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How many movements are needed to reach all arrangements of a Rubik’s Cube?

According to my Google searches, there are $43,252,003,274,489,856,000$ possible arrangements, and the maximum number of moves required to solve is $20$.

Is that right, and how was $20$ calculated?

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    $\begingroup$ 20 is calculated by google with brute force (Except that they reduced the large number by some factor using symmetries ans such) $\endgroup$ – Coolwater Feb 24 '17 at 20:44
  • $\begingroup$ if I want to know the number of movement for each face to get arranged face(have the same color ) how can I do $\endgroup$ – rose Feb 24 '17 at 20:54
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The number of legal positions is indeed $$ \frac{12! \cdot 8!}2 \cdot 2^{11} \cdot 3^{7} = 43\,252\,003\,274\,489\,856\,000 $$ which is derived in most mathematically-minded introductions to the cube.

The fact that each of these positions can be solved in $20$ moves (where turning a side 180° counts as one move) was discovered only in 2010 after an exhaustive computer search for positions that would need more. This used a combination of raw computer power (donated by Google, equivalent to one CPU running for 35 years) and clever tricks to speed up the search. There are details on http://cube20.org/

(If you only count 90° turns as moves, there are positions that require 26 moves to solve, found by similar methods 4 years later).

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  • $\begingroup$ thanks a lot, why 20 how calculated? $\endgroup$ – rose Feb 24 '17 at 20:57
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    $\begingroup$ He answered that - "A lot of computer time." @rose $\endgroup$ – Thomas Andrews Feb 24 '17 at 20:58
  • $\begingroup$ The first position has proven to require 20 moves, this for one face? if for 6 faces how many times $\endgroup$ – rose Feb 24 '17 at 21:10
  • $\begingroup$ @Peter Sho, can you answer if I want to know the number of movement for each face to get arranged face(have the same color ) how can I do $\endgroup$ – rose Feb 25 '17 at 14:03
  • $\begingroup$ @rose: 20 moves are for solving the cube completely. I'm not aware of any analysis of how many moves it can take to solve just one face. $\endgroup$ – hmakholm left over Monica Feb 25 '17 at 14:14

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