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1/1296. I tried to link the post where this was answered but I can't put in a link without the message needing moderator approval. The last layer skip chance is 1/933120 if you wanted to know that too

1/1296. I tried to link the post where this was answered but I can't put in a link without the message needing moderator approval. The last layer skip chance is 1/933120 if you wanted to know that too

For square-1 random state scrambles, does every cubeshape have an equal probability? I don’t know if some show up more than others, or if something like square and star cases show up less because they’re symmetrical.

For square-1 random state scrambles, does every cubeshape have an equal probability? I don’t know if some show up more than others, or if something like square and star cases show up less because they’re symmetrical.

The shapes have equal probability (1/3678) if you treat different AUF/ADFs as different shapes.

But if you treat different AUF/ADFs as still being the same shape, then you need to adjust the probability by how many of the AUF/ADFs let you do a slice move. For example, with square-square, there are four combinations of AUF/ADF, so it has a probability 4/3678. (Since square is rotationally symmetric, doing a (3,0) does not produce a different shape, so we don't count that.) Shield-square has 3 choices of AUF and 2 choices of ADF, so the probability is 6/3678. (Note that this is treating shield-square and square-shield as different shapes; if you want to treat those as the same too, the combined probability is 12/3678.) And so on.

This page has a list of the shapes and probabilities:

I know this has probably been answered elsewhere, but how is decided how long a scramble should be?

3*3 is usually 20 moves, but how do we know that would be enough, rather than 30 moves, or 50, etc? Is it because 20 is God's Number for the 3x3? That number is unknown for bigger cubes though, so that can't be a factor in scramble length for them.

I know this has probably been answered elsewhere, but how is decided how long a scramble should be?

3*3 is usually 20 moves, but how do we know that would be enough, rather than 30 moves, or 50, etc? Is it because 20 is God's Number for the 3x3? That number is unknown for bigger cubes though, so that can't be a factor in scramble length for them.

Cube scrambles are always random state if feasible. For 3x3, since gods number is 20, scrambler programs can easily find a solution in 20 moves or less. This takes much less time than finding an optimal solution like 17 or 18 moves, so scrambles can be generated incredibly quickly.

For bigger cubes, random state is a bit harder. You don’t need God’s number to generate random state scrambles, because you can just set the searching range for an upper bound of gods number. This is why we have random states for 4x4 (and I think 5x5).

For 6x6 and 7x7 random state is possible but it would be so long and so many moves more than the scrambles we use now that it would be impractical, since those take so long to scramble already.