# 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

>

>