elrog
Member
I posted this post a while back in this thread. It is the last post on page 7 if you have 10 posts per page.
I thought about this a little more and realized something. If any position of the cube can be reached in exactly 35 (this is just an example) moves (HTM because that's how God's Number was found), then you can reach any position starting at any position, and any position can be reached with any number of moves above 35 moves because you can just apply any number of moves to the starting state and apply a different set of 35 moves.
After realizing this, I was wondering just what that number may be. To get an idea I took a look at the superflip (a well known position that requires 20 moves optimally HTM). An optimal solution for the superflip is U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2. Because of the symmetry of this position, any move you make makes the position closer to solved.
If you do a single move to the superflip position, you make a 19 move solution possible, but there is also a 20 move position possible as well because you can cancel at the beginning or end of the superflip alg because it has a 90 degree turn on one end and a 180 degree turn on the other. Doing any two moves on adjacent sides also can cancel out to give a 20 move solution. If you do any two moves on opposite sides, you will not be able to cancel both of them out and no 20 move solution is possible.
This would verify that 21 moves is the lower bound as long as there is no other optimal solution for the superflip that is not an inverse or mirror of the given algorithm (I don't think there is). Going by this logic, I could go further but I would likely run into other 20 move optimal positions.
I would like to clarify something in your post. You said that X moves have the same chance to give a random state, but some states are more likely to occur than others due to symmetries. On a supercube, every state has equal probability. I think what you were meaning though is that X moves and Y moves both have the same chance to come up with the same state.
Lets say you start at any particular random state on the cube. If you apply exactly 20 moves to the cube, there may be cases that are still unreachable. Take a state that is 19 moves optimally from the state you started at. That state is only reachable if it is solvable in 20 moves as well as the optimal 19 moves. I am fairly certain that there are 19 move optimal solutions that cannot be solved in exactly 20 moves because there are also 5 move solutions which cannot be solved in 6 moves.
It is true that the probability would eventually be the same, but to define the point, you would have to find a number big enough that every state can be solved in exactly that many moves. There may also be a few numbers of moves above this number of moves which cannot cover every position, but it seems that eventually you would reach a point where any number of moves above this number, whatever it may be, would be able to cover every position. I cannot be to sure though as prime numbers would seem the same way, but they really only become further and further apart.
Oh, and to answer the original question disregarding the point that the chances are so small, I have not and may not ever attend a competition, and that the scramble would surely be thrown out, I would solve the cube in 1 move STM if possible and give up if it wasn't possible.
I thought about this a little more and realized something. If any position of the cube can be reached in exactly 35 (this is just an example) moves (HTM because that's how God's Number was found), then you can reach any position starting at any position, and any position can be reached with any number of moves above 35 moves because you can just apply any number of moves to the starting state and apply a different set of 35 moves.
After realizing this, I was wondering just what that number may be. To get an idea I took a look at the superflip (a well known position that requires 20 moves optimally HTM). An optimal solution for the superflip is U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2. Because of the symmetry of this position, any move you make makes the position closer to solved.
If you do a single move to the superflip position, you make a 19 move solution possible, but there is also a 20 move position possible as well because you can cancel at the beginning or end of the superflip alg because it has a 90 degree turn on one end and a 180 degree turn on the other. Doing any two moves on adjacent sides also can cancel out to give a 20 move solution. If you do any two moves on opposite sides, you will not be able to cancel both of them out and no 20 move solution is possible.
This would verify that 21 moves is the lower bound as long as there is no other optimal solution for the superflip that is not an inverse or mirror of the given algorithm (I don't think there is). Going by this logic, I could go further but I would likely run into other 20 move optimal positions.