# Virtual Cubes

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### M12 Cube

The M12 Cube is based on the Mathieu M12 group, a sporadic simple group known from Group Theory.

The goal is to scramble and restore the cube by using combinations of just two basic moves ('Invert Move' and 'Merge Move'). These moves only affect edges, which are numbered from 1 through 12. In the inital state, any edge and the two neighbouring corners show a same number. Since corners stay at the same place, they can be used as a reference frame to move edges back to their initial locations.

The layout of the M12 Cube was created in 2010 by André Boulouard and Walter Randelshofer.

#### Invert Move (I)

R2 F2 R' D2 U2 R2 L' F2 R F U F' D2 U2 B U' B' L (18* ftm, 26 qtm)
(ur,br) (dr,bl) (dl,lf) (rf,lu) (fu,fd) (bu,bd)

The 'I' move swaps 2 edges in each of 6 pairs, which reverses the order of the numbers on the cube:
(1,12) (2,11) (3,10) (4,9) (5,8) (6,7)

#### Merge Move (M)

R2 F D2 F B' D B U B' D2 F' U' F' D' R B2 R' B' R2 (19* ftm, 24 qtm)
(rf,df,lf,bu,bl,lu,bd,br,dr,dl,uf)

The 'M' move cycles 11 out of 12 edges, which literally shuffles the numbers:
(9,8,10,6,11,4,7,12,2,3,5)

The Mathieu groups were among the first sporadic groups that have been discovered. Mathieu group M12 contains 95'040 members and is a subgroup of alternating group A12 which contains 239'500'800 members. M12 is sharply 5-transitive, meaning that any permutation maps a set of 5 points to another set of 5 points and given 2 sets of 5 points, there is only one permutation that will do the mapping (uniqueness). Two distinct permutations of M12 differ at least on 8 points.

For more information, see this feature of Scientific American.

Any position of the M12 Puzzle can be solved in just 29 moves or less. This has been shown using an exhaustive search technique called Breadth First Search (BFS) to compute all optimal combinations of 'I' and 'M' moves. The M12 group shows a total of 95'040 members, ie. his order is just 95'040 where 95'040 is 12! / 7!. Only 12 antipodes or sequences with a maximal length of 29 moves have been found.

© Walter Randelshofer. All rights reserved.