# How long would it take to get to all permutations of a cube?



## 2180161 (Jan 12, 2015)

Say I turn my cube at 100 TPS (this is just hypothetical) and in order to get to every legal position, (Using R,U,L,F,B,D, M, r,u,l,f,b,d) and each algorithm to get to those positions is 20 moves (QTM), How long would it take to get every single one complete?
I dont know, So please post the math of how you got that.


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## obelisk477 (Jan 12, 2015)

43,252,003,274,489,856,000 positions / 5 Positions per second ~ 274 million millenia


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## 2180161 (Jan 12, 2015)

274 millenia? 
in numbers please. 
Wouldnt that be 274 million years?


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## qqwref (Jan 12, 2015)

43,252,003,274,489,856,000 positions / 100 Positions per second = 13.7 billion years

If you just want to get to every position, you don't need to do a full solution every time, just one move to get to the next position. If you did do the full solution (and back again, right? unless you have 4.3 * 10^19 cubes) you would do ~35.4 moves per cube.


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## obelisk477 (Jan 12, 2015)

qqwref said:


> 43,252,003,274,489,856,000 positions / 100 Positions per second = 13.7 billion years
> 
> If you just want to get to every position, you don't need to do a full solution every time, just one move to get to the next position. If you did do the full solution (and back again, right? unless you have 4.3 * 10^19 cubes) you would do ~35.4 moves per cube.



Wouldn't you have to do a knight's tour kinda thing like in chess? Like this is assuming that you perfectly know each move necessary to hit each position only once

EDIT: and that such a sequence of moves even exists


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## qqwref (Jan 12, 2015)

Yes, Bruce Norskog found such a sequence of moves a while ago: http://bruce.cubing.net/ham333/rubikhamiltonexplanation.html


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## ketchuphater999 (Jan 12, 2015)

Now, how about this? Every person on the planet solves the cube into 100 positions every second.

43,252,003,274,489,856,000 / (7,000,000,000*100) = 61788576.10641408 seconds = 1.958 years

7 billion is approximate


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## Smiles (Jan 12, 2015)

qqwref said:


> Yes, Bruce Norskog found such a sequence of moves a while ago: http://bruce.cubing.net/ham333/rubikhamiltonexplanation.html



I didn't give it a good read so idk how long that algorithm is but that would be inefficient to cycle through all the positions on the cube. For a lot of the positions, when you first start doing this, you can get to the next one in one move. It's only much later on that that is no longer possible, and you have to use multiple moves to create a unique position.
Basically we'd have to look for one algorithm (that is at least 43 quintillion moves long) that takes us through all the positions.
i.e. the answer to OP's question is crazy.


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## obelisk477 (Jan 12, 2015)

Smiles said:


> Basically we'd have to look for one algorithm (that is at least 43 quintillion moves long) that takes us through all the positions.



That's precisely what the link is referencing... exactly 43 quintillion moves to get to all 43 quintillion positions only once each


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## qqwref (Jan 12, 2015)

Smiles said:


> I didn't give it a good read


You should've, because it's exactly what you think it can't be  It's an algorithm that goes to a new position every move, and goes to every position eventually.


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## Smiles (Jan 12, 2015)

obelisk477 said:


> That's precisely what the link is referencing... exactly 43 quintillion moves to get to all 43 quintillion positions only once each



oh HAHA i'm so good at this.
thanks for clarifying


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## Ordway Persyn (Jan 12, 2015)

qqwref said:


> 43,252,003,274,489,856,000 positions / 100 Positions per second = *13.7 billion years[/B}*


*

Just relized that if you did that since the big bang you would of gotten every permutation by now (give or take a few million years)*


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## cmhardw (Jan 13, 2015)

I looked at another take on the question. Say you have Cube Explorer or some other random state generator give you a random scrambled state every time you "click". Assume the generator has a uniform distribution for providing scrambled states to you.

The expected number of clicks it would take for you to see all 43 252003 274489 856000 states of the cube is:

1980 543437 497805 935904 clicks.

clicky

Assuming you can click to generate a position at 100 positions per second, it would take about 628 billion years to see every possible cube state.

clicky


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