SenileGenXer
Member
We know there are 12 orbits of the cube. It is assumed that the probability of randomly assembling the cube and having a solvable puzzle is 1 in 12.
Have we checked that?
I was thinking of all the ways it's possible to misassemble the cube.
1) Clockwise corner twist (1 in 3 chance)
2) Counter-Clockwise corner twist (1 in 3 chance)
3) Edge flipped (1 in 2 chance)
4) Two edges swapped (1 in ? chance)
5) Two corners swapped (1 in ? chance)
The last part is a mistake you could make assembling the cube. Now because of 4-cycle moves like the T-Perm and F-Perm two corners swapped is indistinguishable from two edges. They will show up as the same PLL parity error.
If you make mistake #4 and mistake #5 randomly assembling the cube they cancel each other out - there is no observable mistake. A cube assembled with two edges swapped and two corners swapped is solvable with a F-Perm/T-perm.
I was thinking that instead of 12 orbits there were 14 orbits but two pairs of orbits were conjoined. That is the solvable orbit is conjoined with the two edges and two corners swapped orbit. The two edges swapped orbit is conjoined with the two corners swapped orbit. I hypothesized that this produces a 2/14 chance of randomly assembling a solvable cube. 1/7 when simplified.
To test it I disassembled a cube into a brown paper paper bag, shook up the bag, reassembled the cube randomly and tried to see if it was solvable. Kept track of 50 random assemblies. If my theory was correct I should get 7.1 solvable cubes. If the conventional theory was correct I should get 4.1 solvable assemblies.
I got six.
Data available here.
Have we checked that?
I was thinking of all the ways it's possible to misassemble the cube.
1) Clockwise corner twist (1 in 3 chance)
2) Counter-Clockwise corner twist (1 in 3 chance)
3) Edge flipped (1 in 2 chance)
4) Two edges swapped (1 in ? chance)
5) Two corners swapped (1 in ? chance)
The last part is a mistake you could make assembling the cube. Now because of 4-cycle moves like the T-Perm and F-Perm two corners swapped is indistinguishable from two edges. They will show up as the same PLL parity error.
If you make mistake #4 and mistake #5 randomly assembling the cube they cancel each other out - there is no observable mistake. A cube assembled with two edges swapped and two corners swapped is solvable with a F-Perm/T-perm.
I was thinking that instead of 12 orbits there were 14 orbits but two pairs of orbits were conjoined. That is the solvable orbit is conjoined with the two edges and two corners swapped orbit. The two edges swapped orbit is conjoined with the two corners swapped orbit. I hypothesized that this produces a 2/14 chance of randomly assembling a solvable cube. 1/7 when simplified.
To test it I disassembled a cube into a brown paper paper bag, shook up the bag, reassembled the cube randomly and tried to see if it was solvable. Kept track of 50 random assemblies. If my theory was correct I should get 7.1 solvable cubes. If the conventional theory was correct I should get 4.1 solvable assemblies.
I got six.
Data available here.
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