Is there a rule/method to know how "random" should the cube be when solving it? in order for it to be a "valid" score?
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The World Cube Association has regulations regarding the official scrambling algorithm used for their competitions.
An official scramble sequence must produce a random state from those that require at least 2 moves to solve.
Source: https://www.worldcubeassociation.org/regulations/#article-4-scrambling
Basically, as long as it takes at least 2 moves to solve, it's a valid scramble. All scrambles are computer-generated.
Most scrambles will require a minimum of 18 moves to solve. The number of possible permutations for a certain number of moves is given on http://www.cube20.org.
Although a 2-move scramble is in theory possible, the chance of it happening is about 243 in 43 quintillion.
I seem to recall that the standard for competition scrambles before this was 40 random turns, but I don't have a source for this.
Kendall Frey
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An official scramble sequence must produce a random state from those that require at least 2 moves to solve. — I assume "random" means "uniformly at random". But then how does the WCA actually scramble cubes to guarantee uniformity? – JeffE May 29 '14 at 01:14
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@JeffE They use a computer program dedicated to this purpose. The details are on the WCA page linked above. (I don't think it's specified to be uniformly random, but I expect it is. If not, probably a random series of moves, which is still pretty random.) – Kendall Frey May 29 '14 at 02:04
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1You can't ever get exactly uniform via random moves, but you can get as close as you like to uniform with enough random moves. (See Persi Diaconis' seven-shuffle theorem.) I'm a little surprised that they don't disassembled the cube, throw the pieces in a bag, reassemble the cube blindfolded, and then check whether the resulting scramble is solvable. That would be uniform. – JeffE May 29 '14 at 04:23
