A new contribution towards our Ring variations completion:

6 Rings (r,u,l,b,d,f) (29 m).

It is the inverse of the listed missing 6 Rings (r,f,d,b,l,u).

Per

Another Rings patterns contribution.

6 Rings (f,b,u,d,r.l) (29m).It is simply composed of 2 already posted ring patterns. 2 turns cancelled out

Per

I improved an already posted 4 Rings pattern.

4 Rings (u,f,d,r) (33 m).

Again combining 2 ring patterns and cancelling 2 turns.

Per

@ Per

Algorithms similar to your 3 last ones were already published on Walter's website (same length - transformed by inversion/symmetry).

The group of rings algorithms was already complete, so it would be best to concentrate on improving the published algorithms.

The 2 longest algorithms that could certainly be improved are those of:
(u,r,b,l,d) (36 moves)
(r,f,b,l,d,u) (39 moves)
Unfortunately, I haven't found any shorter solutions for these two, up to now

Here is a list of a few slightly improved algorithms (1 move less only):

5 Rings (r,f)(d,l,b) (23 moves)
WR WF' WR' VF WR' F2 WR NF' NB' WR' F2 WR NF WR' F WR MF' WR' F' WR MR MF MR'
5 Rings (r,f)(d,l,b) (23 moves)


6 Rings (r,l)(b,f,u,d) (24 moves)
WU' WR2 WU WF' MU F MU WF2 MU' F' MU' WF' NR2 U2 B2 NL SF2 NR' F2 U2 NL2 MU' MR2 MU
6 Rings (r,l)(b,f,u,d) (24 moves)


6 Rings (r,f,u,l,b,d) (12 moves)
WR WF' WR VF2 NL2 MF2 VR2 NF' NB' MR MF MR'
6 Rings (r,f,u,l,b,d) (12 moves)


6 Rings (r,f,u,d,l,b) (23 moves)
WR WF' WR' VF WU F2 WU' NF' NB' WU F2 WU' NF WU F WU' MF' WU F' WU' MR MF MR'
6 Rings (r,f,u,d,l,b) (23 moves)


6 Rings (r,d,u,b,f,l) (28 moves)
VB' R2 NB NF R2 NB' WU' NF' R2 NB NF R2 NB' MR D MR' WU' MR D' MR' WF' VL2 NU2 MF MR2 MF' VD2 NL2
6 Rings (r,d,u,b,f,l) (28 moves)

Hmmm.

I cannot find the professor Ring patterns on that webpage. Give me a link please

Kind regards,

Per

Hi Per,

Click the link below for displaying the 3 Rings (u,r,f) page:
Ring Patterns - 5x5x5 Cube

Other patterns can be selected in the left-handside window.

Hi!

I found an improvement to an existing case.

5 Rings (r,d,f,u,l) (33 m).

It is equivalent to 5 Rings (u,r,b,l,d) by symmetry. So quite a nice improvement - by just combining 2 known ring patterns ...

Per

I guess this is my last Rings contribution for a while ...

I improved an existing case. Equivalent to (r,f,b,l,d,u) ...

6 Rings (f,b,u,r,l,d) (35 m).

Yet again combining 2 other ring patterns. Quite a lot of transformation work though ...

Per

I just found an alternative solution for the 'Gridlock' pattern which saves 5 moves:

Gridlock
MR' U' MR MF D' MF' MR' U MR MF D MF2
R' MF MU L' MU' MF' R MF MU L MU2
F' MU MR B' MR' MU' F MU MR B MR' (34 btm)


While we're on the 4x4x4, here are two more patterns for this cube:

Winding Hexagon
R MD' L' U' L MD MU L' U L MU' R'
L' MD' R D R' MD MU R D' R' MU' L
TF MU' L2 MU MR MU' SR2 MU ML MU' R2 MU MR' ML' TF' (39 btm)

4 Woven Rings
SD R' L' U2 F' B' U' F2 WR2 F2
MD TD TL TR TU2 TB TF SD2
MD TR' U MR D MR' U' WR D' R D ML D' TL' U ML D ML' U' WR' D' L D MR TD' (43 btm)

I slightly improved the '4 Woven Rings':

4 Woven Rings
WF' WR WF WR'
F' B' L2 (U D ML2 SD MR2 MF2 MR2 D2) L2 B F
TR D2 F' U SF MD2 SF' U' SF MD2 B SD2 B' D SB MU2 SF D' SF' MU2 F U2 TR' (41 btm)

With some additional moves a scalable algorithm can be created:

4 Woven Rings (scalable version)
SD R' L' U2 F' B' U' F2 SR2 B2
NU TB2 TR2 TL2 TF2 TU TB TF TU2 TR TL TU' TD SD2
ND TR' U NR D NR' U' NL' NR D' R D NL D' TL' U NL D NL' U' NL NR' D' L D NR TD' (51 btm)

When you display the pattern just click on any of these links below the applet:
Revenge Cube · Professor Cube · V-Cube 6 · V-Cube 7.
The pattern is adapting itself to the corresponding cube size.

Here is a slightly shorter version of the 4 Woven Rings (4x4x4 Cube).
Unfortunately, it is not scalable

WF' WR WF WR'
F2 D2 R2 WU WF' MR2 WF WU' TR2 D2 F2
TF L2 U' R SU ML2 SU' R' SU ML2 D SL2 D' L SD MR2 SU L' SU' MR2 U R2 TF'

4 Woven Rings (38 moves)

I improved the Winding Hexagon (34 m) pattern.

Breakdown:

1. TR- (setup)
2. [NU F2 NU-, TB-]
3. [U F2 U-, NB-]
4. [NU B2 NU-, TF-]
5. [U B2 U-, NF-]
6. TR (undo setup)

Basically only 4 8-move commutators with setup

Per

The Winding Hexagon pattern is scalable like so:

Winding Hexagon (V6, 34m).

This one is scaled unequally. It is also possible to scale equally. I will post that quite shortly

Unfortunately it is impossible to scale to 5x5x5 (core restriction...)

Per

Great algorithm improvements for the '4 Woven Rings' and the 'Winding Hexagon'!

Finding algorithms which can display the same pattern on a range of other cube models is quite amazing.
Often patterns look distorted when applied on other cubes so that new solutions have to be found.


I was looking for 'Winding Hexagon' variations with different color arrangements.
Here is a (not scalable) version with 4 colors per face:

Winding Hexagon (with 4 colors per face):
D'
(U F' D SF MD2 SF' D' F SD B' MD2)
(F R' B SR MB2 SR' B' R SF' L' MB2)
(R U' L SD' ML2 SD L' U SR' D' ML2)
D
R2 D2 MU ML R MU' ML MU R' MU' MF' ML' MF ML' D2 R2 (51 btm or 50 btm actually)

Hi!

I improved the Winding Hexagon (4 Colours per face) (43 m).

First i permute 4 "chicken feet" like so:

TD2 TF TR2 TD- TR2 R- L- D2 R L U R- L- D2 R L U TR2 TD TR2 TF- TD2 (5 first turns are setup).

Then the last 2 "chicken feet" like so:

F2 WD NB2 WD- F- WD NB2 WD- NU2 R- L- U R L NU2 R- L- U- R L F-

Quite a nice improvement

The setup moves for first part allows other patterns to be deduced

Per

Another pattern based on the setup moves TD2 TF TR2 TD- TR2:

Squared Hexagon (21 m).

Per

Another surprise find with the same setup turns:

Sorta Cube-in-Cube (22 m).

(Doable in 18 turns with 3-cycles )

Per

It is possible to "scale" the Winding Hexagon (4 colour version, V6) (53 m).

The last 10 turns are center correction turns Actually the 4x4x4 original version is a degenerate version

Per

@ Per
Hey, you found very cool and short solutions!


I am still fascinated by the hexagon that runs through all six faces of the cube.
So the form of the hexagon is exactly the same but the colorization is different:

Double Hexagon 1
R MU L' U' L MU' MD' L' U L MD R'
L' MU R D R' MU' MD' R D' R' MD L
TF MU' L2 MU MR MU' SR2 MU ML MU' R2 MU MR' ML' TF' (39 btm)

Double Hexagon 2
MR' B' D B MR B' MR' D' MR B
MD2 R' F U2 F' R MD2 R' F2 U' MF U F' U' MF' U' F' R
TB2 SR' B R SF L' MB2 L SF' R' B' SR B2
MU2 R' MU MR MU' R MU MR' MU'
F MU' MF' MB' MU F' MU' MB MF MU' (60 btm)

@ Walter

The algorithms have been very slightly improved:

Double Hexagon 1
Edges
L' MU R D R' MU' MD' R D' R' MD L
L MB R' B' R MB' MF' R' B R MF L'
Centers
TL MD' F2 MD MB MD' SB2 MD MF MD' B2 MD MB' MF' TL'
Double Hexagon 1 (37 btm)


Double Hexagon 2
Centers
MU MF' MB' MU F MU' MB MF MU F' MU MR MU' R' MU MR' MU' R MU2
Edges
MU TB2 SD' B' U' SB D MB2 D' SB' U B SD B' L' B' MU' B MU L MU' B' ML2 U F' R2 F U' ML2 U F2 R MF' R' F R MF R F U'
Double Hexagon 2 (58 btm)

I don't know if anyone has made this. Here's the edge turning dodecahedron mezon. It looks like some kind of face turning dodecahedron.

I haven't applied the stickers yet. But jumbling it is still fun.

I improved the following pattern, by a new approach:

Double Hexagon 1 (29 btm).

The breakdown is like so:

1. D U R2 B WU2 R- WU2 R B- R2 D- U- (cycle dedges)

2. TU NB TR- F2 TR NB- TR- F2 TR TU- (bar 3-cycle)

3. TU TF NR- B2 NR TF- NR- B2 NR TU- (bar 3-cycle)

3 turns cancel out for good measure

Per

The centres on the green face for version 2 are incorrect ... minor mistake i guess

Per

The Impossible Big Edge





Exchange 2 corners and flip one edge.
Both impossible on Rubik's Cube.

These are 2 speceial parity cases on Mixup Plus.

I added setup moves to fix the center piece in the back face.
Well, the pattern is actually a very nice color variation of the Winding Hexagon:

Winding Hexagon 2
B (MU MF' MB' MU F MU' MB MF MU F' MU MR MU' R' MU MR' MU' R MU2) B'
MU TB2 SD' B' U' SB D MB2 D' SB' U B SD B' L' B' MU' B MU L MU' B' ML2 U F' R2 F U' ML2 U F2 R MF' R' F R MF R F U' (61 btm)




Very clever approach! This way the centers doesn't have to be done seperately which saves a lot of moves.

@ Andrea

Your pattern is indeed impossible to achive on a Rubik's Cube.
So, here is a solution for a 4x4x4 cube which requires 15 moves:

Impossible Big Edge
TD R2 TU' R2 TD' B2 TU B2 TD' B2 TU B2 TD' T3L2 WF2 (15 btm)

I assume, your algorithm for the 'Mixup Plus Cube' must be completely different.


@ schuma

Your Dodecahedron pattern looks very nice, even without stickers.
Do you have an algorithm for this?

Hi Waran,


Yes. After the parity, you must re-solve the extended centers. The Mixup Cube works very different to the 4x4x4.

@ Per

Double Hexagon 1
A very tiny gain of 1 move can been obtained from this insertion:
Insertion of TU TF MR' B2 MR TF' MR' B2 MR TU' into TU MB TR' F2 TR MB' TR' F2 TR TU':
(TU) (MB TR' F2 TR MB') (TF MR' B2 MR TF' MR' B2 MR) (TR' F2 TR) (TU')
Double Hexagon 1 (28 btm)

@ Walter

Double Hexagon 2
I forgot to invert the algorithm. This has been corrected and 2 more moves have been cancelled.
I guess Per's approach to the previous pattern could be applied to this one too

Edges:
U F' R' MF' R' F' R MF R' F2 U' ML2 U F' R2 F U' ML2 B MU L' MU' B' MU B L B SD' B' U' SB D MB2 D' SB' U B SD TB2 MU'
Centers
MU2 R' MU MR MU' R MU MR' MU' F MU' MF' MB' MU F' MU' MB MF MU'
Double Hexagon 2 (56 btm)

Yet another hexagon pattern variety:

Double Hexagon 3 (54 btm).

A little bit tricky to avoid center side effects. But basically made up from previously published stuff

Per

Some more hexagon versions:

Fat Hexagon (26 btm):

Single Hexagon (25 btm).

And an improved previously posted version: (this one is simply the first 2 combined )

Double Hexagon 2 (50 btm).

There are many more possibilities, combining many axes of rotation - but i find that too tedious and not really interesting

Per

Fat Hexagon (order 3) (21 btm).

step1: U B2 WR2 L WF WR2 D WF D- L- B2 U- (edges)
step2: U F2 L2 D- NR NB NR- NB- D L2 F2 U- (centers)

I could not resist this easy version

(some insertion might improve this ... Glazik???)

Per

Hi Per,

Some algorithms of hexagon patterns have been improved.
A shorter version of that of your last post couldn't be found though.

Double Hexagon 3
U D R2 B
WU2 R' WU2 R
B' R2 D' U'
U2 B2
B2
R2 MU2 F2 MU2 TF2 MU2 TF2 R2 F2 R2 F2
B2
MF MD U MF' MD MF U' MF' MR' MD' MR MD'
B2 U2
TU
MR' B2 MR (TR' F2 TR MB TR' F2 TR MB') TF MR' B2 MR TF'
TU'
Double Hexagon 3 (53 btm)


Single Hexagon
R'
MU F2 MU' MB' MU F2 MU' MB
B F MR U' MR' U B' L' U' MR U MR' L F'
R
Single Hexagon (23 btm)


Double Hexagon 2
R2
F2 MR2 U2 MR2 TU2 MR2 TU2 F2 U2 F2 U2
D2
MU ML R MU' ML MU R' MU' MF' ML' MF ML'
D2
R2
R'
MU F2 MU' MB' MU F2 MU' MB
B F MR U' MR' U B' L' U' MR U MR' L F'
R
Double Hexagon 2 (48 btm)

Large mezon on Bipolar Cube:



If to put beside both kinds from opposite corners, they look as reflection in a mirror each other!
Such interesting property of a puzzle.

Fat Hexagon (order 6) (45 btm).

Breakdown:

1. F L2 WU2 D- WR WU2 B WR B- D L2 F- (edges, 3x3 style))
2. F2 NR2 U2 NR2 TU2 NR2 NU2 F2 (last 2 edges - "parity")
3. NR2 ND- NR U2 NR- ND NR U2 NR (centers)
4. ND- L2 ND- NR- ND L2 ND- NR ND2 (more centers)
5. ND2 NU NL- NU- L- NU NL NU- L ND2 (final 2 centers)

There should be room for plenty improvement

Per

The algorithm of centers has been shortened.
Further improvement is still feasible.

Fat Hexagon (Order 6)
Centers (composition of 3-cycles):
F2 U2 R2
[MU', MR]
R' [MF2, R' MB' R] R
R2 U2 F2

Fat Hexagon (Order 6) (32 btm)

This should be darn close to optimal. Nice work indeed

EDIT:

I found an alternative way to do the centers:

1. F2 U2 NU- NR NU NR- U2 F2
2. R2 ND- NR2 ND R2 ND- NR2 ND

Fat Hexagon (Order 6) (32 btm).

Less cancelled turns, but still resulting in 32 turns for the pattern
And also i guess it's easier to optimise this one, with some insertion

Per

@ Per
It appears that two centers located on faces R and L should be swapped.
Maybe you could correct the alg.

I think shorter algorithms could be found by searching directly for short 6-cycles of edge pairs, although such a direct search for even cycles doesn't seem quite obvious.
Here is an example of a 8-move alg: [WR, U R B]

Apart from 6-cycles, 4-cycles of pairs of edges or centers may also give simple patterns:

Serpentine (Order 4)
WU' WR' MF WU WR2 WU MF' WR'
MR2 MU2 MR2 WF2 MU' WF2 MU'
Serpentine (Order 4) (15 btm)

Bridge(Order 4)
WU' WR' MF WU WR2 WU MF' WR'
SR2 MR2 MU2 MR2 WF2 MU' WF2 MU' SR2
Bridge (Order 4) (17 btm)

Hi

Yes i noticed i made a mistake in my last post. I was gonna post a correction once i had a solution equalling or improving 32 turns. So far, no luck Maybe i will sticker the centres for some serious hunting later

I improved the

Serpentine (Order 4) (8 btm).

And the

Bridge (Order4) (13 btm).

Per

Hi Per,

Fat Hexagon (Order 6)
No shorter algorithm could be found for edges

Bridge
The algorithm can be improved by insertion of (MF2 MD2 MF2 MD2) into (WF2 MU WU2 MR2 WU2 MR2 MU' WF2 WU)
MB2 MU2 MF2 MD2 MU MR2 WU2 MR2 MU' WF2 WU
Bridge (11 btm)

Fat Anaconda (Order 6)
By re-using the algorithm of edges of the Fat Hexagon, a pattern of order 6 can be obtained (not yet optimized, though):
6 wide edges:
F L2 WU2 D' WR WU2 B WR B' D L2 F MR2 U2 MR2 TU2 MR2 MU2 F2
6 big dots:
MF2 WF WR2 MF2 WU' WR2 WF' WU
F L2 WU2 D' WR WU2 B WR B' D L2 F MR2 U2 MR2 TU2 MR2 MU2 TF2 WF WR2 MF2 WU' WR2 WF' WU
Fat Anaconda (Order 6) (26 btm)

Yet another hexagon pattern, this time on 6x6x6.

Double Hexagon 6x6x6 (order 3) (45 btm).

Breakdown:

1. R L B2 U VL- U2 VL U NF VD NF- VD- VF ND VF- ND- U- VL- U2 VL U- B2 L- R- (centers)
2. R N3U- F- U2 F N3U N3D F- U2 F N3D- R- (edges)
3. R- N3B- U F2 U- N3B N3F U F2 U- N3F- R (final edges)

I will explain part 1.

First i do the dots with 2 overlapping bar 3-cycles ([NF,VD] + [VF,ND]).
I conjugate this with U- VL- U2 VL U first to make the dots on U,R,F faces cycle the correct way. Then i conjugate further with R L B2 U2 to finalise center positions

3 turns cancel out (as planned )

Per

This is a very interesting way of conjugating centers in just 2 phases

A gain of 2 moves can be obtained by shortening the first setup phase, if (U VL' U2 VL U) is replaced with (VB' R2 VB U2) to give:
(R L B2) (VB' R2 VB U2) (NF VD NF' VD' VF ND VF' ND') (U2 VB' R2 VB) (B2 L' R')
Double Hexagon (Order 3) (6x6x6 Cube) (43 btm)

The following is a list of new/alternate/improved algorithms of Big Dots (4x4x4 cubes) that could be added to Walter's website:

(2c)
(d,u)
SR2 MF2 SR2 MB2 (4 m)


(2c)(2c)
(d,u) (b,f)
WR WU2 WR' WU2 (4 m)

(l,r) (b,f)
SR2 WU SR2 WU (4 m)

(b,d) (u,f)
MR SU2 MR' ML' SU2 ML (6 m)


(2c)(2c)(2c)
(d,u) (l,r) (b,f)
MR2 WU2 MR2 WF WU2 WF' (6 m)

(b,u) (l,r) (d,f)
MR WU2 WR' WF WU2 WF' ML' (7 m)


(2c)(3c)
(d,r) (l,u,b)
WR WU WR' MD WR U2 WR' MU' MD' WR U2 WR' MD (13 m)


(5c)
(u,d,f,b,l)
WR MD' R2 MD MU R2 MD' WR' MD R2 MD' MU' R2 MD (14 m)

(r,b,l,f)
MD' WF2 MD2 WR2 MD WF2 MD2 WR2

(r,u)
MF WU' B2 WU MB' MF' WU' B2 WU MB

(r,b,l,f)(r,u) = (r,b,l,f,u) = (u,r,b,l,f)
MD' WF2 MD2 WR2 MD WF2 MD2 WR2 MF WU' B2 WU MB' MF' WU' B2 WU MB (18 m)


(6c)
(u,l,f,d,r,b)
MB2 WU2 MB2 WB' WU' WB WU' (7 m)

Slight improvement:

Double Hexagon (6x6x6) (42 btm).

Primary setup: VU F2 VU2 R2 VU
Center cycles: [VU- , NB-] and [NU- , VB-]

etc etc ....

Per

And another version based on same setup turns:

Wide Double Hexagons (6x6x6) (47 btm).

This time doing 3 center cycles: [NU- , N4B-] , [N3U- , N3B-] and [N4U- , NB-].

Cheers

Per

Hi,

A slight improvement is obtained by replacing [NU' , N3F] with VU' [N3F, NU] VU:
Wide Double Hexagons (6x6x6 Cube) (45 btm)

Here are two simple patterns on a 6x6x6 cube:

2 Zeros (f,b) (6x6x6 Cube) (8 btm)

4 Zeros (f,b)(r,l) (6x6x6 Cube) (8 btm)

Good job Glazik

Her is another version:

Fat Double Hexagon (6x6x6) (47 btm).

Decomposition should be familiar

Per

I improved the:

2 Zeros (f,b) (6x6x6) (6 btm).

Per

Double Hexagon
By replacing setup (D2 VU F2 VU2 R2 VU) with (VD R2 T3D2 F2) a 2-move gain can be obtained
Double Hexagon (6x6x6) (40 btm)

Big Dots (4x4x4)
Here is another (unoptimized) algorithm:
T3R2 D2 MB' SU2 MF U2 R2 MF WR2 MF'
Big Dots (l,r,d,u) (4x4x4) (10 btm)

As for Rings, the minimum set of cycles is 33 for Big Dots or Zeros: just too many cases but a few could still be considered

Excellent setup Glazik! I guess shorter than 40 would be hard to beat

Here is a new pattern:

Double Fat Triangle (38 btm).

Other triangle patterns like this can surely be found

Per

Thanks, Per.
Setups of less than 4 moves could not be found, so I think this 4-mover may well be optimal

The Double Fat Triangle pattern has been very slightly improved by replacing:
1- R (N3B U2 N3B' N3U' N3B U2 N3B' N3U) R' with R' (N3F N3D' F2 N3D N3F' N3D' F2 N3D) R
2- R' (V3F ND' F2 ND V3F' ND' F2 ND) R with R' N3D' (V3F ND' F2 ND V3F' ND' F2 ND) N3D R
Double Fat Triangle (6x6x6) (37 btm)


Here are simple variations of the 'Zero' pattern on a 6x6x6 cube:

Four Zeros (f,b)(r,l)
Insertion of 6-mover (r,l) into (f,b):
NR2 NF2 NU2 M2F2 V3U2 V3F2 V3R2 V3U2 M2R2 NU2
Four Zeros (f,b)(r,l) (10 btm)

Four Chained Zeros
NR2 NF2 NU2 M2F2 V3U2 V3F2 V3R2 V3U2 M2R2 NU2
N3U2 SF2 SL2 N3D2 SF2 SL2
Four Chained Zeros (15 btm)

Two Chained Half-Zeros
B2 TR2 M2U NR2 WU' N3B2 ND' NU N3B2 R2 B2
N3D B F N3R N3L SF R2 SF' N3R' N3L' SF R2 F2 N3D'
Two Chained Half-Zeros (25 btm)

Shorter versions can surely be found