=C2=A0 =20=20=20=20=20=20
=20=20=20=20=20=20
=20=20
=20=20
Hi gang,
It's becoming clearer to me that the mini-puzzle does indeed
correspond to a 4D virtual puzzle, still not precisely defined, that
is some flavor of 2x2x2x1 cuboid.=C2=A0 As you'll see by my discuss=
ion of
gyro rotations below, the connection is looking pretty deep.=C2=A0=C2=
=A0=C2=A0 In
fact, I started messing around with the MC4D 2^4 hypercube puzzle
(thinking of various colorings, projections, and allowed moves) in
the hopes that I could figure out what 4D puzzle the mini-puzzle
emulates.=C2=A0 Can anyone see what it is yet?=C2=A0 I'm not quite =
there.
Gambling that this correspondence will pay off ... I hereby name the
mini-puzzle the 2x2x2x1=C2=A0 (or depending on context, "Melinda&#=
39;s
2x2x2x1" or "the physical 2x2x2x1", but please never &qu=
ot;Marc's
2x2x2x1" because I prefer to stay under the radar, name-wise).=C2=
=A0=C2=A0
My latest bit of insight was prompted by Luna's discussion of the
face swap versus the pure face swap:
[ I will call M S the face swap.] =C2=A0 The =
pure
face swap is M S R2 F2 R2 Y'.=C2=A0=C2=A0=C2=A0=C2=A0=C2=
=A0=C2=A0=C2=A0 ~Luna
The pure face swap that Luna describes looks just like a gyro
rotation for the 2x2x2x1 !=C2=A0 (Actually, I might propose that a Z2
rotation be added to the end of his pure face swap, making it=C2=A0 M S
R2 F2 R2 Y' Z2 , so that both the F-B and the L-R face pairs are
completely unchanged.=C2=A0=C2=A0 This is the 4 dimensional "FR ro=
tation",
holding the F-B and R-L axes fixed while rotating the other two.=C2=A0=
=C2=A0
So, this modified gyro for the 2x2x2x1 would sensibly be called
FRro.
Here's a different gyro for the 2x2x2x1, FUro, that corresponds to
the mini version of my favorite gyro, the ROIL version of the FUro
gyro.=C2=A0 The connection is deep enough that there are even three
versions of each rotation:=C2=A0 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:=C2=A0=C2=A0=C2=A0 the short exchange of the L-R and I-=
O axes is Iy
Oy', while the pure rotation, FUro, is Iy Oy' Rx2 BO2 UO2 Rx2.=
=C2=A0=C2=A0 The
short half-exchange is Iy.
For the 2x2x2x1:=C2=A0=C2=A0=C2=A0 the short exchange of the L-R and I-=
O axes is E
X E, while the pure rotation, FUro, is=C2=A0 E X E F2 U2 F2 Y2.=C2=A0 T=
he
short half-exchange is E R E.
Sweeeeet.
--Marc
=20=20
=20=20=20=20=20
=20=20=20=20
=20=20
--000000000000b9e3e60574fb347a--
From: Luna Harran <scarecrowfish@gmail.com>
Date: Mon, 3 Sep 2018 19:13:33 +0100
Subject: Re: [MC4D] 2x2x2x1: Gyro rotations, and seeking the equivalent 4D cuboid
--0000000000007628cd0574fb8009
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Are these equivalent positions?
(M S R2 F2 R2 Y' Z2 on physical puzzle)
(FRro' UF2 IF2 UF2 on virtual puzzle)
~Luna
On 3 Sep 2018 18:52, "Luna Harran" wrote:
Nice. I tried a while ago to get the 2x2x2x1 cuboid in MPU, but I can't run
it at all at the moment unfortunately. Maybe someone with a better
understanding of the puzzle definition could try it out.
Also, could we perhaps find a way to simulate the 2x2x2x1 on the full 2^4?
That might help matters. I'm unsure quite what needs bandaging.
Could this also lead to a physical 3x3x3x1, or other such cuboids? Valid 4d
puzzles that are easier to construct.
Also, I'm a she. =F0=9F=98=89=F0=9F=98=81
~Luna
On Mon, 3 Sep 2018, 18:22 Marc Ringuette ringuette@solarmirror.com
[4D_Cubing], <4D_Cubing@yahoogroups.com> wrote:
>
>
> Hi gang,
>
> It's becoming clearer to me that the mini-puzzle does indeed correspond t=
o
> a 4D virtual puzzle, still not precisely defined, that is some flavor of
> 2x2x2x1 cuboid. As you'll see by my discussion of gyro rotations below,
> the connection is looking pretty deep. In fact, I started messing arou=
nd
> with the MC4D 2^4 hypercube puzzle (thinking of various colorings,
> projections, and allowed moves) in the hopes that I could figure out what
> 4D puzzle the mini-puzzle emulates. Can anyone see what it is yet? I'm
> not quite there.
>
>
> Gambling that this correspondence will pay off ... I hereby name the
> mini-puzzle the 2x2x2x1 (or depending on context, "Melinda's 2x2x2x1" or
> "the physical 2x2x2x1", but please never "Marc's 2x2x2x1" because I prefe=
r
> to stay under the radar, name-wise).
>
>
> My latest bit of insight was prompted by Luna's discussion of the face
> swap versus the pure face swap:
>
> [ I will call M S the face swap.] The pure face swap is M S R2 F2 R2
> Y'. ~Luna
>
>
> The pure face swap that Luna describes looks just like a gyro rotation fo=
r
> the 2x2x2x1 ! (Actually, I might propose that a Z2 rotation be added to
> the end of his pure face swap, making it M S R2 F2 R2 Y' Z2 , so that bo=
th
> the F-B and the L-R face pairs are completely unchanged. This is the 4
> dimensional "FR rotation", holding the F-B and R-L axes fixed while
> rotating the other two. So, this modified gyro for the 2x2x2x1 would
> sensibly be called FRro.
>
>
> Here's a different gyro for the 2x2x2x1, FUro, that corresponds to the
> mini version of my favorite gyro, the ROIL version of the FUro 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 axes is Iy Oy',
> while the pure rotation, FUro, 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.
>
>
> Sweeeeet.
>
> --Marc
>
>=20
>
--0000000000007628c90574fb8007
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Are these equivalent positions?
v>
(M S R2 F2 R2 Y' Z2 on physical puzzle)=C2=A0
=
(FRro' UF2 IF2 UF2 on virtual puzzle)
=3D"auto">
~Luna
extra">
On 3 Sep 2018 18:52, "Luna Harra=
n" <
scarecrowfish@gmail.=
com> wrote:
e=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">r=3D"auto">Nice. I tried a while ago to get the 2x2x2x1 cuboid in MPU, but =
I can't run it at all at the moment unfortunately. Maybe someone with a=
better understanding of the puzzle definition could try it out.
"auto">
Also, could we perhaps find a way to sim=
ulate the 2x2x2x1 on the full 2^4? That might help matters. I'm unsure =
quite what needs bandaging.=C2=A0
=3D"auto">Could this also lead to a physical 3x3x3x1, or other such cuboids=
? Valid 4d puzzles that are easier to construct.=C2=A0
o">
Also, I'm a she. =F0=9F=98=89=F0=9F=98=
=81
=3D"gmail_quote">
=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
=20
=C2=A0 =20=20=20=20=20=20
=20=20=20=20=20=20
=20=20
=20=20
Hi gang,
It's becoming clearer to me that the mini-puzzle does indeed
correspond to a 4D virtual puzzle, still not precisely defined, that
is some flavor of 2x2x2x1 cuboid.=C2=A0 As you'll see by my discuss=
ion of
gyro rotations below, the connection is looking pretty deep.=C2=A0=C2=
=A0=C2=A0 In
fact, I started messing around with the MC4D 2^4 hypercube puzzle
(thinking of various colorings, projections, and allowed moves) in
the hopes that I could figure out what 4D puzzle the mini-puzzle
emulates.=C2=A0 Can anyone see what it is yet?=C2=A0 I'm not quite =
there.
Gambling that this correspondence will pay off ... I hereby name the
mini-puzzle the 2x2x2x1=C2=A0 (or depending on context, "Melinda&#=
39;s
2x2x2x1" or "the physical 2x2x2x1", but please never &qu=
ot;Marc's
2x2x2x1" because I prefer to stay under the radar, name-wise).=C2=
=A0=C2=A0
My latest bit of insight was prompted by Luna's discussion of the
face swap versus the pure face swap:
x">
[ I will call M S the face swap.] =C2=A0 The =
pure
face swap is M S R2 F2 R2 Y'.=C2=A0=C2=A0=C2=A0=C2=A0=C2=
=A0=C2=A0=C2=A0 ~Luna
The pure face swap that Luna describes looks just like a gyro
rotation for the 2x2x2x1 !=C2=A0 (Actually, I might propose that a Z2
rotation be added to the end of his pure face swap, making it=C2=A0 M S
R2 F2 R2 Y' Z2 , so that both the F-B and the L-R face pairs are
completely unchanged.=C2=A0=C2=A0 This is the 4 dimensional "FR ro=
tation",
holding the F-B and R-L axes fixed while rotating the other two.=C2=A0=
=C2=A0
So, this modified gyro for the 2x2x2x1 would sensibly be called
FRro.
Here's a different gyro for the 2x2x2x1, FUro, that corresponds to
the mini version of my favorite gyro, the ROIL version of the FUro
gyro.=C2=A0 The connection is deep enough that there are even three
versions of each rotation:=C2=A0 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:=C2=A0=C2=A0=C2=A0 the short exchange of the L-R and I-=
O axes is Iy
Oy', while the pure rotation, FUro, is Iy Oy' Rx2 BO2 UO2 Rx2.=
=C2=A0=C2=A0 The
short half-exchange is Iy.
For the 2x2x2x1:=C2=A0=C2=A0=C2=A0 the short exchange of the L-R and I-=
O axes is E
X E, while the pure rotation, FUro, is=C2=A0 E X E F2 U2 F2 Y2.=C2=A0 T=
he
short half-exchange is E R E.
Sweeeeet.
--Marc
=20=20
=20=20=20=20=20
=20=20=20=20
=20=20
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--0000000000007628cd0574fb8009--
From: Andrew Farkas <ajfarkas12@gmail.com>
Date: Mon, 3 Sep 2018 17:26:22 -0400
Subject: Re: [MC4D] 2x2x2x1: Gyro rotations, and seeking the equivalent 4D cuboid
--0000000000008820750574fe33e8
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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.
>
>=20
>
--=20
"Machines take me by surprise with great frequency." - Alan Turing
--0000000000008820750574fe33e8
Content-Type: text/html; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
t ms,sans-serif">Hey all!
amily:trebuchet ms,sans-serif">
=3D"font-family:trebuchet ms,sans-serif">I'm very=C2=A0tentative=
to call the twisty-stacky a 2x2x2x1. I've already half-written and del=
eted multiple emails trying to work through the behavior of a 4D cuboid, bu=
t I could never come to a concrete conclusion. Here is one line of thinking=
I ran down:
et ms,sans-serif">
one;padding:0px">t ms,sans-serif">Some observations about the NxNx1, with the Z axis as the =
short axis:
mily:trebuchet ms,sans-serif">
- The puzzle has 2D cuts along the =
XZ and YZ planes. - 180-degree moves are possible in two planes: XZ a=
nd 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.
<=
blockquote style=3D"margin:0 0 0 40px;border:none;padding:0px">
=3D"gmail_default" style=3D"font-family:trebuchet ms,sans-serif">
The 2=
^4 has 2D cuts alo-- well no, they would be 3D cuts along each of the ... 1=
2? 24 possible 3D slices? But how does that correspond to which moves are a=
llowed?
family:trebuchet ms,sans-serif">
Let me try another one:=
ne;padding:0px"> ms,sans-serif">To use a 2x2x2 like a 2x2x1, use only U/D-invariant moves i=
n a plane containing the U/D (short) axis.
" style=3D"font-family:trebuchet ms,sans-serif">
" style=3D"display:inline">
le=3D"font-family:trebuchet ms,sans-serif">
le=3D"display:inline">This looks like something we can work with
class=3D"gmail_default" style=3D"display:inline">. To turn an N^4 into an (=
N^3)x1, use only I/O-invariant moves in a plane containing the I/O axis. Fo=
r the (virtual) 2^4 and 3^4, that means moves like Ru2, Rf2, and its symmet=
ries on the other faces. Doing these on a virtual puzzle certainly seems be=
lievable. 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 br=
oke?
<=
div class=3D"gmail_default" style=3D"font-family:"trebuchet ms",s=
ans-serif;display:inline">
et ms, sans-serif">trebuchet ms",sans-serif;display:inline">The more I think about this, =
the more confused I become.
ms, sans-serif">
buchet ms",sans-serif;display:inline">
t face=3D"trebuchet ms, sans-serif">
ont-family:"trebuchet ms",sans-serif;display:inline">- Andy
=
<=
blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1px=
#ccc solid;padding-left:1ex">
=20
=C2=A0 =20=20=20=20=20=20
=20=20=20=20=20=20
CORRECTION -- I mixed up some axes, so the best part of my last me=
ssage
ended up quite scrambled.=C2=A0=C2=A0 I got all confused because the corner=
axes
of the regular and mini puzzle are named differently (R-L versus I-O).=C2=
=A0=C2=A0
Here's a better version of the comparison.
For the 2x2x2x2:=C2=A0=C2=A0=C2=A0 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=
..=C2=A0=C2=A0
The short half-exchange is Iy.
For the 2x2x2x1:=C2=A0=C2=A0=C2=A0 the short exchange of the U-D and corner=
(I-O) axes
is M y M, while the pure rotation, FRro, is=C2=A0 M y M R2 F2 R2 z2.=C2=A0 =
The
short half-exchange is M U M.
Note that I'm using a tweaked version of Luna's gyro for the 2x2x2x=
1,
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.=C2=A0 The connection is deep enough that there =
are even three versions of each rotation:=C2=A0 the short one on the physic=
al
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 r.com" target=3D"_blank">ringuette@solarmirror.com
[4D_Cubing] wrote:
> Here's a different gyro for the 2x2x2x1, FUro, that corresponds r>
> (CORRECTION, NOT REALLY) to the mini version of my favorite gyro, the =
> ROIL version of the FUro (CORRECTION, FOro) gyro.=C2=A0 The connection=
is
> deep enough that there are even three versions of each rotation:=C2=A0=
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:=C2=A0=C2=A0=C2=A0 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.=C2=A0=C2=A0 The sho=
rt
> half-exchange is Iy.
> For the 2x2x2x1:=C2=A0=C2=A0=C2=A0 the short exchange of the L-R and I=
-O axes is E X
> E, while the pure rotation, FUro, is=C2=A0 E X E F2 U2 F2 Y2.=C2=A0 Th=
e short
> half-exchange is E R E.
=20=20=20=20=20
=20=20=20=20
=20=20
--
class=3D"gmail_signature" data-smartmail=3D"gmail_signature">
tr">
r>
"1">"Machines take me by surprise with great frequency." - Alan T=
uring
--0000000000008820750574fe33e8--
From: Joel Karlsson <joelkarlsson97@gmail.com>
Date: Tue, 4 Sep 2018 09:13:23 +0200
Subject: Re: [MC4D] 2x2x2x1: Gyro rotations, and seeking the equivalent 4D cuboid
--00000000000066a2b8057506651a
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
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 m=C3=A5n 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 th=
e
> ... 12? 24 possible 3D slices? But how does that correspond to which move=
s
> 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 proper=
ty
> 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
>=20
>
--00000000000066a2b8057506651a
Content-Type: text/html; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable
Hi everyone,
Although an inter=
esting puzzle, I don't think that the twisty stacky 2^3 is a representa=
tion 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.
r>
Firstly, an n-dimensional twisty cuboid usually has the same nu=
mber 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 c=
uboid should have two 2x2x2 faces and six 2x2x1 faces but the twisty stacky=
2^3 has only one 2x2x2 face and six 2x2x1 faces.
Secondl=
y, on a 2x2x2x1 cuboid only stickers belonging to similar faces can be mixe=
d. So stickers belonging to the 2x2x2 faces can be mixed with each other bu=
t such a sticker can't be on a 2x2x1 face. You could do a bandaged vers=
ion of the 2^4 where this would be possible if you glue together the sticke=
rs 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 r=
eplace two such stickers (forming a 2x2x1 block) with four 1x1x1 stickers f=
rom 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 howeve=
r, 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 banda=
ged 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 b=
oth the bandaged 2^4 described above and the 2^3x1 you cannot twist the I o=
r O face=C2=A0 (this is true for the O face of the twisty stacky 2^3 as wel=
l). Since all pieces on these puzzles have both a sticker belonging to I an=
d a sticker belonging to O, an attempt to twist I or O would result in a ro=
tation.
Best regards,
Joel
=
mail_quote" style=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-l=
eft:1ex">
=20
=C2=A0 =20=20=20=20=20=20
=20=20=20=20=20=20
:trebuchet ms,sans-serif">Hey all!
=3D"font-family:trebuchet ms,sans-serif">
ult" style=3D"font-family:trebuchet ms,sans-serif">I'm very=C2=
=A0tentative to call the twisty-stacky a 2x2x2x1. I've already half-wri=
tten and deleted multiple emails trying to work through the behavior of a 4=
D cuboid, but I could never come to a concrete conclusion. Here is one line=
of thinking I ran down:
mily:trebuchet ms,sans-serif">
0px;padding:0px">t ms,sans-serif">Some observations about the NxNx1, with the Z axis as the =
short axis:
mily:trebuchet ms,sans-serif">
- The puzzle has 2D cuts along the =
XZ and YZ planes. - 180-degree moves are possible in two planes: XZ a=
nd 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.
<=
blockquote style=3D"margin:0 0 0 40px;padding:0px">
ult" style=3D"font-family:trebuchet ms,sans-serif">
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?
<=
/div>
et ms,sans-serif">
Let me try another one:
>
=3D"gmail_default" style=3D"font-family:trebuchet ms,sans-serif">To use a 2=
x2x2 like a 2x2x1, use only U/D-invariant moves in a plane containing the U=
/D (short) axis.
ss=3D"gmail_default" style=3D"font-family:trebuchet ms,sans-serif">
ss=3D"gmail_default">This looks like something we can work with
lass=3D"gmail_default">. To turn an N^4 into an (N^3)x1, use only I/O-invar=
iant 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. D=
oing 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 ther=
e any sure property of (N^M)x1 puzzles that it broke?
" style=3D"font-family:"trebuchet ms",sans-serif">
t>fault" style=3D"font-family:"trebuchet ms",sans-serif">The more I=
think about this, the more confused I become.
face=3D"trebuchet ms, sans-serif">
nt-family:"trebuchet ms",sans-serif">
=
=3D"font-family:"trebuchet ms",sans-serif">- Andy
iv>
left:1px #ccc solid">
=20
=C2=A0 =20=20=20=20=20=20
=20=20=20=20=20=20
CORRECTION -- I mixed up some axes, so the best part of my last me=
ssage
ended up quite scrambled.=C2=A0=C2=A0 I got all confused because the corner=
axes
of the regular and mini puzzle are named differently (R-L versus I-O).=C2=
=A0=C2=A0
Here's a better version of the comparison.
For the 2x2x2x2:=C2=A0=C2=A0=C2=A0 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=
...=C2=A0=C2=A0
The short half-exchange is Iy.
For the 2x2x2x1:=C2=A0=C2=A0=C2=A0 the short exchange of the U-D and corner=
(I-O) axes
is M y M, while the pure rotation, FRro, is=C2=A0 M y M R2 F2 R2 z2.=C2=A0 =
The
short half-exchange is M U M.
Note that I'm using a tweaked version of Luna's gyro for the 2x2x2x=
1,
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.=C2=A0 The connection is deep enough that there =
are even three versions of each rotation:=C2=A0 the short one on the physic=
al
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 r.com" target=3D"_blank">ringuette@solarmirror.com
[4D_Cubing] wrote:
> Here's a different gyro for the 2x2x2x1, FUro, that corresponds r>
> (CORRECTION, NOT REALLY) to the mini version of my favorite gyro, the =
> ROIL version of the FUro (CORRECTION, FOro) gyro.=C2=A0 The connection=
is
> deep enough that there are even three versions of each rotation:=C2=A0=
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:=C2=A0=C2=A0=C2=A0 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.=C2=A0=C2=A0 The sho=
rt
> half-exchange is Iy.
> For the 2x2x2x1:=C2=A0=C2=A0=C2=A0 the short exchange of the L-R and I=
-O axes is E X
> E, while the pure rotation, FUro, is=C2=A0 E X E F2 U2 F2 Y2.=C2=A0 Th=
e short
> half-exchange is E R E.
=20=20=20=20=20
=20=20=20=20
=20=20
--
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nYwK3bkDHKF5m_dH4=3Ds0-d-e1-ft#http://gmail_signature">v>
t>
&qu=
ot;Machines take me by surprise with great frequency." - Alan Turingfont>
=20=20=20=20=20
=20=20=20=20
=20=20
--00000000000066a2b8057506651a--