# 43 Quintillion is Wrong



## luckysolve (Apr 2, 2018)




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## Duncan Bannon (Apr 2, 2018)

Ill let the pros argue this out. I did learn alot. So thanks for that.


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## BenBergen (Apr 2, 2018)

Really interesting video!

Could the reason the reduction factor is only about 48 be because some scrambles have axes of symmetry? For example, the superflip. No matter which orientation you apply a superflip scramble from, it will give you the same cube permutation.


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## Username (Apr 2, 2018)

you're claiming it is wrong but basing it on your own personal opinion of what a scramble is

the cube states are different even though you might not consider them to be


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## xyzzy (Apr 3, 2018)

Obvious clickbait title is clickbait, but whatever, you got me to watch it. At two points in the video you blithely throw out something like 1% of the "scrambles" just based on move count, but this is completely bollocks and not a thing you should be doing when counting possibilities, regardless of whether you want to take symmetry into account. Just because a particular position/configuration/state/whathaveyou falls in a class that is less likely to be hit doesn't mean the position itself is less likely to be hit than the others. (Analogously, most of the H-set ZBLLs have the same probabilities as the others; H-set is less likely to occur because there are fewer cases in the set, but this _does not contradict the previous statement_.)

(Also, don't forget the 2-move WCA filtering!!!!!!! (I'm obviously trolling here, but to be fair, so were you with the clickbait title.))



Username said:


> the cube states are different even though you might not consider them to be


The actually correct way of phrasing this objection is that there are different notions of "different".



BenBergen said:


> Really interesting video!
> 
> Could the reason the reduction factor is only about 48 be because some scrambles have axes of symmetry? For example, the superflip. No matter which orientation you apply a superflip scramble from, it will give you the same cube permutation.


This is correct.


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## alwin5b (Apr 3, 2018)

I'll just leave this here, for anyone who wants to know more:

http://www.math.rwth-aachen.de/~Mar...rs/Dan_Hoey__The_real_size_of_cube_space.html
https://en.wikipedia.org/wiki/Octahedral_symmetry
https://en.wikipedia.org/wiki/Burnside's_lemma


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## Herbert Kociemba (Apr 7, 2018)

And if you do not take only symmetry into account but also the fact that for example U F' D2 and D2 F U' are positions which are just inverse positions of each other and hence have in a certain sense the same structure and need the same number of moves the number of "true different positions" can be shrunk by almost another factor 2 to 450.541.810.590.509.978

http://cubezzz.homelinux.org/drupal/?q=node/view/22

Only almost because things are again more complicated. The inverse of a position can for example be a rotation of that position. So we cannot divide the true number of positions reduced by the 48 symmetries just by 2.


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## Bruce MacKenzie (Apr 15, 2018)

The positions a 3x3x3 may be placed in by rotating the faces may be represented as a mathematical group. The size of this group is 43,252,003,274,489,856,000 and may be calculated in the manner shown in the video. This is the "real" size of the group.

When one is performing "god's algorithm" calculations one may reduce the number of calculations using cubic symmetry and inversion arguments as the video and Herbert Kociemba have pointed out above. If one grinds out an optimal solution for a position, one may easily and quickly generate optimal solutions for potentially 96 different positions using symmetry and inversion. Nevertheless, these are different positions and one hasn't really "reduced the size of the group".


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