Message #3845

From: Bob Hearn <bob.hearn@gmail.com>
Subject: Re: [MC4D] Melinda’s 2x2x2x2 solved
Date: Mon, 27 Nov 2017 14:26:07 -0600

Hi Marc,

> Good stuff! I’m glad to see that you put your puzzle together and made
> progress on it. Unless we unexpectedly identify a problem, I think
> you’ve got the first valid solution recipe sketched out. Congrats!

Thanks! Great to see we had the same ideas.

> When I had worked through it before, I was worried that I would still
> occasionally end up with the single-piece-double-twist problem at the
> end. I think I was hallucinating, since it seems clearly impossible to
> be in that state with two faces oriented.

Right…

> However, I would be reassured
> if you had a look at the following.
>
> Question: if you take a single piece of your puzzle and give it a 180
> degree twist along any axis, how would you solve the resulting puzzle
> state? … What’s the fewest number of
> full puzzle reorientations required to undo a single-piece-double-twist
> leaving the rest of the puzzle unchanged? I think it might be three,
> and that the reason will become clear to me after a bit. But it might
> be two, given how confused I still am about how the 12 orientations work.

Great question. Three is clearly sufficient, because this state is an instance of where you can be in step 3a in my solution, and from there three reorientations are required.

I’m about convinced you can’t do it in two, though I don’t have a tidy proof yet. You need to show that no matter what you do before the first reorientation, that reorientation will leave you in a state where no opposing color pairs can be solved into opposite faces, because of corner twist parity.

> It would be fun to see a video of you manipulating your puzzle. I’m
> curious what feels natural for you.

I’m still traveling for Thanksgiving, but will try to put something together when I get home.

> I’m also looking forward to jointly tackling some entries on the list of
> open questions about the physical 2^4. After writing down the list of
> open questions, which I have not done. :) Do you have any entries for
> it? Directions you’d like to explore?

For me the most interesting thing about Melinda’s 2x2x2x2 is that it exists at all. It seems kind of a miraculous accident. I wondered what the equivalent 2d representation of a 2x2x2 would be, and realized that it doesn’t exist, because squares do not have 3-fold rotational symmetry. We luck out here that the tetrahedral group is a subgroup of the octahedral group. I.e., that you can get the required 4-fold isotropic symmetry in a cube. Likewise, you can’t make a similar 3x3x3x3 — the three-facet pieces can’t be instantiated as cubes.

So what kinds of 4d puzzles can be implemented this way? Is this it?

Bob