Professor's Cube

The Professor's Cube is a mechanical puzzle, a 5×5×5 version of the Rubik's Cube. It has qualities in common with both the original 3×3×3 Rubik's Cube and the 4×4×4 Rubik's Revenge, and knowing the solution to either can help when working on the 5×5×5 cube.

Naming

Early versions of the 5×5×5 cube sold at Barnes and Noble were marketed under the name "Professor's Cube," but currently, Barnes and Noble sells cubes that are simply called "5×5×5." Mefferts.com offers a limited edition version of the 5×5×5 cube that is identified as the Professor's Cube. This version has colored blocks rather than stickers. Verdes Innovations sells a version which is identified as the V-Cube 5.

Workings

The original Professor's Cube design by Udo Krell works by using an expanded 3×3×3 cube as a mantle with the center edge pieces and corners sticking out from the spherical center of identical mechanism to the 3×3×3 cube. The non central center pieces are fitted into spaces on the surface of the 3×3×3 mantle, and the non central edges slotted between them. All non-central pieces have extensions that fit into allotted spaces on the outer pieces of the 3×3×3, which keeps them from falling out of the cube while making a turn. The Eastsheen version of the puzzle uses a different mechanism. The fixed centers hold the center cubelets next to the central edges in place, which in turn hold the edge cubelets. The non-central edges hold the corners in place, the opposite of the original mechanism. The V-Cube 5 mechanism, designed by Panagiotis Verdes, has elements in common with both. The pieces partially clip into the mechanism (like the original mechanism) and the fixed center piece holds the center cubelets in place, allowing smooth and fast rotation and creating arguably the fastest and most durable version of the puzzle.

Permutations

There are 8 corner cubelets, 36 edge cubelets (two types), and 54 center cubelets (48 movable of two types, 6 fixed).

Any permutation of the corner cubelets is possible, including odd permutations, giving 8! (40,320) 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 center cubelets of each type of each colour are indistinguishable, there are 24! ways to arrange each type, divided by 4!⁶. This reducing factor results from the fact that there are 4! ways to arrange the four cubelets of each color, raised to the sixth power because there are six colors. The total permutations for all of the movable center cubelets is (24!/(4!⁶))² or 24!²/4!¹², all of which are possible, independently of the corner cubelets.

Identically coloured pairs among the 24 outer edge cubelets cannot be flipped, since the interior shape of those pieces is asymmetrical. The two cubelets in each matching pair are distinguishable, since the colours on a cubelet are reversed relative to the other. Any permutation of the outer edge cubelets is possible, including odd permutations, giving 24! arrangements, independently of the corner cubelets and indistinguishable center cubelets. The 12 central edge cubelets can be flipped. Eleven can be flipped and arranged independently, giving 12!/2 × 2¹¹ or 12! × 2¹⁰ possibilities (an odd permutation of the corner cubelets implies an odd permutation of the central edge cubelets, and vice versa, thus the division by 2). Counting the outer edge cubelets, there are 24! × 12! × 2¹⁰ possibilities.

This gives a total number of permutations of :$frac\{8!\; imes\; 3^7\; imes\; 12!\; imes\; 2^\{10\}\; imes\; 24!^3\}\{4!^\{12\; approx\; 2.83\; imes\; 10^\{74\}$

The full number is precisely 282 870 942 277 741 856 536 180 333 107 150 328 293 127 731 985 672 134 721 536 000 000 000 000 000 possible permutations (about 283 duodecillion on the long scale or 283 trevigintillion on the short scale).

Some variations of the Professor's Cube have one of the fixed center pieces marked with a logo, which can be put into four different orientations. This increases the number of permutations by a factor of four to 1.13×10⁷⁵, although any orientation of this piece could be regarded as correct.

Durability

The Professor's Cube is inherently more delicate than the 3×3×3 Rubik's Cube due to the considerable additional movable parts. It is not recommended that it be used for speedcubing. The puzzle should not be excessively forced to twist and it must be aligned properly before twisting to prevent damage. It is far more likely to break due to twisting misaligned rows.If twisted while not fully aligned, it may cause the pieces diagonal to the corners to almost fully come out. It is simply fixed by turning the face back to where it was, causing the piece to go back to its original position. Excessive force may cause the colored tile to break off completely. In such a case, the cubelet will stay in place, yet the color would be gone. Both the Eastsheen 5x5x5 and the V-Cube 5 are designed with a different mechanism in an attempt to remedy the fragility of the Professor's Cube.

olution

People able to rapidly solve puzzles like this usually favour the strategy of grouping similar edge pieces into solid strips, and centres into one-colored blocks. This allows the cube to be quickly solved with the same methods one would use for a 3×3×3 cube. Because the centers, middle edges and corners can be treated as equivalent to a 3×3×3 cube, the parity errors sometimes seen on the 4×4×4 cannot occur on the 5×5×5 unless the cube has been tampered with.

Another frequently used strategy is to solve the edges of the cube first. The corners can be placed just as they are in any previous order of cube puzzle, and the centres are manipulated with an algorithm similar to the one used in the 4×4×4 cube.

World Records

The current record for solving the Professor's Cube in an official competition is 1 minute 18.59 seconds, set by Erik Akkersdijk at the European Championship 2008 in Bilbabo, Spain. [http://www.worldcubeassociation.org/results/c.php?i=Euro2008&allResults=1#555] .

Erik Akkersdijk also holds the record of 1 minute 26.86 seconds for the mean of five solves, set at the Brussels Summer Open 2008.

ee also

* Pocket Cube – A 2×2×2 version of the puzzle
* Rubik's Cube – The original 3×3×3 version of this puzzle
* Rubik's Revenge – A 4×4×4 version of the puzzle
* V-Cube 6 - A 6×6×6 version of the puzzle
* V-Cube 7 - A 7×7×7 version of the puzzle
* Combination puzzles

References

External links

* [http://www.bigcubes.com/5x5x5/5x5x5.html How to solve Professor's Cube]
* [http://wiki.playagaingames.com/tiki-index.php?page=5x5x5+Cube+Solution Professor's Cube text solution]
* [http://www.rubiks-zauberwuerfel.de Professor's Cube interactive solution]
* [http://www.geocities.com/abcmcfarren/math/rp/RubPro1.htm Courtney McFarren's Professor's Cube solution]