Professor's Cube

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The Professor's Cube (also known as Rubik's Professor) is a mechanical puzzle invented by Udo Krell. It is 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.

Contents

[edit] Workings

[edit] 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!/(4!6))2 or 24!2/4!12 arrangements, all of which are possible, independently of the corner cubelets.

Identically coloured pairs among the 24 outer edge cubelets cannot be flipped. 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 or 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.

The corner, central edge and fixed center cubelets together form a 3×3×3 Rubik's Cube. The remaining cubelets can be arranged independently of it.

This gives a total number of permutations of

\frac{8! \cdot 3^7 \cdot 12! \cdot 2^{10} \cdot 24!^3}{4!^{12}} \approx 2.8 \cdot 10^{74}

The full number is 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.

[edit] 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 kitty-corner 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.

[edit] Solution

People able to rapidly solve puzzles like this usually favor the strategy of pairing similar edge pieces into solid strips, and centers into one-coloured blocks. This allows the cube to be quickly solved with the same methods one would use for a 3×3×3 cube.

Others solve the puzzle according to mobility of pieces. That is, once the centers of the cube are placed, one may be able to solve the remainder of the cube using only the most outward twists of the cube.

Another frequently used strategy is to solve the edges of the cube first. This prevents parity errors similar to those first encountered in the Rubik's Revenge. The corners can be placed just as they are in any previous order of cube puzzle, and the centers are manipulated with an algorithm similar to the one used in the 4×4×4 cube.

[edit] Records

The current world record for solving the Professor's Cube in an official competition is 1:46.28, set by Frank Morris at the G.WIZ Fall 2006 competition in Sarasota, Florida. Additionally, Morris holds the record of 1:55.24 for the mean of three solves, also set at the G.WIZ competition. Chris Hardwick holds the world record for solving the Professor's Cube while blindfolded, with a time of 26:19.00 set at the G.WIZ competition.

[edit] See also

[edit] External links

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