• Welcome to the Speedsolving.com, home of the web's largest puzzle community!
    You are currently viewing our forum as a guest which gives you limited access to join discussions and access our other features.

    Registration is fast, simple and absolutely free so please, join our community of 40,000+ people from around the world today!

    If you are already a member, simply login to hide this message and begin participating in the community!

Are all Megaminx states possible with WCA scrambles?

I'm scared of your brain.

The U/U' is dependent on the previous move
This is actually funny because if you switch the direction of the U/U' from current scrambles (i.e. always use D++ U' and D−− U), then it actually improves scramble quality a bit.

All 2/5 turns is (much) better than all 1/5 turns at mixing up pieces on a megaminx, but it turns out that introducing a small fraction of 1/5 turns is good too. (There's a huge edge orientation bias when fully restricted to 2/5 turns; it takes only three 1/5 turns to flip an edge but five 2/5 turns to do the same.)

(Changing the U direction is a fairly small improvement and I have alternatives to Pochmann scrambles that are much bigger improvements. Still working on a writeup for this!)

However, will you always see a new scrambled position? More than likely. […] In order to see every scrambled position of the mega, you would literally need to live and stay awake for years longer than possible. I think that the WCA scramble works fantastically. I will admit that during at home solves, I usually do a few extra moves to separate stuff in order to make it look more scrambled.
This is somewhat veering off-topic, so spoiler box:
Merely having a large number of possible positions isn't good enough. It does (practically) guarantee that repeat scrambles won't happen, but that's about it.

Consider ⟨U, L, F, R, BR, BL⟩. These six moves will cover around 2.0e42 positions (comparable to two WCA scrambles, with 2^140 ~ 1.4e42 scramble sequences), but they're all extremely heavily biased towards being easy – after all, they guarantee F2L skip every time!

So the real question should be: do WCA scrambles exhibit noticeable biases? For example, are free pairs too common? Unfortunately, the answer is "yes" (e.g. corner-edge pairs are ~33% too common, and as mentioned above, flipped edges are too rare), and it takes many more moves to reduce these biases to an unnoticeable level. (120 or more moves, depending on what exactly you measure and what you consider "unnoticeable".)

What you shouldn't do is to do a normal WCA scramble, then look at the puzzle and deliberately break up pairs. This introduces human biases! Instead, try either of these:
1. Increase scramble length. 100 moves still doesn't take that long to do. (Doesn't fully fix the problem (the biases are still detectable), but it's a start.)
2. Deliberately break up pairs, then do a normal WCA scramble.
 
Coincidentially, I will be developing a random state scrambler in the framework of my studies later this year. Conceptionally it will be a 3-phase scrambler, similar to the one that xyzzy made for kilominx in the weekly comp. However it will be a relatively primitive version that mostly relies on blockbuilding, since the projects scope isnt too big and just coding the basics is already a lot of work. I might hang onto the topic for a masters thesis, if that seems like a good option.

If the results show that the runtime is decent and the scramble sequences are not too long, then it might be a fit replacement for Pochmann scrambles, thanks to the 7gen aspect of the 3-phase.
 
70 moves is nowhere near enough, but it is possible to reach every state with enough moves.

You can verify this for yourself with GAP (which presumably uses Schreier-Sims internally). You can use Twizzle to produce GAP files for megaminx (and other puzzles supported by Twizzle).

Or, here's a less computation-heavy proof.

By repeating any move sequence enough times, we will eventually return to the initial state; stopping at just one iteration before thus lets us effectively do the inverse sequence. We will use this trick to build up simpler moves.

(R−− D++ R++ D++ R++ D++ R++ D++ R++ D++ U) (R++ D++ R++ D++ R++ D++ R++ D++ R++ D++ U)' = R+
(Note that "(R++ D++ R++ D++ R++ D++ R++ D++ R++ D++ U)' " really means (R++ D++ R++ D++ R++ D++ R++ D++ R++ D++ U)1679, which is possible with long enough scrambles.)

Now that we have isolated R+ (and by the same logic, R−), we can get:
R− R− (R++ D++ R++ D++ R++ D++ R++ D++ R++ D++ U) = D++ R++ D++ R++ D++ R++ D++ R++ D++ U
and
R− R− (R++ D−− R++ D++ R++ D++ R++ D++ R++ D++ U) = D−− R++ D++ R++ D++ R++ D++ R++ D++ U

From these we can also isolate D+ and D− with the same process:
(D−− R++ D++ R++ D++ R++ D++ R++ D++ U) (D++ R++ D++ R++ D++ R++ D++ R++ D++ U)' = D+

Then we can isolate U:
(D− D− R− R−)5 (R++ D++ R++ D++ R++ D++ R++ D++ R++ D++ U) = U

Then we have y rotations:
y = U D−
y' = U' D+

From which we create a rotation around a different face:
R+ y2 R− y2 R− y2 R+ y2' = Fv'

Fv and y generate all rotations of the dodecahedron, so ⟨U, Fv, y⟩ = ⟨all face turns, all rotations⟩.

The U, Fv and y moves created by this process are like a million moves long if you fully expand them in WCA scramble notation, so this is a very poor (but finite) upper bound on God's number for WCA-style scrambles.


This is true only if the scrambler can do turns on every side, which is not the case with WCA scrambles.
How can we use Twizzle to generate GAP files for megaminx?

I am sorry for sounding dumb here. You have shared some useful insights into this mathematical topic.
 
Just a historical note - when the current scramble format was invented by Stefan Pochmann, he was fully aware that it only reached a tiny fraction of the total possible states. This was all heavily debated, and the eventual decision was that it didn't matter because it would still generate fair scrambles. There were certainly a few people who disagreed with the decision, but it did make running Megaminx as an event practical; the old scrambles were simply too hard to execute.
 
I'm scared of your brain.
Basically it's just an explicit construction of the generators of the megaminx group. As an existence proof so as xyzzy says no effort into optimizing move count.
Like how the GAN robot can only turn 5 sides, but you can make a U move from the other moves, so it can solve any scramble. But going through all the steps to make a single U isn't going to be the most efficient.
 
Last edited:
Back
Top