Message #4119

From: Joel Karlsson <joelkarlsson97@gmail.com>
Subject: Re: [MC4D] 2x2x2x1: Gyro rotations, and seeking the equivalent 4D cuboid
Date: Tue, 04 Sep 2018 09:13:23 +0200

Hi everyone,

Although an interesting puzzle, I don’t think that the twisty stacky 2^3 is
a representation of a 2x2x2x1. I haven’t thought this through properly yet
so feel free to criticise my arguments and point out things that I’ve
missed.

Firstly, an n-dimensional twisty cuboid usually has the same number of
colours (2*n) as an n-dimensional twisty cube. However, when we cut the
physical 2^4 into two halves we remove one of the colours. A 2x2x2x1 cuboid
should have two 2x2x2 faces and six 2x2x1 faces but the twisty stacky 2^3
has only one 2x2x2 face and six 2x2x1 faces.

Secondly, on a 2x2x2x1 cuboid only stickers belonging to similar faces can
be mixed. So stickers belonging to the 2x2x2 faces can be mixed with each
other but such a sticker can’t be on a 2x2x1 face. You could do a bandaged
version of the 2^4 where this would be possible if you glue together the
stickers of the F, B, R, L, U and D faces in the in/out direction of the
virtual 2^4. In this puzzle, the stickers on the F, B, R, L, U and D faces
would be 1x1x2 blocks instead of 1x1x1 blocks. A twist could then be
performed to replace two such stickers (forming a 2x2x1 block) with four
1x1x1 stickers from the I and O face. Thus, two stickers from I and O could
take the place of one sticker from any of the other faces. On the twisty
stacky 2^3 however, one sticker on O can take the place of one sticker on
F, B, R, L, U or D. The "ordinary" 2x2x2x1 cuboid would be the subset of
this bandaged 2^4 which allows all moved except the ones that mix F, B, R,
L, U and D stickers with I and O stickers.

Observation: In both the bandaged 2^4 described above and the 2^3x1 you
cannot twist the I or O face (this is true for the O face of the twisty
stacky 2^3 as well). Since all pieces on these puzzles have both a sticker
belonging to I and a sticker belonging to O, an attempt to twist I or O
would result in a rotation.

Best regards,
Joel

Den mån 3 sep. 2018 kl 23:27 skrev Andrew Farkas ajfarkas12@gmail.com
[4D_Cubing] <4D_Cubing@yahoogroups.com>:

>
>
> Hey all!
>
> I’m *very* tentative to call the twisty-stacky a 2x2x2x1. I’ve already
> half-written and deleted multiple emails trying to work through the
> behavior of a 4D cuboid, but I could never come to a concrete conclusion.
> Here is one line of thinking I ran down:
>
> Some observations about the NxNx1, with the Z axis as the short axis:
>
>
> - The puzzle has 2D cuts along the XZ and YZ planes.
> - 180-degree moves are possible in two planes: XZ and YZ.
> - Moves are not possible in the XY plane, because there are no cuts
> in that plane.
> - In a 3x3x3 emulating a 3x3x1, pieces in the E layer never mingle
> with pieces in the U and D layers.
>
> The 2^4 has 2D cuts alo– well no, they would be 3D cuts along each of the
> … 12? 24 possible 3D slices? But how does that correspond to which moves
> are allowed?
>
>
> Let me try another one:
>
> To use a 2x2x2 like a 2x2x1, use only U/D-invariant moves in a plane
> containing the U/D (short) axis.
>
> This looks like something we can work with
> . To turn an N^4 into an (N^3)x1, use only I/O-invariant moves in a plane
> containing the I/O axis. For the (virtual) 2^4 and 3^4, that means moves
> like Ru2, Rf2, and its symmetries on the other faces. Doing these on a
> virtual puzzle certainly seems believable. But what about a move like Ro?
> It’s not in a plane containing the I/O axis, but is there any sure property
> of (N^M)x1 puzzles that it broke?
>
>
> The more I think about this, the more confused I become.
>
> - Andy
>
> On Mon, Sep 3, 2018 at 2:36 PM Marc Ringuette ringuette@solarmirror.com
> [4D_Cubing] <4D_Cubing@yahoogroups.com> wrote:
>
>>
>>
>> CORRECTION – I mixed up some axes, so the best part of my last message
>> ended up quite scrambled. I got all confused because the corner axes
>> of the regular and mini puzzle are named differently (R-L versus I-O).
>> Here’s a better version of the comparison.
>>
>> For the 2x2x2x2: the short exchange of the U-D and corner (R-L) axes
>> is Iy Oy’, while the pure rotation, FOro, is Iy Oy’ Rx2 Bz2 Uy2 Rx2…
>> The short half-exchange is Iy.
>> For the 2x2x2x1: the short exchange of the U-D and corner (I-O) axes
>> is M y M, while the pure rotation, FRro, is M y M R2 F2 R2 z2. The
>> short half-exchange is M U M.
>>
>> Note that I’m using a tweaked version of Luna’s gyro for the 2x2x2x1,
>> FRro, where I put the y rotation in between the restacks. This version
>> draws out the parallels between the 2x2x2x1 FRro and my favorite gyro
>> for the 2x2x2x2, ROIL FOro. The connection is deep enough that there
>> are even three versions of each rotation: the short one on the physical
>> puzzle, the longer one with cleanup moves to correspond to a one-click
>> MC4D rotation, and the short half-exchange, that re-aligns only half of
>> the puzzle corners.
>>
>> Whatever the details, the parallel between these is still sweeeet.
>>
>> –Marc
>>
>> On 9/3/2018 10:22 AM, Marc Ringuette ringuette@solarmirror.com
>> [4D_Cubing] wrote:
>> > Here’s a different gyro for the 2x2x2x1, FUro, that corresponds
>> > (CORRECTION, NOT REALLY) to the mini version of my favorite gyro, the
>> > ROIL version of the FUro (CORRECTION, FOro) gyro. The connection is
>> > deep enough that there are even three versions of each rotation: the
>> > short one on the physical puzzle, the longer one with cleanup moves to
>> > correspond to a one-click MC4D rotation, and the short half-exchange,
>> > that re-aligns only half of the puzzle corners.
>> >
>> >
>> > For the 2x2x2x2: the short exchange of the L-R and I-O (CORRECTION,
>> > U-D and L-R) axes is Iy Oy’, while the pure rotation, FUro
>> > (CORRECTION, FOro), is Iy Oy’ Rx2 BO2 UO2 Rx2. The short
>> > half-exchange is Iy.
>> > For the 2x2x2x1: the short exchange of the L-R and I-O axes is E X
>> > E, while the pure rotation, FUro, is E X E F2 U2 F2 Y2. The short
>> > half-exchange is E R E.
>>
>>
>
> –
>
> "Machines take me by surprise with great frequency." - Alan Turing
>
>