- #1
Antiphon
- 1,685
- 4
I've always been fascinated by Rubik's cube. I have developed solutions for it and
all the related cubes 2x2, 3x3, 4x4, 5x5. For me the cube it is to group theory
(of a partcular type of group) what a slide rule is to real arithmetic. Even "laboratory"
might not be too stong a label for it.
For example it's immediately obvious how xy \neq yx. If you turn the front of the cube
and then the right you get a very different set of faces than the right followed by front.
Also, you can discover marvelous "operators" (my terminology) by doing some random
series of twists (abc) followed by a particular twist (Z) then undoing the first
twists (via cba), that is: abcZcba where the letters stand for some particular oriented
twist. What happens is that most of the cube is unperturbed except for some
marvelous little permutation like a twisted corner in place or three swapped edges.
My solutions then consist of applying these "operators" in sequence by inspection
to see which one is "needed" next.
Alas however, I am not formally trained in group theory and I would like
to know: How would one go about using GT to develop a more effective
or efficient solution to something like Rubik's cube? I know it has been
done, but my question is very specifically: Can anyone explain to the group theory novice
(but Rubik's cube expert) how one would actually go about using GT to
devise (more) efficient solutions to such a puzzle?
all the related cubes 2x2, 3x3, 4x4, 5x5. For me the cube it is to group theory
(of a partcular type of group) what a slide rule is to real arithmetic. Even "laboratory"
might not be too stong a label for it.
For example it's immediately obvious how xy \neq yx. If you turn the front of the cube
and then the right you get a very different set of faces than the right followed by front.
Also, you can discover marvelous "operators" (my terminology) by doing some random
series of twists (abc) followed by a particular twist (Z) then undoing the first
twists (via cba), that is: abcZcba where the letters stand for some particular oriented
twist. What happens is that most of the cube is unperturbed except for some
marvelous little permutation like a twisted corner in place or three swapped edges.
My solutions then consist of applying these "operators" in sequence by inspection
to see which one is "needed" next.
Alas however, I am not formally trained in group theory and I would like
to know: How would one go about using GT to develop a more effective
or efficient solution to something like Rubik's cube? I know it has been
done, but my question is very specifically: Can anyone explain to the group theory novice
(but Rubik's cube expert) how one would actually go about using GT to
devise (more) efficient solutions to such a puzzle?