I knew someone who had a ton of algorithms for very specific cases. So if he needed say an edge 3 cycle that did not disturb anything else he knew a ton of different edge 3 cycle for all kinds of different 3 edge combinations.
One day several of us were sitting around playing with our cubes, and he hit some position where none of his algorithms worked and was stuck while he tried to find a new algorithm for that position. We asked why he didn't just use a conjugate of one of his other algorithms.
That was how the rest of us did it. I for example knew one edge 3 cycle that didn't disturb anything else. Call it T. If I needed any other edge 3 cycle I'd just find some transformation g such that gTg' did what I wanted, where g' is the inverse of g. gTg' is called a conjugate of T, and the set of all gTg' is called the conjugacy class of T.
It's easy to find g, because it doesn't matter how much g messes up the rest of the cube. All that matters is that it moves the 3 edges I want to permute into the position they need to be for T. Whatever else g messes up will be restored by g'.
He was astounded when we showed him that approach. It had never even occurred to him that such a thing was possible.
The funny thing is he was a math major, and had taken (and passed with a good grade) an abstract algebra course. He definitely knew about conjugates of group elements. He had just never noticed that group theory was applicable to Rubik's Cube.
One day several of us were sitting around playing with our cubes, and he hit some position where none of his algorithms worked and was stuck while he tried to find a new algorithm for that position. We asked why he didn't just use a conjugate of one of his other algorithms.
That was how the rest of us did it. I for example knew one edge 3 cycle that didn't disturb anything else. Call it T. If I needed any other edge 3 cycle I'd just find some transformation g such that gTg' did what I wanted, where g' is the inverse of g. gTg' is called a conjugate of T, and the set of all gTg' is called the conjugacy class of T.
It's easy to find g, because it doesn't matter how much g messes up the rest of the cube. All that matters is that it moves the 3 edges I want to permute into the position they need to be for T. Whatever else g messes up will be restored by g'.
He was astounded when we showed him that approach. It had never even occurred to him that such a thing was possible.
The funny thing is he was a math major, and had taken (and passed with a good grade) an abstract algebra course. He definitely knew about conjugates of group elements. He had just never noticed that group theory was applicable to Rubik's Cube.