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Cube probability and coin flips

campos20

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Have you ever considered the probability of getting a random state cube solvable optimally with n moves? Well, I have. Since normal probability are not that readable when talking about cube, one can picture the scene with exciting coin flips.

The chance you get a cube that can be solved with 10 moves is approximately the same as getting 28 Heads in a row in a coin flip (or 5.37e-07%. See, coin flips are better).

Here's a table:

MovesFlips

[TR1][TD1] 0 [/TD1][TD1] 66 [/TD1][/TR1]
[TR2][TD1] 1 [/TD1][TD1] 62 [/TD1][/TR2]
[TR1][TD1] 2 [/TD1][TD1] 58 [/TD1][/TR1]
[TR2][TD1] 3 [/TD1][TD1] 54 [/TD1][/TR2]
[TR1][TD1] 4 [/TD1][TD1] 50 [/TD1][/TR1]
[TR2][TD1] 5 [/TD1][TD1] 47 [/TD1][/TR2]
[TR1][TD1] 6 [/TD1][TD1] 43 [/TD1][/TR1]
[TR2][TD1] 7 [/TD1][TD1] 39 [/TD1][/TR2]
[TR1][TD1] 8 [/TD1][TD1] 35 [/TD1][/TR1]
[TR2][TD1] 9 [/TD1][TD1] 32 [/TD1][/TR2]
[TR1][TD1] 10 [/TD1][TD1] 28 [/TD1][/TR1]
[TR2][TD1] 11 [/TD1][TD1] 24 [/TD1][/TR2]
[TR1][TD1] 12 [/TD1][TD1] 21 [/TD1][/TR1]
[TR2][TD1] 13 [/TD1][TD1] 17 [/TD1][/TR2]
[TR1][TD1] 14 [/TD1][TD1] 13 [/TD1][/TR1]
[TR2][TD1] 15 [/TD1][TD1] 9 [/TD1][/TR2]
[TR1][TD1] 16 [/TD1][TD1] 6 [/TD1][/TR1]
[TR2][TD1] 17 [/TD1][TD1] 2 [/TD1][/TR2]
[TR1][TD1] 18 [/TD1][TD1] 1 [/TD1][/TR1]
[TR2][TD1] 19 [/TD1][TD1] 5 [/TD1][/TR2]
[TR1][TD1] 20 [/TD1][TD1] 37 [/TD1][/TR1]

Now, how was that calculated? Well, first we can go to http://cube20.org/. The number of cubes solvable with 10 moves is c10 = 232,248,063,316 and but the total number is ctotal = 43,252,003,274,489,856,000. So, ctotal is approximately 186,231,922.2686 times c10. Taking log2 (to calculate coin flips) we have 27.4725. Its ceiling is 28. Repeating this process, we can build this table.
 
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