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The part of the proof that shows that you can actually reach all the

configurations in a particular equivalence class is not particularly

elegant. Basically, you have to appeal to the details of a particular

cube solving algorithm.

For example, I have a tool that "flips" two edge cubies in place,

without desturbing anything else. This tool shows that I can orient

the edge cubies in ANY even permutation of the edge cubie faces. The

fact that I cannot obtain any odd permutations is a result of the fact

that a quarter twist is itself an even permutation of the edge cubie

faces.

Most people can examine their own cube solving tools and see that in

fact, they are capable of obtaining all the configurations not

forbidden by the familiar constraints.