
The afternoon lecture by Alexander Konovalov was a huge fun. Aside from all the algebraical goodness in GAP, he demonstrated something really amazing even to generic audience. It is actually easy to represent Rubik’s Cube as a group.
Let’s just unfold the cube and number the faces.
+--------------+
| |
| 1 2 3 |
| |
| 4 top 5 |
| |
| 6 7 8 |
| |
+--------------+--------------+--------------+--------------+
| | | | |
| 9 10 11 | 17 18 19 | 25 26 27 | 33 34 35 |
| | | | |
| 12 left 13 | 20 front 21 | 28 right 29 | 36 rear 37 |
| | | | |
| 14 15 16 | 22 23 24 | 30 31 32 | 38 39 40 |
| | | | |
+--------------+--------------+--------------+--------------+
| |
| 41 42 43 |
| |
| 44 bottom 45 |
| |
| 46 47 48 |
| |
+--------------+
Then you can represent the cube as
gap> cube := Group(
> ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19),
> ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35),
> (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11),
> (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24),
> (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27),
> (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40) );
GAP can tell you, how many different positions the puzzle has:
gap> Size( cube );
43252003274489856000
And that’s pretty damn much. But the best is: you can actually produce a solution from an arbitrary position just in three lines of GAP code. It looks like this: (mehr …)