The laws of a Rubik's Cube

[Cride5]

There's a new page on the Wiki about this here.

 

[JLarsen]

Only an even number of cubie swaps is possible (1/2 of all states)

I'm confused by this. Isn't a swap defined as one piece occupying the location of another, and vice versa? Therefore if you have a piece unsolved, it isn't "swapped" with anything? Help?

 

[rjohnson_8ball]

@Sn3kyPandaMan, The poster meant on a 3x3 there must be an even number of swapped pairs. For example, an H-perm or Z-perm or T-perm or J-perm or R-perm swaps 2 pairs. A U-perm can be thought of swapping one pair and then a second pair to produce the 3 way swap. The only way to have just one swapped pair is to assemble the 3x3 wrong.

 

[JLarsen]

Look like we need a minor edit then. good rules still.

 

[Stefan]

The poster meant on a 3x3 there must be an even number of swapped pairs.

And... that's wrong. Counter-example: F2 U M' U2 M U F2 U'. There's just one swapped pair.

"Only an even number of edges can be flipped" is also bad as we're not told what "flipped edge" means. Same with his third rule and "corner twists".

Though thankfully he linked to Ryan's page which does it all properly.

 

[rjohnson_8ball]

The poster meant on a 3x3 there must be an even number of swapped pairs.

And... that's wrong. Counter-example: F2 U M' U2 M U F2 U'. There's just one swapped pair.

"Only an even number of edges can be flipped" is also bad as we're not told what "flipped edge" means. Same with his third rule and "corner twists".

Though thankfully he linked to Ryan's page which does it all properly.

It looks to me like your U-perm with U' appended gives 1 swap of 2 edges but also a 4-cycle of corner pieces. The 4-cycle can only be fixed by an odd number of swapped pairs. So the total number of swapped pairs would be even.

I don't see how the Wiki made it any clearer. Would it be more clear if we just said a 3x3 cannot have all pieces except 2 in their correct position?

I just re-read http://www.ryanheise.com/cube/cube_laws.html. The first sentence of the permutation section mentions even number of swaps. What did I say differently?

 

[Cride5]

The poster meant on a 3x3 there must be an even number of swapped pairs.

And... that's wrong. Counter-example: F2 U M' U2 M U F2 U'. There's just one swapped pair.

This position is solvable only by an even number of cubie swaps.
URF <-> UFL
URF <-> ULB
URF <-> RBR
UR <-> UB

If you actually meant U-Perm, then again its an even number of swaps:
UR <-> UL
UR <-> UB

Pair swaps is misleading, since you can imagine taking a pair of cubies, and swapping it with another pair of cubies (an odd number of pair swaps)

Is it possible for a cubie swap not to involve a pair of cubies?? The common definition of swap is to exchange one thing for another. The use of pair is completely redundant and implied by the use of the word swap. There is no ambiguity as to what a cubie swap is.

"Only an even number of edges can be flipped" is also bad as we're not told what "flipped edge" means. Same with his third rule and "corner twists".

Fixed..

Though thankfully he linked to Ryan's page which does it all properly.

What's with the constant need to be offensive??

 

[cubeninjaIV]

things will always go out of stock the day before you buy them
seriously cube for you has NO 2x2s

 

[rjohnson_8ball]

@Cride5, I agree that the word "pair" should not be needed. But I used the word to help unconfuse Sn3kyPandaMan. Sometimes people describe a 3-cycle as a "3 way swap", which may be technically improper, but still it is a popular description. I used the word "pair" to help clarify (I thought) that each swap should involve 2 pieces.

 

[Cride5]

@Cride5, I agree that the word "pair" should not be needed. But I used the word to help unconfuse Sn3kyPandaMan.

Its cool ... I think what Stefan was probably getting at is that:
a swapped pair [of cubies]
...is not the same as a...
pair of swapped cubies
..but I appreciate your explanation

I'm not sure if it helps, but to try answering SneakyPanda's question: More than one swap of each piece with another may be required to solve a cube by physically swapping pieces around. Using the idea of swaps doesn't mean the cube has to be a state which is solvable by swapping each piece exactly once.

EDIT: lol, Ninja'ed again!!

I used the word "pair" to help clarify (I thought) that each swap should involve 2 pieces.

You're correct. In the context of the law I wrote in the Wiki, we are counting a swap of exactly two pieces as a swap. A three cycle can be thought of as three pieces replacing each other in a particular direction (like musical chairs), but can also be conceptualised as two individual swaps. My response to Stefan above, describing the number of swaps in a U-perm illustrates how.

 

[Stefan]

It looks to me like your U-perm with U' appended gives 1 swap of 2 edges but also a 4-cycle of corner pieces.

Correct.

The 4-cycle can only be fixed by an odd number of swapped pairs.

Wrong. I can also fix it with a U turn, that's two 4-cycles. And zero swaps, an even number.

So the total number of swapped pairs would be even.

Wrong. There's exactly one swapped pair, just like you yourself said two sentence ago.

I just re-read http://www.ryanheise.com/cube/cube_laws.html. The first sentence of the permutation section mentions even number of swaps. What did I say differently?

Let me provide a longer quote:
"every cube state reachable by legal moves can always be represented by an even number of swaps"
The bolded part is what you're missing. A 4-cycle can be represented by swaps, but it isn't swaps. It is a 4-cycle.

 

[Stefan]

And to be constructive... I suggest to state the minimal effects that are impossible:

1. You can't just swap two pieces
2. You can't just flip one edge
3. You can't just rotate one corner

That might suffice as a short version, giving people an idea of the "laws". For a proper understanding for people interested enough, I think Ryan got about as concise as possible.

Oh and another thing missing on our wiki page that I just noticed:
"Only a 12th of possible cube states is reachable by twisting the 6 faces"
What are "cube states"? Again, see Ryan's page for proper treatment.

 

[Cride5]

Oh and another thing missing on our wiki page that I just noticed:
"Only a 12th of possible cube states is reachable by twisting the 6 faces"
What are "cube states"? Again, see Ryan's page for proper treatment.

I've amended the first sentence to clarify.

And to be constructive... I suggest to state the minimal effects that are impossible:

1. You can't just swap two pieces
2. You can't just flip one edge
3. You can't just rotate one corner

Nice suggestion, thank you. Added to the page..

That might suffice as a short version, giving people an idea of the "laws". For a proper understanding for people interested enough, I think Ryan got about as concise as possible.

The page on the Wiki was not by any means intended to replace Ryan's explanation. I agree that a much fuller understanding of the laws can be gained by reading his page. The reason I put it in the Wiki was to make them easily accessible to anyone browsing our Wiki for information. In the same way the method pages rarely go into detail about the algs/substeps involved, the laws page does not go into huge detail about how the laws were derived etc. It is intended to be used as a quick reference only.

 

[rjohnson_8ball]

by Stephan

The 4-cycle can only be fixed by an odd number of swapped pairs.

Wrong. I can also fix it with a U turn, that's two 4-cycles. And zero swaps, an even number.

I thought it was reasonable that other pieces should not be disturbed when attempting to fix the 4-cycle via swaps. A U-turn would have a side effect on how the edges need to be swapped, so my earlier note about swapping 2 edges would become invalid.

One training case I do for BLD is perform one 90 degree turn on a face, and solve it using 3OP. I do as much as I can using 3OP cycles only, and then the last step involves swapping 2 edges and 2 corners. This is related to the case you gave, because I use an odd number of swaps for edges and odd number of swaps for corners to finish. Of course a person might easily see that a 90 degree twist would solve it, but a computer program which solves by 3OP might not see it.

Okay, I left out the words can be represented.

 

[Cride5]

The 4-cycle can only be fixed by an odd number of swapped pairs.

Wrong. I can also fix it with a U turn, that's two 4-cycles. And zero swaps, an even number.

Stefan, the context of this conversation is about many cubie swaps are required to solve particular positions, assuming only swaps are used. The law in question does not operate on the cube using U turns.

Paying attention to the context of rjhonson's statements, and under the assumption that when he says 'swapped pairs' he's really referring to cubie swaps, he is not wrong.

 

[Stefan]

Stefan, the context of this conversation is about many cubie swaps are required to solve particular positions, assuming only swaps are used.

You just expressed this "context" for the very first time. And it's still wrong to say there must "be" an even number of swaps as that is more likely understood as "be" (on the cube) than as "be in a solution using only swaps by taking apart the cube", unless your "context" is explicitly mentioned.

The actual context was your wiki page at that time plus Sn3kyPandaMan post, who I think saw the same problem I saw:

Only an even number of cubie swaps is possible (1/2 of all states)

I'm confused by this. Isn't a swap defined as one piece occupying the location of another, and vice versa? Therefore if you have a piece unsolved, it isn't "swapped" with anything? Help?

I think he meant "unsolved" as in for example a 4-cycle, where each piece is indeed not swapped with anything.

The wiki page is getting much better, btw, especially the Recognising an Unsolvable Cube section is very well-done. I do appreciate you're working on the page. I thought about doing it myself but knew it would've taken me a lot of time to do it well, time I don't have right now.

You could just use Ryan's titles as the laws:
1. Only half of the permutations are reachable
2. Only half of the edge orientations are reachable
3. Only one third of the corner orientations are reachable
Less confusing, correct, and you wouldn't need your footnotes.

Speaking of the footnotes...

Only an even number of edges can be flipped (1/2 of all states) [1]
[1] Assuming ...

Let me use the same technique for another statement:

Edges cannot be flipped [1]
[1] Assuming you use only <U, D, L R, F2, B2>

 

[Cride5]

You could just use Ryan's titles as the laws:
1. Only half of the permutations are reachable
2. Only half of the edge orientations are reachable
3. Only one third of the corner orientations are reachable
Less confusing, correct, and you wouldn't need your footnotes.

Newton could have described gravity by saying "dropped apples will always fall", and although this is certainly true it is a consequence of gravity. A proper description would describe the way in which objects' masses result in a force of attraction.

The statements you quoted are just consequences of the of the laws. They don't say which orientations/permutations are reachable. The reason I wanted to mention cubie swaps, is that it is how the generalisation is derived. It also enables someone reading the laws to determine whether their cube is solvable, without attempting to solve it first.

Speaking of the footnotes...

Only an even number of edges can be flipped (1/2 of all states) [1]
[1] Assuming ...

Let me use the same technique for another statement:

Edges cannot be flipped [1]
[1] Assuming you use only <U, D, L R, F2, B2>

I know the footnotes are not desirable, but I put them there so that the laws stand out clear and concise. Like a lot of theories, there is always a context, or set of assumptions. Although they should always be stated, mixing them up with the core statement makes them less readable.

I see the problem with the way the edge orientation law/assumptions didn't go together well. I've updated the page to make it clearer.

Let me know if there are any more loopholes

 

[Stefan]

The statements you quoted are just consequences of the of the laws. They don't say which orientations/permutations are reachable.

Ok, I admit they're not so good as "laws" for our purpose here. On the top level, Ryan's page seems more concerned with the "1 in 12", so there they fit his purpose well. I still like "even permutations" better than "even number of swaps", though, but I guess the latter might be more useful for someone who doesn't know permutation parity yet.

And while we're talking about different levels (laws/consequences)... I want to point out that what you're calling laws are actually consequences. The real laws are just how the puzzle moves. Our three "laws" are just observed consequences of that.

Btw, you might want to merge or at least connect your page with this section:
http://www.speedsolving.com/wiki/index.php/General_Information#Permutations
Not sure how to do it best, but there's now redundancy and no connection. And I think the number of assembleable states, the number of possible states, and those three laws really belong together.

I know the footnotes are not desirable, but I put them there so that the laws stand out clear and concise.

Right, but my point was not the formatting but the "Assuming" (sorry I didn't make it clear). It's slightly better now, but it's still... so arbitrary. You make a statement and provide a context but it's not clear why you chose that particular context, and it's not clear whether the statement would still be true for other contexts. This is really bugging me.

 

[Faz]

Meh, I tried to come up with 3 laws that mimic Newtons laws of motion. As you can see, I failed.

1. A cube at rest will stay at rest until it is turned (Meh)

2. PLL = F2L + OLL (Bad)

3. For every 3 cycle, there is an equal and opposite 3 cycle. (I like this one)

 

[Toad]

Meh, I tried to come up with 3 laws that mimic Newtons laws of motion. As you can see, I failed.

1. A cube at rest will stay at rest until it is turned (Meh)

2. PLL = F2L + OLL (Bad)

3. For every 3 cycle, there is an equal and opposite 3 cycle. (I like this one)

I like the third one... for the second how about "Speed of solve = Number of moves / Moves per second" ?