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Tim Browne

Post subject: Cubic identities? Posted: Sun Jul 30, 2006 4:04 pm 

Joined: Sun Dec 19, 1999 3:02 am Location: Canada

The fewest moves challenge thread got me thinking, especially Kirjava's amazing identity algorithms. In light of his patterns, I was wondering, what would be the odds of 25 random turns on the cube returning it to the solved state, so that if it was used in competition, the "World's Record cube solver" would be determined simply by how quickly someone could go between the start and finish positions on the Stackmat? I'd imagine that under such circumstances they'd need to have video records taken from right in front of the competitor in super high speed which could then be slowed down enough to make sure that the person did in fact pick up the cube instead of cheating by simply lifting hands off the stackmat and putting them down right away, or simply touching the cube and returning to stop the stackmat timer (both of which I'd say should count as an automatic disqualification). What would the odds change to if it was done on a picture cube instead of a plain vanilla one? Would this even be calculable right now, or will we still need to wait for computers to become powerful enough that we could simply brute force it? L8r.


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Erik J

Post subject: Posted: Sun Jul 30, 2006 9:25 pm 

Joined: Mon May 29, 2006 10:50 pm Location: New York

oh I love math like that. look at it this way. With 1 random move there are 18 posible moves. for each side there are three different turns that could be done. 2 moves would be 324 combinations. So with 25 moves it would be a 1 in 20,488,659,210,943,104,643,915,283,693,568 chance of getting it solved in 25 random moves. After I got this I thought why is it higher than the number of combinations posible on a 3x3x3 cube. the only reason I could think of is that there are a lot more than one 25 move alg that mixes then solves the cube. but still the chances are basicly zero.
_________________ Average: 14.38 seconds
Individual Times: 14.17, 12.93, (12.89), 13.12, 14.00, 15.59, 16.23, 14.11, 14.03, (17.61), 14.75, 14.89


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Adam Zamora

Post subject: Posted: Sun Jul 30, 2006 11:06 pm 

Joined: Thu Nov 20, 2003 7:29 am Location: San Diego, California

Tim i dont quite get what you are asking here


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Erik J

Post subject: Posted: Sun Jul 30, 2006 11:28 pm 

Joined: Mon May 29, 2006 10:50 pm Location: New York

the odds of the cube being solved after 25 random moves.
_________________ Average: 14.38 seconds
Individual Times: 14.17, 12.93, (12.89), 13.12, 14.00, 15.59, 16.23, 14.11, 14.03, (17.61), 14.75, 14.89


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David J

Post subject: Cubic identities Posted: Mon Jul 31, 2006 12:25 pm 

Joined: Tue Mar 04, 2003 11:17 am

Odds are a tricky thing. Relatively few people understand them.
If 17 turns of a scramble move away from the solved position and 8 turns move toward the solved position then the scramble would be only nine moves from solved.
A while ago I asked Tyson if he would set a standard, and I suggested that he run the scrambles he was going to use at tournaments through a few solvers, and eliminate those scrambles which are too easy.
I don't know if he took up the suggestion.
David J


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Tim Browne

Post subject: Re: Cubic identities Posted: Mon Jul 31, 2006 12:49 pm 

Joined: Sun Dec 19, 1999 3:02 am Location: Canada

David J wrote: Odds are a tricky thing. Relatively few people understand them.
OK, let's see if I can simplify the understanding, then. The first slice can be turned any one of 18 ways. To prevent the pattern from being completely degenerative, the remaining turns can be made one of 15 ways, so any random mix of 25 turns gives you 18*15^24 ways of doing it, and yes, this WILL result in a lot of patterns such as U L R L R2 L2 R L2... but that can't really be avoided with a truly random pattern. Out of all of these 18*15^24 patterns, how many of them result in a solved cube? Divide one into the other and you'll get the probability, and from there you can work out the odds. L8r.


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away

Post subject: Re: Cubic identities Posted: Mon Jul 31, 2006 3:30 pm 

Joined: Tue Oct 07, 2003 10:00 pm

Tim Browne wrote: patterns such as U L R L R2 L2 R L2... but that can't really be avoided with a truly random pattern.
Why not? Particularly if you acknowledge the factor 15 already...
The (reasonable) hope is that 25 moves is enough to get very close to a uniform distribution so the probability to get a solved cube should be about 1/numberOfPossibleCubeStates.


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TimReynolds

Post subject: Re: Cubic identities? Posted: Sat Aug 12, 2006 10:08 pm 

Joined: Wed May 10, 2006 8:48 pm Location: New York

Tim Browne wrote: I'd imagine that under such circumstances they'd need to have video records taken from right in front of the competitor in super high speed which could then be slowed down enough to make sure that the person did in fact pick up the cube instead of cheating by simply lifting hands off the stackmat and putting them down right away, or simply touching the cube and returning to stop the stackmat timer (both of which I'd say should count as an automatic disqualification)
Sorry, but why would someone have to even touch the cube? There is nothing in the regulations which states that a competitor must pick up the cube (correct me if that is incorrect). It does say (in Article A: Ending the Solve) that "The competitor ends the solve by releasing the puzzle and then stopping the timer", though I think the meaning of that is "the competitor ends the solve by stopping the timer with the cube no longer in his hand". I'd expect to see some 0.03's in that case.
CaptianCrash44, I'm not sure I understand you fully. You say that there's a 1 in 18^25 (?) chance that it will be solved; that is, there's only one 25 move scramble that solves the cube. Is that true?
Also, Tim Browne, you can't say there's 15 choices per turn. For the first turn, it doesn't really matter which side you turn since the starting position is symmetric, so there's only really 3 possibilities. The R L R' L' R2 L2... case is avoided by most scrambling programs since such a pattern decreases the number of turns being applied to the cube, making it less random rather than more random.
Of course, this whole matter could be avoided by scrambling with 25 moves QTM. Then there will be an odd permutation of both edges and corners, meaning the scramble can't create a solved cube. Though blindfoldsolvers would hate this...


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