Pontificating on megaminx scrambles
Channel: hurfcharacterlimit
Date: 19/02/2024

What if Pochmann megaminx scrambles didn't scramble enough?

Clipped from last Friday's stream.

tl;dw summary: Pochmann scrambles have noticeably too many free pairs.

Extra notes:

If two (finite-support) distributions differ in any way at all, then you can distinguish them with a large enough sample size. Thus, we shouldn't decide whether a scramble method is good merely based on whether it can be distinguished from uniform via a statistical test, but instead come up with a more precise definition of what's "good enough".

An earlier version of Pochmann scrambles used mixed 1/5 and 2/5 turns, rather than all 2/5 turns as we do today. It turns out that neither is clearly better than the other: pure 2/5 is better at breaking up corner-edge pairs, while mixed is better at breaking up edge-centre pairs. By either of these two metrics, pure 1/5 is much worse than both mixed and pure 2/5.

Besides Daniel Hayes (in 2009) and myself (in 2024), I'm not aware of anyone else who has detected major statistical flaws in Pochmann scrambles. Stefan Pochmann analysed the centre distribution (it's roughly uniform), and Kit Clement analysed the number of qt to solve the star edges up to flipping (it's very close).

Gil Zussman's and Lucas Garron's very recent podcast on scrambles is recommended watching: https://www.youtube.com/watch?v=su5MjUdSgZs