The megaminx notation

[Electrical]

I've always thought the WCA notation for megaminx is really weird. I never thought it really scrambled the puzzle properely, as it only turns 3 layers.

Anyway my question is, where did it come from and who thought of it?

 

[DGCubes]

It was suggested by Stefan Pochmann in 2007 in this thread. You're right that it can't reach all possible states (in fact, it can only reach ~10^21 of the ~10^68 states); that was discussed in this thread.

 

[xyzzy]

You're right that it can't reach all possible states (in fact, it can only reach ~10^21 of the ~10^68 states); that was discussed in this thread.

It can't reach all possible states with the standard scramble length of 70 moves, but fun fact: if you allow arbitrarily many rows of moves instead, it can reach all possible states. (I managed to get GAP to verify this with a lot of very boring manual effort making sure I typed in the facelet permutations for R++/D++/U moves correctly. I probably wasn't the first to do this, but I can't find evidence that anyone else bothered.)

I have a vague suspicion that 70 moves isn't really enough to scramble the puzzle "fully" (in the sense of various statistics (e.g. number of corner-edge pairs, moves to solve the white star, etc.) being very close to the ideal values with true random states), but I've never bothered to actually write some code to check. (Then again, I did this for 5×5×5 and demonstrated that 60 moves wasn't enough to mix up the centres, and to this day we still use 60 moves anyway. :thinking: )

 

[whatshisbucket]

It can't reach all possible states with the standard scramble length of 70 moves, but fun fact: if you allow arbitrarily many rows of moves instead, it can reach all possible states. (I managed to get GAP to verify this with a lot of very boring manual effort making sure I typed in the facelet permutations for R++/D++/U moves correctly. I probably wasn't the first to do this, but I can't find evidence that anyone else bothered.)

Then this brings up the issue that the pseudorandom number generators used only have 128-bit seeds, so the scrambler wouldn't reasonably get all the scrambles anyway.

 

[CurlyFries]

Though it does not provide scrambles for every possible permutation, does it really matter? 10^21 is equal to 1 sextillion, roughly the number of grains of sand on earth. According to a cubeorithms video, we will probably never solve all the possible permutations of a 3x3, which is a measly 1000th of that of a megaminx, so if that’s the case, will we ever need a scrambler with easy access to all 10^68 possible permutations?

 

[qwr]

Though it does not provide scrambles for every possible permutation, does it really matter? 10^21 is equal to 1 sextillion, roughly the number of grains of sand on earth. According to a cubeorithms video, we will probably never solve all the possible permutations of a 3x3, which is a measly 1000th of that of a megaminx, so if that’s the case, will we ever need a scrambler with easy access to all 10^68 possible permutations?

It matters if the permutations that appear in scrambles are somehow significantly easier than "fully random" permutations. For example if xyzzy's suspicion is correct, maybe by some metric the current scrambles are actually 1.5x easier (or whatever number) than fully random permutations. I think we can all agree in theory that the puzzle scrambles should be representative of what the puzzle is capable of scrambling into. As an extreme example: we wouldn't want competitions where the 3x3 is only scrambled with R, U, and L moves, or it is only scrambled with 10 moves. It doesn't match what we think of as "solving" a puzzle.