# Theoretical Question - Rubik's Cubes

I have a very basic question on Rubik's Cubes.

Take any completely solved cube or skewb. Now pick out (with hand) any two similar pieces and swap their positions, without changing positions/orientation of any other pieces in the cube/skewb. This new cube/skewb is also completely solved except for the swapped pieces. Can this new cube/skewb be achieved from the old solved cube/skewb by using only moves, without any hand picking?

I have an intuition that it is not possible to do so, but I am unable to prove it formally.

One small correction first: It's Rubik's Cube, not Rubix. ;)

And I will get back to you/edit my answer for Skewb when I found out. And also no for a Skewb.

## 3x3x3 Cube:

Let's start with the 3x3x3 Cube. The only moves available are face turns and slice turns. However, slice turns like M, can be deducted into two face turns RL'. Because of that, we can say the only available moves on a 3x3x3 Cube are face-turns.

So what does a face-turn do?

• It 4-cycles four edges;
• It 4-cycles four corners;
• And it orients the center by 90 degrees.

You can't do one, without the other.

When we look at how we solve twisty puzzles, we always have to have an Even State to solve them. So when we swap a set of two corners, and a set of two edges, it means we do two swaps, which is even. When we look at the 4-cycles of a face-turn, we can say we need three corner swaps AND three edge swaps to accomplish the two 4-cycles. Odd + Odd = Even, so a face-turn is an Even turn.
A 3x3x3 Cube is in an Even State when it's solved (zero swaps left). So since we started with Even, no amount of turns will change that. Even + Even = Even, and therefore a regular 3x3x3 Cube is always solvable.

But when we look at what you proposed and have manually swapped two corners, we start with an Odd State, and solving it is an Even State. But, Odd + Even = Odd, and this will always remain Odd, and therefore it will remain unsolvable no matter how many turns you make.

## Skewb:

EDIT: Actually, the same applies to Skewb.

Let's look at what a corner turn on a Skewb does:

• It 3-cycles three corners;
• It 3-cycles three centers;
• It orients a corner by 120 degrees.

When we look at a 3-cycle of a corner turn, we can saw we need two corner swaps AND two center swaps to accomplish the two three-cycles. Even + Even = Even, so a corner turn is an Even turn.
So the same applies to the Skewb as to the 3x3x3 Cube.

# Parity:

Parity on twisty puzzles are seemingly impossible situations when we follow the basic law of cubes explained above. An example of parity is when we are left with a single swap in order to solve the puzzle.

On a 4x4x4 Cube for example, we can have everything solves except for two swapped corners, or except for a single incorrectly oriented edge, or two reduced edges that need to swap.

Since this is outside of your question, I won't go to deep into this. But this post on the TwistyPuzzles Forum explains the different kind and causes of parity pretty clearly.

• Just a proof of why product of two even swaps is always even . In other words, why the parity of a permutation is uniquely defined. faculty.luther.edu/~macdonal/EvenOdd.pdf – kasa Jun 19 '17 at 14:30