# Rubik’s Cube arrangement

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?

• 20 is calculated by google with brute force (Except that they reduced the large number by some factor using symmetries ans such) – Coolwater Feb 24 '17 at 20:44
• if I want to know the number of movement for each face to get arranged face(have the same color ) how can I do – rose Feb 24 '17 at 20:54

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/