Help with proving property of Rubik's cube.

Question

I am looking at these notes, and am trying to prove the following on page 13:

Let $C_1$ and $C_2$ be two different unoriented corner cubies, and let $C'_{1}$ and $C'_{2}$ be two different unoriented corner cubicles. Prove that there is a move of the Rubik’s cube which sends $C_{1}$ to $C'_{1}$ and $C_2$ to $C'_2$. Since we are talking about unoriented cubies and cubicles, we only care about the positions of the cubies, not their orientations. (For example, if $C_1$ = dbr, $C_2$ = urf, $C'_{1}$ = dlb, and $C'_{2}$ = urf, then the move $D$ sends $C_1$ to $C'_{1}$ and $C_2$ to $C'_{2}$.)

My first thought was to use induction. The general idea was show that you can move any particular cube to a desired cubicle, then do this for one cubie, then the next cubie, and so on... There was no (relatively simple) base case I could come up with. Likewise, I couldn't think of a good inductive step. What if a necessary rotation for one cubie displaced other cubie and vice versa?

Honestly, I don't have any good ideas after that. I've tried messing around with conjugates and commutators hoping something will minimize the support between two cubies and allow me to obtain a solution. Even with the Rubik's cube in front of me I can't seem to visualize of kinesthetically think of where I might go from here. Any ideas of what I should try would be greatly appreciated.

Answer 1

It is easy to find a move which sends $C_1$ to $C_1'$:

1. if $C_1$ and $C_1'$ are on the same face, just rotate that face until $C_1$ goes on $C_1'$
2. otherwise $C_1$ is diagonally opposite to $C_1'$. Rotate any face containing $C_1$ to reduce to the previous case.

Now you need a sequence of moves which sends $C_2$ to $C_2'$ without moving $C_1$. There are three faces containing $C_1'$ and you can achieve your task withouth moving them:

3. if $C_2$ is on a face containing $C_2'$ and not containing $C_1'$ you can rotate this face untile $C_2$ goes on $C_2'$
4. otherwise there is a face containing $C_2$ but not $C_1$ and moving that face you reduce to case 3.