Pancakes and the two-click property

[Raymond Zhao]

The other day on the Hypercubers Discord, I brought up how the 11-move parity algorithm tetrian22 found is 16 moves HTM (half-turn metric) if one only allows 2c turns (i.e. on a program like MC7D). Blobinati Cuber then mentioned that there should be a new turn metric for hypercubes.

Immediately, I wondered about what should be considered a single twist, and asked whether sequences like RU2 and RU RF should be considered a single twist or not.

Later in the chat, Luna explained that all twists on a cubic cell (on the n^4) can be done in one of the following ways: a single 2c click, a double 2c click, a single 3c click, or a single 4c click. This result implies that one can get a single cell of an n^4 puzzle oriented how they want in at most two clicks on MC4D.

In this thread, I'll say a cell satisfies the two-click property if it can be oriented as desired in at most two clicks in MC4D (technically, one can also start checking for this feature on MPUlt puzzles). From the previous paragraph, we see that cubic cells for n^4 puzzles satisfy the two-click property.

I was interested to see whether the result applied to cells of other MC4D puzzles. For {p}x{q} duoprisms, the prismatic cells can undergo pancake-like flips, and it follows pretty quickly that those cells all satisfy the two-click property. For example, I attached a video showing how TU5 on the {10}x{4} 3 has a two-click equivalent.

I also confirmed with pen and paper that the two-click property also applies to dodecahedral cells, which are present in the 120-cell and dodecaprism.

Therefore, if anyone wants to try setting a fewest moves record for the 120-cell, dodecaprism, or a large duoprism, that's a way to go about it.

[Raymond Zhao]

Edit: I meant tetrian22's 12-move 4c 3-cycle instead of the 11-move parity. Both algs can be found here: https://groups.google.com/g/hypercubing/c/WkMx35w6Jbs

[Melinda Green]

Those are interesting properties, though I suspect there is a more general way to define twists in all dimensions. I think we talked about this, but that was long ago, and before some of our more recent members. In essence, I would like to stop thinking about operations on twist axes, and instead focus on face reorientations, where twist axes are simply the means to reorientate a face. So in this interpretation, you should be free to make as many sequential twists as you like on a face and have it count as only one twist. Your pancake example makes this clear. If a pancake face is a 100-gon, and you twist it 50 times in the plane of the pancake, you clearly haven't changed the puzzle much at all, compared with 50 random twists.

I've been wanting to try changing the twist counter to work this way, but I'd also feel badly about invalidating the shortest twists records. Another problem is deciding what to do with the shift masks. We clearly can't mix different masks on the same face. One solution might be to require the masks be the same in order to merge twists like I'm thinking. Or we could simply look at the result, and merge all sequences that resolve to a single face reorientation, though that sounds like a bit of a coding nightmare. And what if someone alternates between twisting of two faces that do not interact with each other? You could argue that that should resolve to exactly 2 twists.

Another fun thing that could come out of this could be the ability to click and drag a face which smoothly spins around along with it's neighboring stickers. Then when you let go of the mouse, the face can snap into the closest fit and count as one move.

Best,
-Melinda

On 6/12/2021 12:41 PM, Raymond Zhao wrote:
> The other day on the Hypercubers Discord, I brought up how the 12-move 4c 3-cycle algorithm tetrian22 found is 16 moves HTM (half-turn metric) if one only allows 2c turns (i.e. on a program like MC7D). Blobinati Cuber then mentioned that there should be a new turn metric for hypercubes.

[Andrew Farkas]

Hello all,

This is something I thought about when I was briefly considering a VR version of MC4D. In that context, the smoothly spinning & snapping idea would make the most sense. For the sake of fewest-move solving, it seems to me that any reorientation of a face should be considered one move. There is some considerable cognitive load in trying to optimize a face reorientation to take fewer clicks (to reduce the move count), which is a property of the software interface rather than the puzzle itself.

I think the simplest way to implement this in software would be to simply alter the displayed move count, but leave undo/redo history intact. Consecutive moves of the same face would simply count as a single move. This speedsolving wiki page about turn metrics is relevant, in case anyone hasn't seen it.

Invalidating the existing fewest-moves record is an important consideration. It would be good to hear thoughts from anyone who has made fewest-move attempts/solves in the past.

- Andrew

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[Rty Euzch]

Dear Melinda,

One of my ideas on this is that one can switch turning metrics on the menu, but once they make their first move, the option is locked, unless they reset it.

I believe a way to make the reorientation move count work is that one can press ALT+Click on the cell, reorient the face with various twists, and press ALT+Click again to declare it as one move. If one tries to twist another face while the former cell has only been ALT+Clicked once, the program cancels the reorientation of the former, and instead twists the latter. It can also be made so that one can only use one slice mask at a time when reorienting a face.

For the predicament of invalidating the Fewest-Move records, I believe there could be separate fewest move records, with one record for the standard move metric, and another for the face-reorientation metric. If the newly proposed metric were to be added, in which I hope it would, there would basically be another fewest move game all over again. 

As for the alternating of twisting two sides, I believe that although one could argue that it had only taken two twists, the alternation essentially breaks the flow of simply reorienting a face in one move, therefore, it would count as more than two.

What do you think of my ideas?

Rty

Vào lúc 14:58:22 UTC-5 ngày Chủ Nhật, 13 tháng 6, 2021, ajfarkas12 đã viết:

[Melinda Green]

Dear Rty,

The more I think about it, the more problems I see. I do think however that some partial stepwise implementations could be useful and practical. The first could affect nothing but the way twists are counted. Anytime two consecutive twists can be counted as one they should be. That would have the added benefit of making some existing shortest records a little shorter.

Then completely independently from the above, we could add the drag-snapping twists which we decompose into twist pairs. They'll get saved as two twists, but with the above they'll only be counted as one.

Best,
-Melinda

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