Imposible sates on the 3x3x3

[Deleted member 36445]

I searched this on the internet but coudn't find.

Why its impossible to twist a corner, flip one edge or permute two pieces?

I showed some people how to solve the cube and explained the impossible states, but when they asked why it was impossible I just said it was mathematically impossible to do it, but coudn't explain further.

Anyone has the explanation for this?

Thanks.

 

[Lucas Garron]

Search for "parity". There are already some extensive discussions and explanations on this.

 

[cmhardw]

This website is a good place to start.

 

[Deleted member 36445]

Thanks guys for the fast replies

 

[porkynator]

It can be proven by (mathematical) induction.
I'll try to explain the "permute two pieces" intuitively.
Start with the cube solved; if you perform one turn, you can see that, trying to memorize like you would do for a blind solve, the numbers of needed edge-swaps and corner-swaps are either both odd (if you have performed a quarter turn move) or both even (if you have performed a half turn move); thus, the total numbers of swaps needed is even, in both cases. Now you can perform another move and, whatever number of swaps you need to solve the cube you can see that you need an even number to "undo" the last move (because it is only one move, and we've already seen that after one move you need an even number of swaps to get back) and then another even number to solve the cube. You can then do another move ad so on, and since even + even + even + ... = even you will always need an even number of swaps to solve the cube (assuming you have started from the solved state). Thus, cases like "two edges swapped" are "impossible" because they can't be reached from the solved state.
This is not a rigorous mathematical proof, but this is the main idea. The other two cases are relatively similar; the "one edge flipped case" is even easier if you know ZZ (or Petrus at least).

EDIT: ouch, double (triple?) ninja'd

 

[Christopher Mowla]

I also remember this thread being very good.

 

[GlowingSausage]

I just said it was mathematically impossible to do it, but coudn't explain further.

incorrect, more like physically

 

[qqwref]

No, it is mathematically impossible. You can represent the cube as a permutation group generated by certain allowed operations, and then show that positions like a single corner twist are not in that group.