four edge pieces which show two coloured sides each, and eight corner pieces which show three colours. Each piece (or pair of edge pieces) shows a unique colour combination, but not all combinations are present (for example, there is no edge piece with both white and yellow sides, if white and yellow are on opposite sides of the solved Cube). The location of these cubes relative to one another can be altered by twisting an outer fourth of the Cube 90°, 180° or 270°, but the location of the coloured sides relative to one another in the completed state of the puzzle cannot be altered: it is fixed by the relative positions of the centre squares and the distribution of colour combinations on edge and corner pieces.

For most recent Cubes, the colours of the stickers are red opposite orange, yellow opposite white, and green opposite blue. However, there also exist Cubes with alternative colour arrangements. These alternative Cubes have the yellow face opposite the green, and the blue face opposite the white (with red and orange opposite faces remaining unchanged).

[edit] Permutations

Rubik's Revenge with a tilted side

There are 8 corner cubelets, 24 edge cubelets and 24 centre cubelets.

Any permutation of the corner cubelets is possible, including odd permutations, giving 8! possible arrangements. Seven of the corner cubelets can be independently rotated, and the eighth cubelet's orientation depends on the other seven, giving 3⁷ combinations.

Assuming the 4 centre cubelets of each colour are indistinguishable, there are 24!/(4!⁶) arrangements, all of which are possible, independently of the corner cubelets. (An odd permutation of the corner cubelets implies an odd permutation of the centre cubelets, and vice versa; however, even and odd permutations are indistinguishable because of identically coloured centre cubelets.)

The 24 edge cubelets cannot be flipped. The two edge cubelets in each matching pair are distinguishable, since the colours on a cubelet are reversed relative to the other. Any permutation of the edge cubelets is possible, including odd permutations, giving 24! arrangements, independently of the corner or centre cubelets.

Assuming the cube does not have a fixed orientation in space, and that the permutations resulting from rotating the cube without twisting the cube are considered identical, the number of permutations is reduced by a factor of 24.

This gives a total number of permutations of

$\frac{8! \cdot 3^7 \cdot 24! \cdot 24!}{4!^6 \cdot 24} \approx 7.4 \cdot 10^{45}$

The full number is 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 possible combinations.

[edit] Solutions

There are a number of methods that can be used to solve a Rubik's Revenge. The layer by layer method that is often used for the 3×3×3 cube is not usually used on the Rubik's Revenge. One of the most common methods is to first group the center pieces of common colors together, then to pair edges that show the same two colors. Once this is done, turning only the outer layers of the cube allows it to be solved as the 3×3×3 cube would be solved. However, certain positions that cannot be solved on a standard 3×3×3 cube may be reached. For instance, a single pair of edges may be inverted, or the cube may appear to have odd permutation (that is, two pieces must be swapped, which is not possible on the 3×3×3 cube). These situations are known as parity errors. These positions are still solvable; however, special sequences must be applied to fix the errors.

Another similar approach to solving this cube is to first pair the edges, and then the centers. This, too, is vulnerable to the parity errors described above.

Some methods are designed to avoid the parity errors described above. For instance, solving the corners and edges first and the centers last would avoid such parity errors.

[edit] Records

The current world record for solving the Rubik's Revenge in an official competition is 51.16 seconds, set by Michael Fung at the Dutch Open 2006. Fung also set the record for the fastest average of 5 solves, with a time of 59.21 seconds. Chris Hardwick holds the world record for solving the Rubik's Revenge while blindfolded, with a time of 8:04.90 set at the G.WIZ Fall 2006 competition in Sarasota, Florida. [1]