# The laws of a Rubik's Cube



## xXdaveXsuperstarXx (Aug 4, 2009)

I used the search function and I think it would be helpful to make a thread like this. So I'll say what I know. *Please* no silly stuff, this is serious.

1. You can only flip an even amount of edges.

2. You can never flip 2,4,5,7, or 8 corners in the same direction. 

3. You can only do an even amount of swaps. 

4. You can only cycle an odd amount of pieces. 

5. You can't flip a single corner

Um, I think I got all of them. Is there any more that I'm forgetting?


----------



## Stefan (Aug 4, 2009)

Quite flawed. Just take this: http://www.ryanheise.com/cube/cube_laws.html


----------



## xXdaveXsuperstarXx (Aug 4, 2009)

Hm. I wrote down things that seem impossible on my cube.


----------



## Stefan (Aug 4, 2009)

Here's one hint: You forgot the following law:

You can never flip 4 corners in the same direction.


----------



## Kickflip1993 (Aug 4, 2009)

StefanPochmann said:


> Here's one hint: You forgot the following law:
> 
> You can never flip 4 corners in the same direction.



Sune/Anti-Sune + U-Perm from that position = 1 corner flipped


----------



## Stefan (Aug 4, 2009)

Kickflip1993 said:


> Sune/Anti-Sune + U-Perm from that position = 1 corner flipped


Huh?


----------



## xXdaveXsuperstarXx (Aug 4, 2009)

> You can never flip 4 corners in the same direction.



Oh, so it should should be you can never flip an even amount of corners in the same direction?


----------



## watermelon (Aug 4, 2009)

xXdaveXsuperstarXx said:


> > You can never flip 4 corners in the same direction.
> 
> 
> 
> Oh, so it should should be you can never flip an even amount of corners in the same direction?



If you do a Sune on U, then a Sune on D, you have rotated 6 corners in the same direction.


----------



## xXdaveXsuperstarXx (Aug 4, 2009)

Okay, I fixed that. Still something wrong?


----------



## Stefan (Aug 4, 2009)

xXdaveXsuperstarXx said:


> Okay, I fixed that. Still something wrong?


Yeah. Your approach.

Another hint: You forgot this rule:

You can never flip 5 corners in the same direction.


----------



## Waffle's Minion (Aug 4, 2009)

StefanPochmann said:


> xXdaveXsuperstarXx said:
> 
> 
> > Okay, I fixed that. Still something wrong?
> ...



Wait, i think i have the rule. You cannot flip an amount of corners in the same direction unless the amount is divisible by 3.


----------



## Stefan (Aug 4, 2009)

Now that Waffle did it, too, it's pissing me off too much.

http://web.archive.org/web/20080206044316/www.wsu.edu/~brians/errors/amount.html


----------



## xXdaveXsuperstarXx (Aug 4, 2009)

Yeah, so if you multiply that number buy 2, if it's legal it should be divisible by 3. So if you multiply 2, 4, or 5 by 2 it won't be divisible by 3.


----------



## waffle=ijm (Aug 4, 2009)

StefanPochmann said:


> Now that _Waffle _did it, too, it's pissing me off too much.
> 
> http://web.archive.org/web/20080206044316/www.wsu.edu/~brians/errors/amount.html



it's my minion, not me 
just to clarify on things, I'm not into the laws of cubes


----------



## Stefan (Aug 4, 2009)

xXdaveXsuperstarXx said:


> 2. You can never flip 2,4,5, or 7 corners in the same direction.


Wow. You forgot 8.


----------



## xXdaveXsuperstarXx (Aug 4, 2009)

At least I got the right equation.


----------



## Stefan (Aug 4, 2009)

Hint: You forgot this law:

You can never flip 2 corners in the same direction and 1 in the other direction.


----------



## elcarc (Aug 4, 2009)

easiest rule. 

its impossible to get sub 3 sec unprepared solve. humans just dont move fast enough


----------



## xXdaveXsuperstarXx (Aug 4, 2009)

Okay, so let me guess, you can never flip 2,4,5,7,8 corners in the same direction and flip 1 corner in another direction.


----------



## jcuber (Aug 4, 2009)

elcarc said:


> easiest rule.
> 
> its impossible to get sub 3 sec unprepared solve. humans just dont move fast enough



Yet. Thinking like that will get us nowhere. The sky's the limit!


----------



## Stefan (Aug 4, 2009)

xXdaveXsuperstarXx said:


> Okay, so let me guess, you can never flip 2,4,5,7,8 corners in the same direction and flip 1 corner in another direction.


False. And more importantly, you're still missing the point.


----------



## xXdaveXsuperstarXx (Aug 4, 2009)

The point being I suck at cube theory and should get this thread deleted? If so you can just delete this thread. BTW I thought you used to be a moderator?


----------



## Stefan (Aug 4, 2009)

I'll give you one final hint.

You forgot this law:
You can never flip 4 corners in the same direction and 3 in the other direction.

The point is that I could go on and on and on like this and you'd add more and more and more laws. The list would grow huge. Instead, come up with one general description that enlightens and covers all cases. Or... just read it on Ryan's page which I showed earlier. It's btw the #1 result if you google this thread's title.

Edit: Even if you listed all impossible corner orientation combinations, you'd still only talk about pure orientations. You'd still not cover combinations of orientation and permutation.


----------



## Lucas Garron (Aug 5, 2009)

xXdaveXsuperstarXx said:


> *Please* no silly stuff, this is serious.


Then stop posting.



xXdaveXsuperstarXx said:


> You can never flip ...4... corners in the same direction.


R U2 R' U2' R U R' U' R U2 R' U2' R U R' U'


----------



## FredM (Aug 14, 2009)

Come on !

You can clearly see what the cube restrictions are when you assemble the cube in a scrambled state. You can have three impossibilities : 

Permutation case : Odd amount of swaps (adding edges and corners, considering centers as fixed)
Edge orientation : Odd amount of edges flipped
Corner orientation : Overall orientation of corners not equal to 0 (modulo 360°)


----------



## mrCage (Aug 14, 2009)

xXdaveXsuperstarXx said:


> I used the search function and I think it would be helpful to make a thread like this. So I'll say what I know. *Please* no silly stuff, this is serious.
> 
> 1. You can only flip an even amount of edges.
> 
> ...


 
How about when you disassemble? 

Point 3 and 4 are not phrased strictly enough. One can clearly swap only 2 edges (if 2 corners also swapped for instance ...)

Per


----------



## fanwuq (Aug 15, 2009)

A cube may not injure a human being or, through inaction, allow a human being to come to harm.
A cube must obey any orders given to it by human beings, except where such orders would conflict with the First Law.
A cube must protect its own existence as long as such protection does not conflict with the First or Second Law.


----------



## Escher (Aug 15, 2009)

fanwuq said:


> A cube may not injure a human being or, through inaction, allow a human being to come to harm.
> A cube must obey any orders given to it by human beings, except where such orders would conflict with the First Law.
> A cube must protect its own existence as long as such protection does not conflict with the First or Second Law.



This post gets +1 gazillion.


----------



## Sa967St (Aug 15, 2009)

Lucas Garron said:


> xXdaveXsuperstarXx said:
> 
> 
> > *Please* no silly stuff, this is serious.
> ...


+1


----------



## krazedkat (Aug 28, 2009)

You can never make it so that two or more of the same colour aren't on the same side.


----------



## cmhardw (Aug 28, 2009)

krazedkat said:


> You can never make it so that two or more of the same colour aren't on the same side.



In all seriousness, I like this train of thought.

Let's make it a little more precise. Assume we have a standard 3x3x3 cube with 6 distinct colors used, and each of the 6 faces having 9 of the same distinct color per face.

Using the above law, can we derive all of the standard laws of the cube? I purposefully do not state the standard laws, as the thread is still leading others to correctly state these laws. I do not want to spoil this. Perhaps those working on this new question I am proposing can use spoiler tags?



Spoiler



I am wondering if this law, as stated by krazedkat, is enough (given my assumption above) to actually derive all of the standard laws of the cube. This problem interests me, I might take a look at this a little more seriously in my free time this weekend.



Chris


----------



## Jigsaw (Aug 28, 2009)

fanwuq said:


> A cube may not injure a human being or, through inaction, allow a human being to come to harm.
> A cube must obey any orders given to it by human beings, except where such orders would conflict with the First Law.
> A cube must protect its own existence as long as such protection does not conflict with the First or Second Law.



lol we all know how flawed these laws are


----------



## shelley (Aug 28, 2009)

xXdaveXsuperstarXx said:


> 4. You can only cycle an odd amount of pieces.



No.


----------



## rahulkadukar (Aug 28, 2009)

Golden Rule:

You cannot flip one Edge


----------



## shelley (Aug 28, 2009)

rahulkadukar said:


> Golden Rule:
> 
> You cannot flip one Edge



Or one corner. You also cannot swap only two pieces.


----------



## happa95 (Aug 28, 2009)

fanwuq said:


> A cube may not injure a human being or, through inaction, allow a human being to come to harm.
> A cube must obey any orders given to it by human beings, except where such orders would conflict with the First Law.
> A cube must protect its own existence as long as such protection does not conflict with the First or Second Law.



Yay for isaac asimov references!


----------



## *LukeMayn* (Aug 28, 2009)

We seem just to be stating the obvious in this thread :/


----------



## cmhardw (Aug 28, 2009)

rahulkadukar said:


> Golden Rule:
> 
> You cannot flip one Edge



Not to be a complete stickler for semantics, but I completely disagree. I can easily flip the UF edge using U' R' F'. Now, doing so has a massively destructive *side effect* to the other pieces, but I have flipped the one edge I intended to. Please phrase your laws more precisely.

Chris


----------



## JLarsen (Sep 12, 2009)

xXdaveXsuperstarXx said:


> Yeah, so if you multiply that number buy 2, if it's legal it should be divisible by 3. So if you multiply 2, 4, or 5 by 2 it won't be divisible by 3.


As Pochmann was nudging at here, you need to succinctly summarize the laws of the cube, but you cannot do that, as you have purely memorized laws, rather than understanding the reason behind them. I don't pretend to think I understand all the reasons, but then again, I don't make threads such as this. 



cmhardw said:


> rahulkadukar said:
> 
> 
> > Golden Rule:
> ...



I.E "You cannot flip *one and only one* edge." Or, "You cannot have one edge flipped while the rest of the puzzle in a solved state"


----------



## daniel0731ex (Sep 12, 2009)

three laws of cubing:

1. the best and worst solve will appear at the same time
2. others' cubes always turn better than yours
3. discount happens just after you bought it


i saw this on PTT


----------



## Cride5 (Sep 15, 2009)

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


----------



## JLarsen (Sep 15, 2009)

Cride5 said:


> 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 (Sep 15, 2009)

@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 (Sep 15, 2009)

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


----------



## Stefan (Sep 16, 2009)

rjohnson_8ball said:


> 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 (Sep 16, 2009)

StefanPochmann said:


> rjohnson_8ball said:
> 
> 
> > The poster meant on a 3x3 there must be an even number of swapped _pairs_.
> ...



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 (Sep 16, 2009)

StefanPochmann said:


> rjohnson_8ball said:
> 
> 
> > The poster meant on a 3x3 there must be an even number of swapped _pairs_.
> ...


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.



StefanPochmann said:


> _"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..



StefanPochmann said:


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


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


----------



## cubeninjaIV (Sep 16, 2009)

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


----------



## rjohnson_8ball (Sep 16, 2009)

@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 (Sep 16, 2009)

rjohnson_8ball said:


> @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!!


rjohnson_8ball said:


> 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 (Sep 16, 2009)

rjohnson_8ball said:


> 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.



rjohnson_8ball said:


> 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.



rjohnson_8ball said:


> 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.



rjohnson_8ball said:


> 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 (Sep 16, 2009)

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 (Sep 17, 2009)

StefanPochmann said:


> 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.




StefanPochmann said:


> 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..




StefanPochmann said:


> 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 (Sep 17, 2009)

> by Stephan
> 
> 
> rjohnson_8ball said:
> ...


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 (Sep 17, 2009)

StefanPochmann said:


> rjohnson_8ball said:
> 
> 
> > The 4-cycle can only be fixed by an odd number of swapped pairs.
> ...



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 (Sep 18, 2009)

Cride5 said:


> 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:


Sn3kyPandaMan said:


> Cride5 said:
> 
> 
> > Only an even number of cubie swaps is possible (1/2 of all states)
> ...


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 (Sep 18, 2009)

StefanPochmann said:


> 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.



StefanPochmann said:


> Speaking of the footnotes...
> 
> _*Only an even number of edges can be flipped* (1/2 of all states) [1]
> [1] Assuming ..._
> ...



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 (Sep 19, 2009)

Cride5 said:


> 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. 



Cride5 said:


> 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 (Sep 19, 2009)

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 (Sep 19, 2009)

fazrulz said:


> 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)
> 
> ...



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


----------



## miniGOINGS (Sep 19, 2009)

randomtoad said:


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



I have a question, for the determining factors of the speed of a solve so far I have:

-Number of Moves
-Recognition
-Look Ahead
-Turning Speed

Are there any that I missed?


----------



## Toad (Sep 19, 2009)

miniGOINGS said:


> I have a question, for the determining factors of the speed of a solve so far I have:
> 
> -Number of Moves
> -Recognition
> ...



Look ahead surely comes under Recognition if you're looking at it that broadly?


----------



## miniGOINGS (Sep 19, 2009)

randomtoad said:


> miniGOINGS said:
> 
> 
> > I have a question, for the determining factors of the speed of a solve so far I have:
> ...



I'm not sure what you mean, I was just listing them.


----------



## JLarsen (Sep 19, 2009)

This thread needs to die.


----------



## Toad (Sep 19, 2009)

Sn3kyPandaMan said:


> This thread needs to die.



+1


----------

