# Virtual Cubes

Install the Java Plug-in to see the Virtual Cubes!

### M24 Cube

The M24 Cube is based on the Mathieu M24 group, a sporadic simple group known from Group Theory.

The goal is to scramble and restore the cube by using combinations of just three basic moves ('Left Move', 'Right Move' and 'Switch' Move). These moves only affect edges, which are numbered from 0 through 23. In the inital state, any edge and the neighbouring corner 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 Mathieu groups were among the first sporadic groups that have been discovered. The order of Mathieu group M24 is equal to: 244'823'040 = 3 · 16 · 20 · 21 · 22 · 23 · 24. The M24 group is a subgroup of A24, the order of which is equal to 24!/2, which is a very large number: 3.1 × 1023.

The layout of the M24 Cube was created in 2010 by André Boulouard and Walter Randelshofer.

#### Left Move (L)

MR B2 MR F2 MR' B2 MR F2 MR2 MB F R' D ML' D' L2 R D ML D' L2 F' R2 D' ML' D SR2 D' ML D L2 MB' MD2 R2 F ML F' SR2 F ML' F' L2 WD2 R2 F ML F' SR2 F ML' F' L2 MU2 D2 ML' D2 ML D2 ML' D2 ML D2 ML' D2 ML (65 btm)
(ur1,br1,rb2,lb1,bl2,fl1,lf2,rf1,fr2,df1,fd2,bd1,db2,ub1,bu2,fu1,uf2,lu1,ul2,dl1,ld2,rd1,dr2)

The 'L' move cycles 23 out of 24 edges, which literally shuffles the numbers. The 'L' move is the inverse of 'R':
(23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1)

#### Right Move (R)

ML' D2 ML D2 ML' D2 ML D2 ML' D2 ML D2 MU2 L2 F ML F' SR2 F ML' F' R2 WD2 L2 F ML F' SR2 F ML' F' R2 MD2 MB L2 D' ML' D SR2 D' ML D R2 F L2 D ML' D' R' L2 D ML D' R F' MB' MR2 F2 MR' B2 MR F2 MR' B2 MR' (65 btm)
(dr2,rd1,ld2,dl1,ul2,lu1,uf2,fu1,bu2,ub1,db2,bd1,fd2,df1,fr2,rf1,lf2,fl1,bl2,lb1,rb2,br1,ur1)

The 'R' move cycles 23 out of 24 edges, which literally shuffles the numbers. The 'R' move is the inverse of 'L':
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)

#### Switch Move (S)

SL' D2 L MU2 L' D2 SL D R MU2 R' D' F' L2 F MR2 F' L2 SF L' B MR2 B' L B ML' D ML D MB U MB' D' ML' D' ML MB U' MB MF R' MF' L F2 L' MF R MB' MF' SL F2 SL' MB' MD SR' D' R' L' MU2 L R D R' L' MU2 R2 MD' MU SL' U' L' R' MD2 R L U L' R' MD2 L2 MU' (81 btm)
(ur1,ru2) (br1,dr2) (rd1,ld2) (ub1,ul2) (lu1,uf2) (fu1,bu2) (lb1,db2) (bd1,fd2) (rf1,fr2) (fl1,lf2) (df1,bl2) (dl1,rb2)

The 'S' move swaps 2 edges in each of 12 pairs, which reverses the order of the numbers on the cube:
(1,0) (23,2) (4,3) (22,5) (11,6) (8,7) (10,9) (21,12) (14,13) (20,15) (17,16) (19,18)

Notation