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Alexander's Star is a puzzle similar to the Rubik's Cube, in the shape of a great dodecahedron.

History

Alexander's Star was invented by Adam Alexander, an American mathematician, in 1982. It was patented on 26 March 1985, with US patent number 4,506,891, and sold by the Ideal Toy Company. It came in two varieties: painted surfaces or stickers. Since the design of the puzzle practically forces the stickers to peel with continual use, the painted variety is likely a later edition.
'Description

The puzzle has 30 moving pieces, which rotate in star-shaped groups of five around its outermost vertices. The purpose of the puzzle is to rearrange the moving pieces so that each star is surrounded by five faces of the same color, and opposite stars are surrounded by the same color. This is equivalent to solving just the edges of a six-color Megaminx. The puzzle is solved when each pair of parallel planes is made up of only one colour. To see a plane, however, one has to look "past" the five pieces on top of it, all of which could/should have different colours than the plane being solved.

If considering the pentagonal regions as faces, like in the great dodecahedron represented by Schläfli symbol {5,5/2}, then the requirement is for all faces to be monochrome (same color) and for opposite faces to share the same color.

The puzzle does not turn smoothly, due to its unique design.[1]
Permutations

There are 30 edges, each of which can be flipped into two positions, giving a theoretical maximum of 30!×230 permutations. This value is not reached for the following reasons:

Only even permutations of edges are possible, reducing the possible edge arrangements to 30!/2.
The orientation of the last edge is determined by the orientation of the other edges, reducing the number of edge orientations to 229.
Since opposite sides of the solved puzzle are the same color, each edge piece has a duplicate. It would be impossible to swap all 15 pairs (an odd permutation), so a reducing factor of 214 is applied.
The orientation of the puzzle does not matter (since there are no fixed face centers to serve as reference points), dividing the final total by 60. There are 60 possible positions and orientations of the first edge, but all of them are equivalent because of the lack of face centers.

This gives a total of 30 \( {\frac {30!\times 2^{{15}}}{120}}\approx 7.24\times 10^{{34}} \) possible combinations.

The precise figure is 72 431 714 252 715 638 411 621 302 272 000 000 (roughly 72.4 decillion on the short scale or 72.4 quintilliard on the long scale).
See also

Rubik's Cube
Combination puzzles
Mechanical puzzles

External links

Description and solution

References

Wray, C. G. (1981). The cube: How to do it. Totternhoe (, Church Green, Totternhoe, Beds. ): C.G. Wray.

vte

Rubik's Cube
Puzzle inventors

Ernő Rubik Larry Nichols Uwe Mèffert Tony Fisher Panagiotis Verdes Oskar van Deventer



Rubik's Cubes

Overview Rubik's family cubes of all sizes 2×2×2 (Pocket Cube) 3×3×3 (Rubik's Cube) 4×4×4 (Rubik's Revenge) 5×5×5 (Professor's Cube) 6×6×6 (V-Cube 6) 7×7×7 (V-Cube 7) 8×8×8 (V-Cube 8)

Cubic variations

Helicopter Cube Skewb Dino Cube Square 1 Sudoku Cube Nine-Colour Cube Gear Cube Void Cube

Non-cubic
variations
Tetrahedron

Pyraminx Pyraminx Duo Pyramorphix BrainTwist

Octahedron

Skewb Diamond

Dodecahedron

Megaminx Pyraminx Crystal Skewb Ultimate

Icosahedron

Impossiball Dogic

Great dodecahedron

Alexander's Star

Truncated icosahedron

Tuttminx

Cuboid

Floppy Cube (1x3x3) Rubik's Domino (2x3x3)

Virtual variations
(>3D)

MagicCube4D MagicCube5D MagicCube7D Magic 120-cell

Derivatives

Missing Link Rubik's 360 Rubik's Clock Rubik's Magic
Master Edition Rubik's Revolution Rubik's Snake Rubik's Triamid

Renowned solvers

Yu Nakajima Édouard Chambon Bob Burton, Jr. Jessica Fridrich Chris Hardwick Kevin Hays Rowe Hessler Leyan Lo Shotaro Makisumi Toby Mao Prithveesh K. Bhat Krishnam Raju Gadiraju Tyson Mao Frank Morris Lars Petrus Gilles Roux David Singmaster Ron van Bruchem Eric Limeback Anthony Michael Brooks Mats Valk Feliks Zemdegs Collin Burns Max Park

Solutions
Speedsolving

Speedcubing

Methods

Layer by Layer CFOP method Optimal

Mathematics

God's algorithm Superflip Thistlethwaite's algorithm Rubik's Cube group

Official organization

World Cube Association

Related articles

Rubik's Cube in popular culture The Simple Solution to Rubik's Cube 1982 World Rubik's Cube Championship

 

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Graduate Studies in Mathematics

Mathematics Encyclopedia

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Index

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