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43 Quintillion is Wrong

Joined
Nov 17, 2015
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Location
Canada!
WCA
2015BERG10
#3
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.
 
Joined
Dec 24, 2015
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#5
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.))

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".

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.
 
Joined
Nov 29, 2008
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#7
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.
 
Last edited:
Joined
Sep 3, 2017
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#8
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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