# 3x3 cube number of positions with a different alg to solve



## kvaele (Dec 23, 2010)

Sorry for the confusion probably caused be the title, but I don't know how else to put it in that few words. I know that there are 43 and some odd quintillion positions, but what I'm wondering is how many of those arent reapeats? There are quite a few of those cases that have the same solution, just with different rotations, but how many? If you still don't get what I mean here is an example:
Two of those 43 quintillion positions can be made by either doing an R or L turn. They are considered 2 different positions, but you can solve them by doing *R'* and *Y2* *R'* respectively (Y2 may be the wrong rotation but you get the point).
Is there a way to figure this out? If it is obvious please dont :fp at me. I really don't know how to go about doing this. Any help, or an answer, would be great!


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## Julian (Dec 23, 2010)

I believe you divide 43 quintillion by 24 (number of cube orientations) to find the number of different possibles states. Unless the calculation has already been done to arrive at the number 43 quintillion.


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## Christopher Mowla (Dec 23, 2010)

I believe you are wondering if the number of scrambled positions would be less than what it is said to be because you said "how many of those aren't repeats". Well, sorry, but there are still approximately 43 quintillion *different* positions. How you can solve back these positions is different than the number of positions themselves. But what I think you are trying to say is, "How many distinct positions are there that are not mirror images of each other". I am sure someone can answer this question to your satisfaction.


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## cuBerBruce (Dec 23, 2010)

Julian said:


> I believe you divide 43 quintillion by 24 (number of cube orientations) to find the number of different possibles states. Unless the calculation has already been done to arrive at the number 43 quintillion.


 
Well that would be approximately the right value for the number of positions that would be considered "distinct" under what the original poster was trying to express. But the calculation is much more complicated than simply dividing the number of states by 24. This is because some states have symmetry and don't have 23 other equivalent states. For example L is only equivalent to R, U, D, F, and B (and itself, of course), so only 6 states rather than 24.

If you consider symmetry of reflection and not just cube rotations, then the correct answer would be approximately the number of states divided by 48. (For example L would also be considered equivalent to L', R', U', D', etc.) The exact value for this was calculated quite some time ago as 901,083,404,981,813,616. See this link.


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## Stefan (Dec 23, 2010)

cmowla said:


> Well, sorry, but there are still approximately 43 quintillion *different* positions.



Depends on how you define "different".

Looks like 1,802,166,805,653,080,256:
http://www.math.rwth-aachen.de/~Mar...ert__Re__Re__Re__Re__Models_for_the_Cube.html


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## kvaele (Dec 23, 2010)

Cmowla, I do not mean mirror images, I mean things that are the same scramble, just with different orientations, like doing an R turn on the red face and doing an R turn on the orange face.


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## kvaele (Dec 23, 2010)

Julian said:


> I believe you divide 43 quintillion by 24 (number of cube orientations) to find the number of different possibles states. Unless the calculation has already been done to arrive at the number 43 quintillion.


 
Yes, there are 24 orientations, but if you do a D turn it is the same exact thing as doing X F, so that will reduce the orientations.


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## kvaele (Dec 23, 2010)

Stefan said:


> Depends on how you define "different".
> 
> Looks like 1,802,166,805,653,080,256:
> http://www.math.rwth-aachen.de/~Mar...ert__Re__Re__Re__Re__Models_for_the_Cube.html



Are you sure? It seems to me like that is far too few. This reduces the number of total positions by about 42/43, implying that there are over 40 orientations to each case.


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## Stefan (Dec 23, 2010)

kvaele said:


> This reduces the number of total positions by about 42/43


 
No.


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## Christopher Mowla (Dec 23, 2010)

kvaele said:


> Cmowla, I do not mean mirror images, I mean things that are the same scramble, just with different orientations, like doing an R turn on the red face and doing an R turn on the orange face.



There does exist a formula to tell you the number of "the same scramble*s*, just with different orientations" as in corners and edges are swapped in the same slots (in the same fashion), but they are just twisted different. For that situation, Chris Hardwick (his username is cmhardw) said to make the following adjustment to the formula that tells the number of possible scrambled positions to have:


cmowla said:


> *Here is his adjusted formula that ignores orientations *(that is, if I understood the changes he said to be made).


From this thread.

_Just substitute n=3 in the formula in wolfram alpha to get the result for the 3x3x3.

_Also,
This


kvaele said:


> I mean things that are the same scramble, just with different orientations


and this


kvaele said:


> like doing an R turn on the red face and doing an R turn on the orange face.


don't make sense to describe the same thing. Doing "R turn on the red face and doing an R turn on the orange face" is just the inverse of the mirror image. So what you want is related to the mirror image.


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## Ravi (Dec 24, 2010)

kvaele said:


> Are you sure? It seems to me like that is far too few. This reduces the number of total positions by about 42/43, implying that there are over 40 orientations to each case.


 
Punch it into a calculator. 43252003274489856000/1802166805653080256 = 23.999999966049719... So almost all positions have 24 distinct orientations, and a relatively tiny number (perhaps only a few billion) have fewer distinct orientations due to rotational symmetry.

It's also worth noting that some people have calculated the "real size of cube space" by modding out by rotations, reflections, and inverses. This gives a number just over a quarter of the number quoted above: about half because of reflections and about half again because of inverses.


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## kvaele (Dec 24, 2010)

Wow sorry I have no idea what I was looking at.


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## maggot (Dec 24, 2010)

if youre asking about applying the same scramble to 2 cubes using different orientations, i.e. scramble one cube as U-white F- green and scramble another as U-yellow, F-Blue, i believe that is considered the same orientation. You can arrive at the solved state with the same solution, by applying the solution to each cube in the starting position of scrambles orientation. I do not believe that these are considered unique solutions.


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## kvaele (Dec 24, 2010)

maggot said:


> if youre asking about applying the same scramble to 2 cubes using different orientations, i.e. scramble one cube as U-white F- green and scramble another as U-yellow, F-Blue, i believe that is considered the same orientation. You can arrive at the solved state with the same solution, by applying the solution to each cube in the starting position of scrambles orientation. I do not believe that these are considered unique solutions.


 
I'm almost positive they are. The formula for the total number of positions takes into account *all* possible permutations and orientations and doesnt take into account rotations, so if you have a scramble of R and a scramble of L they are different positions.


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## uberCuber (Dec 24, 2010)

kvaele said:


> Wow sorry I have no idea what I was looking at.


 
You don't? Because I am pretty sure I know what you were looking at: the fact that the big number is 43 quintillion, and the small number started with a 1.


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## kvaele (Dec 24, 2010)

uberCuber said:


> You don't? Because I am pretty sure I know what you were looking at: the fact that the big number is 43 quintillion, and the small number started with a 1.


 
I guess... Sorry got an 8th grade mind running on <5 hours of sleep. Easy way out: look at the "big numbers" only.


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## kvaele (Dec 25, 2010)

This is sort of just a post to "bump" this thread because I am really curious. Can we agree on approx 1.8 quintillion for the number?


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## Lucas Garron (Dec 25, 2010)

kvaele said:


> This is sort of just a post to "bump" this thread because I am really curious. Can we agree on approx 1.8 quintillion for the number?


No.

Do we need to?


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