Hi, A few days ago ChrisH asked me for some quesitons about the exact edge distributions for the 4x4. I'd like to report my findings here.
Due to the rather technical nature of how I obtained these number (quadruple nested summations and products)..., I omit the details. Besides I don't want to ruin it for ppl who'd like to solve this problem on their own. For the 3x3, the cross *exact *probabilities on a random cube is: 0 84.43% 160489/190080 1 14.51% 6893/47520 2 1.03% 65/6336 3 0.046% 17/47520 4 (5.26*10^-4)% 1/190080 For the 4x4, the chances of getting a specific number of solved edges on a random scramble or after doing centers randomly is: 0 59.38% 1 30.92% 2 8.08% 3 1.41% 4 0.186% 5 0.0198% 6 etc..... 7 8 9 10 11 12 For the 5x5, .... well you get the idea... Ok it's better if you just look at the following pages if you are intersted in this discrete distribution: http://www-personal.umich.edu/~dlli/3x3_ExactCross.jpg http://www-personal.umich.edu/~dlli/4x4_EdgePairs.jpg http://www-personal.umich.edu/~dlli/5x5_EdgePairs.jpg That is Mathematica code, btw (but those functions are certainly not built in, lol). These took me several hours to figure out, the 5x5 probabilites where based on the other two and easily found after I figured out the cross stuff... Everything adds to 1, so it checks out. I also matched the numbers to doing it by case count for some base cases so I'm confident it's correct. In case you're interested there is a great deal of probability theory involved and what Chris showed me: Inclusion-Exculsion Principle. Also I had to resort to what I would call "the Theory of Highly-Generalized Partial-Derangments" (lots of Subfactorials involved and most math ppl don't even know what that is). I can show you how I setup the summations if soneone's interested. Plus I can generalized it to NxNxN cubes. Tell me if someone already did this, I'd like to know if I was the first to bother doing this. -Doug Yahoo! Groups Links <*> To visit your group on the web, go to: http://groups.yahoo.com/group/speedsolvingrubikscube/ <*> To unsubscribe from this group, send an email to: [EMAIL PROTECTED] <*> Your use of Yahoo! Groups is subject to: http://docs.yahoo.com/info/terms/
