Yet another ring pattern:

5 rings (5x5x5).

This takes more turns than 6x6x6 equivalent And i'm sure it can be optimised!

Per

One more outstanding Gridlock variety:

Grid - Locked Whirls.

I think i achieved some good cancellation with this. But i still believe better can be found. It is symmetric and not too many pieces are involved

Per

This is a pattern i posted on speedsolving a long time ago:

Both ways tiger stripes.

It's getting harder to find good names

Per

Increasing opposite spots.

I'm collecting all my old nice big cubes patterns here ...

Per

One of Walter's old favourites:

Psychedelic waves.

The start is his own contribution (less 4 twisted corners).

Per

I optimized your approach a little bit and saved 13 moves.

Psychedelic Waves
F R2 U' R2 U D B' D2 SR D2 F U' D' L2 U L2 B'
TF' TR2 TU' TR2 TU2 TR' TU2 TF (25 btm)


Here is another way to create 3 rings:

3 Rings
NF R2 NF' NB' R2 NB WD NF R2 NF' NB' R2 NB WD'
WD' MR U MR' WD MR U' MR' (22 btm or 21 btm actually)

Some spot contributions:

3 large spots.

6 large spots.

6 large spots (order 2).

These should be optimal

Per

Nice improvement - due to better breakdown. I was lazy, simply adding twists to another pattern.

3 Rings can also be done on even sized cubes!

Per

@ Per
Fantastic patterns and as always I keep wondering how you find such short algorithms.


This pattern looks similar to the 'Psychedelic Waves' above.
However it isn't quite that extreme and only has 3 colors per face.
Nevertheless it turned out to be tricky to build it.

My current approach requires 114 moves.
Can anyone see a way to further optimize it?

6x6x6 Waves
F (R' D SR' F R F' L' D' F R' F' R2) F'
VB' L2 VF VB L2 VF'
WD VB' L2 VF VB L2 VF' WD'
R F' (NU2 ND2 F U F' VU2 ND2 F U' F' MU2) F R'
B L' (NU2 ND2 L D L' NU2 VD2 L D' L' MD2) L B'
SD2 SR
(NR NL' NF' NB NR' NL NF NB')
(MF MU' MF' MB' MU MD MB MD')
(MD NF NB' MD' NF' NB) (NU' ND MF NU ND' MF')
SR' SD2
L' (MD' R D R' MD MU R D' R' MU') L
B (MU' F' D' F MU MD F' D F MD') B' (114 btm)


Fortunately a 4x4x4 version is much easier to achieve:

4x4x4 Waves
D' B' L MF2 L' B L MF2 L' MF U B U' MF' U B' D MF D' F' D MF' D' F D' F' L2 D F SR SF L' U' F2 L U' D' F R' MF2 R F' R' MF2 R U'
SD2 SR (MB' MU' MB MU) (MD MF MD' MF') SR' SD2 (58 btm)

Maybe this was published here before. The thread is getting long. But i believe it should be an improvement nevertheless:

3 large rings (same slice).

Constructed from 2 parts:

N2R- WU- MF D (N2F2 MR2)2 D- MF- R2 WU N2R N2L WU- R2 WU N2L- (2 adjacent rings)
N2L2 WU2 VL2 N2U2 MR2 VD2 (2 opposite rings - this might be new as well)

I tried other approaches, but did not get any good cancellations.

Per

This solution is composed out of 3-cyles.
Part 1 draws the horizontal lines and part 2 the dots inbetween.

3 Large Rings (Same Slice)
NU2 L2 F2 NU SF2 ND' B2 L2 NU2
R' MD' R' NF2 NB2 R MD R' NF2 NB2 R2 (20 btm)

It might be possible to fuse these two parts to get a shorter sequence.
I don't believe this is already an optimal solution.

@ Walter

3 Rings (f,l,r) (18 m)
Part 2 can be replaced with a 9-mover to gain 2 moves:
NU2 L2 F2 NU SF2 ND' B2 L2 NU2
WF [MR' F MR', WF2] WF'
3 Rings (f,l,r) (18 m)

Here are a few additional (unoptimized) algorithms of rings:

4 Rings (f,l)(r,b) (15 m)
NU' WR2 NU2 WF2 NU' WR2 WF2
MF WR MF' SR' MF WR' MF' SR
4 Rings (f,l)(r,b) (15 m)

4 Rings (f,b,l,r) (22 m)
2 Rings (f,b)
VR2 WU2 NR2 VD2 MR2 NU2
3 Rings (f,l,r)
NU2 L2 F2 NU SF2 ND' B2 L2 NU2 WF MR' F MR' WF2 MR F' MR WF
4 Rings (f,b,l,r) = (f,b)(f,l,r)
4 Rings (f,b,l,r) (22 m)

6 Rings (u,r,f,d,l,b) (13 m)
6 Rings (u,r,f)(d,l,b)
WR WF' WR' MR MF MR' MF' WF
2 Rings (d,u)
WF2 VR2 NF2 MR2 VB2 NR2
6 Rings (u,r,f,d,l,b) = (u,r,f)(d,l,b) (d,u) = (u,r,f) (d,l,b,u) = (u,r,f) (u,d,l,b) = (u,r,f,d,l,b)
6 Rings (u,r,f,d,l,b) (13 m)

Excellent work!
As you just proofed, it is indeed possible to build the '3 Rings (f,l,r)' with less then 20 moves.

I couldn't find any shorter solutions for your other Ring patterns.

Thank you for checking the algs.

Continuing further, by slightly modifying the 2 parts of Per's algorithm of the '6 Rings (u,d)(r,l)(f,b)' pattern, a gain of 2 moves can be obtained:

12 Centers
MR2 NF2 ND2 MR2 MF2 ND2 VF2

12 Bars
VF2 WU2 NB2 NR2 WU2 WF2 NR2

Algorithm (11 moves)
MR2 NF2 ND2 MR2 MF2 VU2 NB2 NR2 WU2 WF2 NR2

6 Rings (u,d)(r,l)(f,b) (11 m)

Slight improvement:

4 Rings (f,l)(r,b)(14m)

Per

This one can also be slightly improved.
6 Rings (u,d)(f,b)(r,l)(10m).

Start with
WF WU2 WF- WU2 and add
WU2 VL2 N2U2 MR2* VD2 N2L2 after that
Then insert MR2 MF MR2 MF- at the *.

I hope to get back to Walter's new (3 colour/side) psychedelic waves. The 4x4x4 looks hardest to improve

Per

One final improvement:

4 Rings (f,b,l,r) (21m).

We are already pushing the limits now...

Per

@ Per
Good work done on optimizing the algs!
It seems that algorithms of rings are indeed nearing the 20 moves border.

As I was'nt able to found much info on the group of the 720 permutations of blocks of centers, I wrote a small program to find all cycles. Providing there were no errors, 33 cycles were found to be sufficient, up to symmetry + inversion. The list may be useful for adding cycles that don't still exist on Walter's website:

(b, d)
(d, u)

(d, l, b)
(l, u, d)

(l, u) (b, d)
(l, r) (d, u)
(d, u) (b, l)
(u, r) (d, l)

(l, u, b, d)
(l, u, d, b)
(u, r, d, l)
(u, r, l, d)

(u, r) (d, l, b)
(l, r) (d, u, b)
(b, d) (u, r, l)

(u, r, b, d, l)
(u, r, b, l, d)
(u, r, l, b, d)

(r, f) (l, u) (b, d)
(r, f) (d, u) (b, l)
(b, f) (l, r) (d, u)

(r, f) (l, u, b, d)
(r, f) (l, u, d, b)
(d, u) (r, f, b, l)
(d, u) (r, f, l, b)

(r, f, u) (d, l, b)
(r, f, u) (b, l, d)
(r, f, l) (d, u, b)

(r, f, b, d, l, u)
(r, f, b, l, d, u)
(r, f, b, d, u, l)
(r, f, d, b, l, u)
(r, f, d, l, b, u)

By pairing up 2 orbits of centers, distinct cycles can be obtained to give simple dual color patterns, such as:
Dual Color Rings (r,f,u)(d,l,b) | (d,f,r)(l,u,b)

One more easy pattern:

Dot-plus-zero.

Quite easy construction. I will ezplain if needed

Per

A minor improvement.

4 Rings (f,u,b,d) (20 m).

Per

I improved it, based on previous improvement:

5 Rings (u,b,d,f,r) (37 m).
Surely better can be found - but probably not by me

Per

The algs have been slightly improved (but not yet optimized).
Except the (5c), they are all less than 20 moves, so much work remains to be done on five-cycles!

Dot-Plus-Zero (11 m)
]MU' ND2 WF2, MR2] [MR', MU2]
Dot-Plus-Zero (11 m)

4 Rings (f,d,b,u) (19 m)
8 Bars:
]NR WU2 NR2, WF2]
8 Centers:
NU' MR2 NU MR' NU MR NU' MR NU' MR NU
4 Rings (f,d,b,u) (19 m)

5 Rings (l,b,u,f,d) (31 m)
(l,b,u,f,d) = (l,b,u,r,f,d)(f,r)
(l,b,u,r,f,d):
WR WF' WR' MR MF MR' MF' WF'
VR2 NF2 MR2 VB2 NR2
(f,r):
MU WR' D WR MU' WR' D' WR
ND' WR' D2 WR NU ND WR' D2 WR NU'
5 Rings (l,b,u,f,d) (31 m)

5 Rings (u,r,f,d,l) is essentially the same pattern as Glazik's last contribution. Saves 1 turn!

1. Same as Glazik start
2. Swap u and b rings

5 Rings (u,r,f,d,l) (30 m).

Per

Edit:

Another essentially similar 5-cycle of rings:

5 Rings (f,l,b,d,r) (25 m).
I can probably delay the 6 spots and cancel even more turns. Quite a great optimization from the initial 39 turns, due to better breakdown of building blocks!!

Edit 2: (I moved the 6-spots and saved yet more 2 turns)

5 Rings (f,l,b,d,r) (23 m).

I believe this one is now optimal!

Great optimization work!

Here are additional (unoptimized) algorithms of 5-cycles, slightly improved compared to those of Walter's website:

2 Rings (f,r) (18 m) (Alternate alg)
[MU, WR' D WR]
ND' WR' D2 WR NU ND WR' D2 WR NU'
2 Rings (f,r) (18 m) (Alternate alg)

5 Rings (f,b)(u,r,l) (25 m)
(f,b)(r,l)
WF2 WU WF2 WU'
MU MF2 MU' MF2
(r,u)
[MF', WR' F WR]
NF' WR' F2 WR NB NF WR' F2 WR NB'
(f,b)(u,r,l) = (f,b)(r,l,u) = (f,b)(r,l)(r,u)
WF2 WU WF2 WU'
MU MF2 MU' MF2
[MF', WR' F WR]
NF' WR' F2 WR NB NF WR' F2 WR NB'
5 Rings (f,b)(u,r,l) (25 m)

5 Rings (l,r,b,f,u) (28 m)
(l,r)(b,f)
MR2 MU' MR2 MU
WF2 WU WF2 WU'
(b,u,l)
WU [WU, MF' U MF] WU'
[WU, NR B2 NL' NR' B2 NL]
(l,r,b,f,u) = (l,r)(l,b,f,u) = (l,r)(b,f,u,l) = (l,r)(b,f)(b,u,l)
MR2 MU' MR2 MU
WF2 WU WF2 WU'
WU [WU, MF' U MF] WU'
[WU, NR B2 NL' NR' B2 NL]
5 Rings (l,r,b,f,u) (28 m)

5 Rings (u,r)(d,b,l) (25 m)
(u,f)
[MR', WU' R WU]
NR' WU' R2 WU NL NR WU' R2 WU NL'
(u,f,r)(d,b,l)
[ML', MF']
[WF', WR]
(u,r)(d,b,l) = (u,f,f,r)(d,b,l) = (u,f)(u,f,r)(d,b,l)
[MR', WU' R WU]
NR' WU' R2 WU NL NR WU' R2 WU NL'
[ML', MF']
[WF', WR]
5 Rings (u,r)(d,b,l) (25 m)

And finally, to end this long list of long 5-cycles, here is a simpler 'exotic' pattern on a 4x4x4 cube:
6 Small Dual Diagonals (f,l,b,u,r,d) (8 m)
[WF2, ML' MR' MU'[ + shift:
MR MU WF2 ML' MR' MU' WF2 ML
6 Small Dual Diagonals (f,l,b,u,r,d) (8 m)

6 Rings (f,r)(b.l)(u,d) (18m).

Construction:

1. ND- WF2 (*)ND NU WF2 NU-
2. WF2 VL2 N2F2 MR2 VB2 N2L2 (insert at *)

And finally do the remaining centers:
3. MR WF- MR- SF MR WF MR- SF-

Note that shorter solution for even sized cubes is possible based on my posted 6 spots 8-mover

Per

Edit:

6 Rings 6x6x6 (f,r)(b,l)(u,d) (15m).

(As already stated above)

Another 4-ring pattern:
4 Rings (u,d)(r,f) (23 m)
(u,d)(r,l)
MR2 MF MR2 MF'
WF' WR2 WF WR2

(r,l,f)
NU2 L2 F2 NU SF2 ND' B2 L2 NU2
WF MR' F MR' WF2 MR F' MR WF

(u,d)(r,f) = (u,d)(r,l,l,f) = (u,d)(r,l)(r,l,f)
NU2 L2 F2 NU SF2 ND' B2 L2 NU2
WF MR' F MR' WF2 MR F' MR MR2 MF MR2 MF' WF
WF' WR2 WF WR2

4 Rings (u,d)(r,f) (23 m)

An attempt to solve the 4x4x4 Waves pattern (unoptimized alg), with a reduction of 23 moves!
4x4x4 Waves (35 m)
Basic Edge 4(3c)
[U' ML2 U D ML2 D', SL]

Edges 8(3c)
WF [U' ML2 U D ML2 D', SL] [SL, U' MR2 U D MR2 D'] WF'
WF U' ML2 U D ML2 D' SL D ML2 D' U' ML2 MR2 U D MR2 D' SL' D MR2 D' U' MR2 U WF'

Centers
(WF WR') (ML MD MR' MD') (MU' ML' MU MR) (WR WF')

4x4x4 Waves (35 m)

@ perfredlund and galzik
Wow, you did a lot of awesome contributions!


Here is a new approach for the 6x6x6 Wave which saves 31 moves:

6x6x6 Waves:
D B D2 L B2 D L2 D2 B2 U B' WR' B' WR U'
WF (U' MR2 U D MR2 D' SL D MR2 U' D' VR2 NL2 U D NR2 NL2 D' SL' D NR2 NL2 U' D' NR2 NL2 U) WF'
VB' L2 VF VB L2 VF'
WD VB' L2 VF VB L2 VF' WD'
SD2 SR
(NR NL' NF' NB NR' NL NF NB')
(MF MU' MF' MB' MU MD MB MD')
(MD NF NB' MD' NF' NB) (NU' ND MF NU ND' MF')
SR' SD2 (90 btm or 83 btm actually)

I tried to improve the 4x4x4 and 6x6x6 waves by using co-transportation, but i was unsuccessful. I did not get the edges into the correct configuration without long combinations.

So, instead i now post waves on the 5x5x5 cube. But due to centre restrictions, the waves must be reflexive - unlike on even-sized cubes.

5x5x5 reflexive waves.

1. CU- WR2 U- WF2 U2 WR2 D- F- WU2 F2 WR- WF- WR- F- (first step edges)
2. CU- MR2 TU- MF2 TU2 MR2 TD- TF- MU2 TF2 MR- MF- MR- TF- (second step edges)
3. SF- NL NU NL- NU- NR- ND- NR ND SF (centres)

Unoptimized ...

Per

I got in the mood for some serious cube-in-cube stuff.

Cube-In-Cube-Waves (44 m).

3 obvious parts. No slice turns for this one

Per

3 swapped pairs of Rings (35 m).

This was the worst case scenario of permuted rings on 5x5x5 I could think of. My solution is not elegant, and I just reuse old stuff already posted. Please improve me

Per

Here is an alternate way of composing already existing permutations, not a great gain though (1 move only).
Shorter solutions do not seem to be quite obvious.

(f,d,l,b,u,r):
WR WF' WR' MR MF MR' MF' WF'
VR2 NF2 MR2 VB2 NR2

(l,f,u):
WD2 MF' U' MF WD' MF' U MF
NR' F2 NL NR F2 NL' WD' NR' F2 NL NR F2 NL'

(f,d)(l,b)(u,r) = (f,d,l,b,u,r) (l,f,u):
WR WF' WR' MR MF MR' MF' WF'
VR2 NF2 MR2 VB2 NR2
WD2 MF' U' MF WD' MF' U MF
NR' F2 NL NR F2 NL' WD' NR' F2 NL NR F2 NL'

6 Rings (f,d)(l,b)(u,r) (34 m)

This solution isn't a breakthrough either, I just canceled out another move:

6 Rings (u,l) (r,b) (f,d)
WF' WR' WF MF' MR MF MR' WR'
VB2 NR2 MF2 VL2 NB2
NB D2 NF' NB' D2 NF WR NB D2 NF' NB' D2 NF WR2
MD L MD' WR MD L' MD' (34 btm or 33 btm actually)

Those images below inspired me to create Pac-Man Monsters on a 7x7x7:

4 Pac-Man Monsters with small eyes:
TR2 S2F2 TL2 D TR2 S2F2 TL2 U
R2 SF2 L2 ND R2 SF2 L2 NU
T3R2 NF2 MF2 NB2 T3R2 D T3L2 NF2 MF2 NB2 T3L2 D'
NF2 MR2 NF2 V3R2 VF2 M2U2 VF2 VR2 M2U2
N3F N3L N3B N3R MD N3R' N3B' N3L' N3F' MD' (47 btm)

4 Pac-Man Monsters with large eyes:
TR2 S2F2 TL2 D TR2 S2F2 TL2 U
R2 SF2 L2 ND R2 SF2 L2 NU
T3R2 NF2 MF2 NB2 T3R2 D T3L2 NF2 MF2 NB2 T3L2 D'
NF2 MR2 NF2 V3R2 VF2 M3D2 VF2 VR2 M3D2
N3F N3L N3B N3R MD N3R' N3B' N3L' N3F' MD' (47 btm)

Here is a way of cancelling a few moves (still improvable):

1- Replace the first 16-mover with this 14-mover
TU R2 SF2 L2 ND R2 SF2 NL2 S2F2 TR2 D TL2 S2F2 TR2

2- Insert (V3F2 VR2 M2U2 VR2 VF2 M2U2 NR2 MF2 NR2) at *
TU * R2 SF2 L2 ND R2 SF2 NL2 S2F2 TR2 D TL2 S2F2 TR2
T3R2 NF2 MF2 NB2 T3R2 D T3L2 NF2 MF2 NB2 T3L2 D'
N3F N3L N3B N3R MD N3R' N3B' N3L' N3F' MD'

4 Pac-Man Monsters with small eyes (43 m)

The following (unoptimized) algorithms are new or have been slightly improved:

4x4x4 Waves (31 m)

6 Rings (f,l)(r,d)(b,u) (31 m)

4 Rings (f,r)(b,u) (30 m)

4 Rings (u,d)(f,r) (22 m)

4 Rings (f,l,b,r) (18 m)

4 Rings (f,b,u,r) (25 m)

5 Rings (u,d)(l,r,f) (22 m)

5 Rings (u,f)(d,l,b) (24 m)

5 Rings (u,d,r,l,f) (25 m)

5 Rings (u,f,d,b,l) (22 m)

6 Rings (u,d)(f,r,b,l) (19 m)

6 Rings (u,d)(f,b,l,r) (25 m)

6 Rings (d,f)(u,l,b,r) (27 m)

6 Rings (d,f)(u,b,r,l) (23 m)

6 Rings (b,f,l,d,r,u) (24 m)

I guess there are still 8 ring algorithms to be found to complete the group of 33 permutations:
(l, u, b, d)
(b, d) (u, r, l)
(u, r, b, l, d)
(r, f, u) (b, l, d)
(r, f, l) (d, u, b)
(r, f, b, l, d, u)
(r, f, b, d, u, l)
(r, f, d, b, l, u)

@ Glazik:

Now i know how you spent your Easter vacation

It's not all that interesting anymore, but we have to complete the 5x5x5 rings variations
When I get the time ...

Per

Let's make an overview of the already known Ring patterns and let's see what's missing. I am just posting the shortest algorithms and exclude invers versions:

2 Rings

2 Rings (u,d)
NR2 WF2 VR2 NF2 MR2 VB2 (6 btm)

2 Rings (r,f)
MD' WR' D WR MD WR' D' WR
ND' WR' D2 WR NU ND WR' D2 WR NU' (18 btm)


3 Rings

3 Rings (u,r,f)
WD2 MF U MF' WD MF U' MF'
NL F2 NR' NL' F2 NR WD NL F2 NR' NL' F2 NR (21 btm)

3 Rings (r,f,l)
NU2 L2 F2 NU SF2 ND' B2 L2 NU2
WF MR' F MR' WF2 MR F' MR WF (18 btm)


I guess the groups of 2 and 3 Rings can be regarded as complete.

@ Walter and Per

Here is a compiled (yet unchecked) list of the 33 cycles for which algorithms are either already known or unknown:

(d, u) --- (6 moves)

(d, l, b) --- (21 moves)
(l, u, d) --- (18 moves)

(l, u) (b, d) --- (30 moves)
(l, r) (d, u) --- (8 moves)
(d, u) (b, l) --- (22 moves)
(u, r) (d, l) --- (14 moves)

(l, u, b, d) ------------------ (UNKNOWN)
(l, u, d, b) --- (25 moves)
(u, r, d, l) --- (18 moves)
(u, r, l, d) --- (21 moves)

(u, r) (d, l, b) --- (24 moves)
(l, r) (d, u, b) --- (22 moves)
(b, d) (u, r, l) ------------------ (UNKNOWN)

(u, r, b, d, l) --- (22 moves)
(u, r, b, l, d) ------------------ (UNKNOWN)
(u, r, l, b, d) --- (25 moves)

(r, f) (l, u) (b, d) --- (31 moves)
(r, f) (d, u) (b, l) --- (18 moves)
(b, f) (l, r) (d, u) --- (10 moves)

(r, f) (l, u, b, d) --- (27 moves)
(r, f) (l, u, d, b) --- (23 moves)
(d, u) (r, f, b, l) --- (25 moves)
(d, u) (r, f, l, b) --- (19 moves)

(r, f, u) (d, l, b) --- (8 moves)
(r, f, u) (b, l, d) ------------------ (UNKNOWN)
(r, f, l) (d, u, b) ------------------ (UNKNOWN)

(r, f, b, d, l, u) --- (24 moves)
(r, f, b, l, d, u) ------------------ (UNKNOWN)
(r, f, b, d, u, l) ------------------ (UNKNOWN)
(r, f, d, b, l, u) ------------------ (UNKNOWN)
(r, f, d, l, b, u) --- (13 moves)

It happens that (yet unknown) cycle (l, u, b, d) could be used to get a few other unknown cycles, simply by composing cycles.


As many patterns are built from double-line sub-patterns, I searched for a few new/shorter/alternate algorithms of double-lines and found these ones, that may possibly be useful:

(f, r)
(10 moves)
ND' WR' D2 WR NU ND WR' D2 WR NU'
ND' WR' D2 WR VU VD WR' D2 WR NU'
VD' WR' D2 WR NU ND WR' D2 WR VU'
VD' WR' D2 WR VU VD WR' D2 WR VU'

(d, f, u, b)
(8 moves)
NR WU WF2 WU NR' WU' WF2 WU'

(d, u) (l, r) (b, f)
(7 moves)
VU2 WR2 NU2 WF WR2 MU2 WF'
WU2 MR2 WF VR2 WU2 NR2 WF'
VR2 WU2 NR2 WF MR2 WU2 WF'
VL2 WF2 NR2 NU2 WR2 WF2 NU2

(b, d) (l, r) (u, f)
(8 moves)
NR WU2 NL' NR' NF2 WU2 NF2 NL

(b, f) (d, l, u, r)
(10 moves)
NR WU2 NL' NR' WU2 NR WU NL NR' WU'

(u, f, l, d, b, r)
(8 moves)
NL' NR' NF2 WU' NF2 NL NR WU

@ Glazik:

Let us not forget this simple (f,r)(b,l) stripes version.

Per

It seems that the group of 4 Rings has more then just 8 members:

4 Rings

4 Rings (r,l) (f,b) -> (l,r) (d,u)
WD WR2 WD' WR2
MR2 MD MR2 MD' (8 btm)

4 Rings (r,b) (f,l) -> (u,r) (d,l)
ND' WR2 NU ND WR2 NU'
MF WR MF' SR' MF WR' MF' SR (14 btm)

4 Rings (u,d) (r,f) -> (d,u) (b,l)
MD' WR' D WR MD WR' D' WR
ND' WR' TD2 VL2 MD2 NR2 VD2 WR' VU VD WR' D2 WR NU' (22 btm)

4 Rings (u,r) (f,d) -> (l,u) (b,d)
WF' R' WF NF WD' F2 NB' L2 NB WD NF' NB' WD' NF L2 F2 NF' WD NB MR' WF' R WF WD' MR U MR' WD MR U' (30 btm)

4 Rings (u,f) (r,d) -> new
WR F WR' NR' WD R2 NL B2 NL' WD' NR NL WD NR' B2 R2 NR WD' NL' MF WR F' WR' WD MF' U' MF WD' MF' U (30 btm)


4 Rings (r,f,l,b) -> (u,r,u,l)
NR' MD2 NR MD' NR MD NR' WR2 ND2 WF2 ND' WR2 ND2 WF2 VD NR' MD NR (18 btm)

4 Rings (r,f,b,l) -> (u,r,l,d)
VL2 MD2 NR2 VU2 T4L2
F2 NU SF2 ND' B2 L2 NU2
WF MR' F MR' WF2 MR F' MR WF (21 btm)

4 Rings (u,r,f,d) -> (l,u,d,b)
NF2 VL2 MF2 NR2 VF2 MD' R' MD WR MD' R MD NB' D2 NF NB D2 NF' WR NB' D2 NF NB D2 NF' (25 btm)

4 Rings (u,f,r,d) -> new
NR2 VB2 MR2 NF2 VR2 MD F MD' WF' MD F' MD' NL D2 NR' NL' D2 NR WF' NL D2 NR' NL' D2 NR (25 btm)

4 Rings (u,r,d,f) -> new
NF' WD B2 WD' · NF NB · WD B2 WD' NB'
WD B' WD' MF' WD B WD' MF
NF R2 · NF' NB' · R2 NB WD NF R2 · NF' NB' · R2 NB
MR' U' MR WD MR' U MR WD2 (39 btm)

4 Rings (u,f,d,r) -> (l,u,b,d)
NR WD' L2 WD · NR' NL' · WD' L2 WD NL
WD' L WD MR WD' L' WD MR'
NR' F2 · NR NL · F2 NL' WD' NR' F2 · NR NL · F2 NL'
MF U MF' WD' MF U' MF' WD2 (39 btm)

Maybe the group of 5 Rings has more than 6 variations as well:

5 Rings

5 Rings (u,r,b,l,f) -> new
WD WR WD' ND2 F2 L2 ND SR2 NU' R2 F2 ND2 WR'
MD' MF MD L MF WR2 MF' L' MF' WR2 (23 btm)

5 Rings (u,r,f,l,b) -> (u,r,b,d,l)
NR2 U2 F2 NR SF2 NL' B2 U2 NR2
WF MD' F MD' WF2 MD F' WD MF MD NF NB' WD' (22 btm)

5 Ringe (u,r,l,f,b) -> (u,r,l,b,d)
MR2 WD' WR2 WD MD' MR2 R MD WR' MD' R' MD NB U2 NF' NB' U2 NF WR' NB U2 NF' NB' U2 NF (25 btm)

5 Rings (u,r,b,l,d)
unknown


5 Rings (l,b) (u,r,f) -> (u,r) (d,l,b)
NL WD' R2 WD NR' NL' WD' R2 WD NR WD' R WD MR' WD' R' WD WR MF' MR WF' MF WR' WF (24 btm)

5 Rings (f,b) (u,r,l) -> (l,r) (d,u,b)
NF2 R2 U2 NF SD2 NB' D2 R2 NF2
WD' MR U MR NB2 VD2 MF2 NU2 VB2 MR' U' MR' WD' (22 btm)

5 Rings (u,b) (r,f,l) -> (b,d) (u,r,l)
NU2 L2 F2 NU SF2 ND' B2 L2 NU2
WF MR' F MR' WF2 MR F' MR WF
MR' WF R WF' MR WF R' WF'
NR' WF R2 WF' NL NR WF R2 WF' NL' (36 btm)

There are 720 items in the group of permutations of face patterns, indexed by the 6 letters {f,r,u,l,d,b}, provided that face pattern twists are not taken into account, ie. are considered the same, which is the case for rings. From these 720 items, only 33 representatives should be considered, from which all remaining permutations can be computed by inversion and symmetry (48 cases).

So, (u,r)(f,d) and (u,f)(r,d) are strictly equivalent, because (u,f)(r,d) can be obtained from (u,r)(f,d) by rotation CF2 CU plus Point Reflection, or equivalently by applying transformation:
F -> R'
R -> F'
U -> U'
L -> B'
D -> D'
B -> L'

I think the challenge would be to find algorithms shorter than say 30 moves for the missing representatives.

Hi!

I have been working one of the missing 6 rings patterns for a few days now. It is the case 6 rings (f,r,l)(b,u.d)(34 m).

My solution is just slightly shorter than the trivial 36 move solution (3 rings done twice). I have not found any way to cancel out turns by that trivial way. So instead i found a new way to do only the missing centers after doing 6 stripes.

The sequence i found was: B SU MF- L MF NR- NL MU NR NL- MU- MF- L- MF SU- B-. Which is [NR- NL,MU] conjugated with B SU MF- L MF.

I am sure further optimisations can be found

Per

Another missing ring variation.

6 Rings (u,f,r)(d,l,b) (24 m).

Per

(Edited because found shorter sequence ... )

And a minor improvement on an already found ring variation:

4 Rings (u,r)(f,d) (29m).

Per

This list of 6 Ring patterns already contains Per's recent algorithms:

6 Rings

6 Rings (u,d) (r,l) (f,b) -> (b,f) (l,r) (d,u)
WF WD2 WF' VL2 NU2 MF MR2 MF' VD2 NL2 (10 btm)

6 Rings (u,d) (r,b) (f,l) -> (r,f) (d,u) (b,l)
ND' VB2 NL2 MF2 VR2 NB2 NU ND WR2 NU'
MF WR MF' SR' MF WR' MF' SR (18 btm)

6 Rings (u,b) (r,d) (f,l) -> (r,f) (l,u) (b,d)
WF' VL B2 NR' NL' B2 NL WD NR B2 NR' NL' B2 NL WD2 MF' U MF WD MF' U' WF MR MF MR' WR' VB2 NR2 MF2 VL2 NB2 (31 btm)


6 Rings (u,r,f) (d,l,b) -> (r, f, u) (d, l, b)
WR WF' WR' NF NB' MR MF MR' (8 btm)

6 Rings (u,r,f) (d,b,l) -> (r,f,u) (b,l,d)
NF' U2 NF2 R2 NF' WD WF WD' VB R2 NF2 U2 NF
MD' MR' MD MR2 U MR' WD' MR U' MR' WD (24 btm)

6 Rings (u,b,d) (r,f,l) -> (r,f,l) (d,u,b)
B SD' MF' L MF MD' NR' NL MD NR NL' MF' L' MF SD B'
NU2 L2 F2 NU SF2 ND' B2 L2 NU2
NR2 D2 B2 NR SF2 NL' F2 D2 NR2 (34 btm)


6 Rings (u,d) (r,f,l,b) -> (d,u) (r,f,l,b)
NU' WR2 NU ND WR2 ND'
MF WR' MF' SR MF WR' MR2 WF' WR2 WF MF' MR2 SR' (19 btm)

6 Rings (u,d) (f,b,l,r) -> (d,u) (r,f,b,l)
VL2 MD2 NR2 VU2 T4L2 F2 NU SF2 ND' B2 L2 NU2 WF MR' F MR' NU2 VF2 MD2 NB2 VU2 MR F' MR WF (25 btm)

6 Rings (l,b) (u,r,d,f) -> (r,f) (l,u,b,d)
WF2 WD WR WD' MD MR' MD' WR' F MR WF' MR' F' MR NU L2 NU' ND' L2 ND WF' NU L2 NU' ND' L2 ND (27 btm)

6 Rings (l,b) (u,d,r,f) -> (r,f) (l,u,d,b)
NR2 D2 B2 NR SF2 NL' F2 D2 NR2
WF' MD B MD WF2 MD' B' MD2 WD MF' MD NB NF' WD' (23 btm)


6 Rings (u,b,r,d,f,l) -> (r,f,d,l,b,u)
WF' WR' WF MF' MR MF MR' WR'
VB2 NR2 MF2 VL2 NB2 (13 btm)

6 Rings (u,r,l,b,d,f) -> (r,f,b,d,l,u)
WF WD WF' MF MD' MF' WD' WF' D WF MD WF' D' WF ND' WF' D2 WF NU ND WF' D2 WF NU' (24 btm)

6 Rings (r,f,b,l,d,u)
unknown

6 Rings (r,f,b,d,u,l)
unknown

6 Rings (r,f,d,b,l,u)
unknown

A minor improvement can be done on this 4 Ring pattern:

4 Rings (u,f,d,r)
NR WD' L2 WD NR' NL' WD' L2 WD NL
WD' L WD MR WD' L' WD VR' F2 NR NL F2 NL' WD' NR' F2 NR NL F2 NL'
MF U MF' WD' MF U' MF' WD2 (38 btm)

Here are additional new/improved/alternate algorithms of rings:

6 Rings (l,r,f) (d,u,b)
(l,r,f) (d,u,b) --- Double Lines (18 m)
NU2 L2 F2 NU SF2 ND' B2 L2 NU2 NR2 D2 B2 NR SB2 NL' F2 D2 NR2
(l,r,f) (d,u,b) --- Midge-Centers (15 m)
WF' MU' B MU' MR' B MR' WF2 MR B' MR MU B' MU WF'
6 Rings (l,r,f) (d,u,b) (33 m)


6 Rings (f,b,d,u,l,r)
(f,b)(d,u)(l,r)
WF' NR2 VU2 MF MR2 MF' ND2 VR2 WF WU2
(l,f,d)
WU2 MF D MF' WU MF D' MF'
NR F2 NL' NR' F2 NL WU NR F2 NL' NR' F2 NL
(f,b,d,u,l,r) = (f,b)(f,d,u,l,r) = (f,b)(d,u,l,r,f) = (f,b)(d,u)(d,l,r,f) = (f,b)(d,u)(l,r,f,d) = (f,b)(d,u)(l,r)(l,f,d)
6 Rings (f,b,d,u,l,r) (29 m)


4 Rings (u,r)(f,d) --- Alternate solution to Per's algorithm
[WU' WR2, NF R2 NF' NB' R2 NB]
MR MF' ND NU' MF2 U MF ND' NU MF' U' MF' MR'
4 Rings (u,r)(f,d) (29 m)


4 Rings (u,f,d,r)
(u,f)
MR' WU' R WU MR WU' R' WU NR' WU' R2 WU NL NR WU' R2 WU NL'
(u,d,r)
WR' MU' R' MU' WR2 MU R MU WR' NF2 U2 R2 NF' SR2 NB L2 U2 NF2
(u,f,d,r) = (u,f)(u,d,r)
4 Rings (u,f,d,r) (35 m)


4 Rings (b,d)(u,r,l) --- Alternate solution to Walter's algorithm
(b,d)
MR' WU R WU' MR WU R' WU'
NR' WU R2 WU' NL NR WU R2 WU' NL'
(u,r,l)
WU MR U MR WU2 MR' U' MR' WU
NF2 R2 U2 NF SU2 NB' D2 R2 NF2
(b,d)(u,r,l)
MR' WU R WU' MR WU R' WU'
WU MR U MR WU2 MR' U' MR' WU
NR' WU R2 WU' NL NR WU R2 WU' NL'
NF2 R2 U2 NF SU2 NB' D2 R2 NF2
4 Rings (b,d)(u,r,l) (34 m)

Shorter solutions should certainly be found, because these are based on composition of already existing algorithms. Anyway, this gives at least an upper bound.

By adding the following algorithms, I think all 33 representatives are now done...
These could of course be further optimized if algorithms of length lower than 30 moves is what would be searched for.

4 Rings (u,f,d,r)
(d,u)
NF2 MR2 VB2 NR2 WF2 VR2
(d,f)(u,r)
WD WR2 NB' R2 NF NB R2 NF' WR2 WD'
NB' R2 NF NB R2 TF' MD' MR' L WD' MR SD' MR' WD MR SD L' MD F
(u,f,d,r) = (u,f,d)(u,r) = (d,u,f)(u,r) = (d,u)(d,f)(u,r)
NF2 MR2 VB2 NR2 WF2 VR2
WU' WR2 NB' R2 NF NB R2 NF' WR2 WU
NB' R2 NF NB R2 TF' MU MR' L WU MR SD' MR' WU' MR SD L' MU' F
4 Rings (u,f,d,r) (33 m)

5 Rings (r,b,l,d,u)
(r,b,l)
ND2 L2 B2 ND SB2 NU' F2 L2 ND2
WF' MR' B MR' WF2 MR B' MR WF'
(r,d,u)
NF2 D2 R2 NF SR2 NB' L2 D2 NF2
WR MU' R MU' WR2 MU R' MU WR
(r,b,l,d,u) = (r,b,l)(r,d,u)
ND2 L2 B2 ND SB2 NU' F2 L2 ND2
WF' MR' B MR' WF2 MR B' MR WF'
NF2 D2 R2 NF SR2 NB' L2 D2 NF2
WR MU' R MU' WR2 MU R' MU WR
5 Rings (r,b,l,d,u) (36 m)

6 Rings (u,r,f,d,b,l)
(u,r,f)(d,b,l)
MU MR' MU' MR2 U MR' WU MR U' MR' WU'
NF' U2 NF2 R2 NF' WU' WF WU VB R2 NF2 U2 NF
(d,u)
NF2 MR2 VB2 NR2 WF2 VR2
(u,r,f,d,b,l) = (u,r,f)(u,d,b,l) = (u,r,f)(d,b,l,u) = (u,r,f)(d,b,l)(d,u)
MU MR' MU' MR2 U MR' WU MR U' MR' WU'
NF' U2 NF2 R2 NF' WU' WF WU VB R2 NF2 U2 NF
NF2 MR2 VB2 NR2 WF2 VR2
6 Rings (u,r,f,d,b,l) (29 m)

6 Rings (f,b,l,d,u,r)
(f,b)(r,l)(d,u)
WF WD2 WF' VL2 NU2 MF MR2 MF' VD2 NL2
(d,r)(l,f)
WR' WD2 NB' D2 NF NB D2 NF' WD2 WR NB' D2 NF NB D2 TF' MR MD' U WR MD SL' MD' WR' MD SL U' MR' F
(f,b,l,d,u,r) = (f,b)(r,l)(d,u)(d,r)(l,f)
WF WD2 WF' VL2 NU2 MF MR2 MF' VD2 NL2
WR' WD2 NB' D2 NF NB D2 NF' WD2 WR NB' D2 NF NB D2 TF' MR MD' U WR MD SL' MD' WR' MD SL U' MR' F
6 Rings (f,b,l,d,u,r) (39 m)

Hi!

I managed to improve a previous contribution. I base it on a 4 Rings sequence : (u,r)(f,d).

I do another 4 Rings : (f,b)(r,l) inserted in two parts.

The result is:

6 Rings (f,b,d)(r,l,u) (31 m).

Per