I want to shuffle the 3x3 cube randomly in quarter turn metric and determine the minimum number of moves of each position with some mathematical solution. I know that the minimum moves from the most shuffled position is 26 in quarter turn metric, but I want to know from any other position. Or know the level of difficulty of any position of the Rubik's cube. Perhaps not the minimum moves but some number that represents the level of difficulty from any position.
Asked
Active
Viewed 777 times
1
-
I don't believe that a practical way of doing this is known. The people who came up with the 26 move overall minimum explain that they did not do this by looking for minimal solutions from each position. – Ben Grossmann Jul 04 '20 at 19:00
-
@AlexeyBurdin 20 moves is if you count $180^\circ$ turns as a single move – Ben Grossmann Jul 04 '20 at 19:01
-
$3\times 3$ is not a cube. – markvs Jul 04 '20 at 19:58
-
1@JCAA I think he means 3x3x3 Rubik's Cube, I think a lot of people just say '3x3' for short – jpthesolver2 Jul 04 '20 at 20:08
-
Yes I'm talking about 3x3x3 Rubik's Cube – doulos Jul 04 '20 at 22:02
1 Answers
2
Tomas Rokicki and Morley Davidson found all the optimal solutions for a cube for quarter turn and half turn metrics. As you mentioned, the maximum number of turns is 20 for half turn metric and 26 for quarter turn metric.
Unfortunately I don't think there is an efficient way to get the guaranteed number of minimum moves, but you can get fairly close. The Kociemba method has a following where some people have made programs that will find close to or exactly optimal solutions.
Most of the solutions, as you can see in the distribution below, lie around 16-19. So even if the solution isn't guaranteed to be optimal, but still lies in that range, there is a good chance that it is optimal or very close to.
Distribution of solutions from cube20
jpthesolver2
- 203
-
Thanks @jpthesolver2 but is it possible to get not the optimal solution (the steps to solve the cube) but the minimum amount of moves without getting the optimal solution? Perhaps with some mathematical equation. Knowing that the numbers are all connected. – doulos Jul 04 '20 at 22:26
-
Unfortunately I'm not sure that one really exists without doing that sort of iteration, at the moment. Some of the methods (like Kociemba and Korf) are based on Group Theory, which rely on the state of the properties of the sub groups of the cube, which means it depends on which permutation they are in. You can read more about the algorithms here: https://en.wikipedia.org/wiki/Optimal_solutions_for_Rubik%27s_Cube – jpthesolver2 Jul 05 '20 at 04:28
-
Thanks @jpthesolver2, and there is no way to know the difficulty level of each position with some mathematical solution? – doulos Jul 17 '20 at 16:32