Well, how we solve the puzzle from the previous post?

## Archive for the ‘puzzle’ Category

### Solve-By-Play

Donnerstag, Juli 9th, 2009### Business As Usual

Mittwoch, Juli 8th, 2009Still near Linz, at the RISC/SCIEnce Summer School. Different courses, interesting people, no time to blog. ;)

We were yesterday in Linz for the sightseeing and the conference dinner. ‚T was nice.

And by the way, I have a new puzzle for you. It was told to me by Padraig Ó Catháin. You have two rectangles with one being 3 times the area of other and other being 3 times the perimeter of the one. Find their sides in integral numbers. More difficult question: how many solutions are there?

### Rubik’s Cube and GAP

Mittwoch, Juli 1st, 2009

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 …)