That's indeed an intersting solution!
So far I just found a minor improvement for part one:

Gift-wrapped Cube (Order 3)
D WR2 U2 WF2 U WR' B WD2 B2 SR2 B WR'
R U F' N3U2 N3D2 F U' F' U2 N3U2 N3D2 F N3U' N3D F' U2 F N3U N3D' R'
L' (D' B N3U2 N3D2 B' D B N3U2 N3D2
D2 B' N3U' N3D B D2 B' N3U2 N3D2
N3R' N3L N3U' N3D N3R N3L') L (58 btm)

I found another small improvement for the

Gift-wrapped Cube (Order 3).

I simply replaced the corner twisting start with

CU2 WU- B- WU2 F2 WR2 F- WU- L- WF2 L2 WU2 R-

Then 2 turns cancelled ...

Per

A few easy contributions, maybe too easy to deserve a place here ...

Orthogonal blocks.

Orthogonal diffusion.

Orthogonal wide stripes.

2X (Order 2). Notice that on a 3x3x3 this requires 6 turns ...

Per

That's a very clever and short solution for the 2X (Order 2) pattern!

Here is an optimized algorithm for the Orthogonal Wide Stripes:
TR2 TF2 S2R2 TF2 MR2 M2D2 T3L2 S2D2 (8 btm)

I was working quite some time on this pattern for the 4x4x4 cube.
My current approach requires 78 moves to build a nice Double Python:

Double Python
R U' (WR2 B' WR F2 B WR F2) U R'
SD2 (TD MB2 TD' TR2 MD' TR2 (MU TF2 MU' TB2)2) SD2
U (WR2 U2 MR2 TU2 MR2 MU2 ML2) (D (WR WF' WR' WF) D') U'
MF' D2 MF WD MF' D2 MF WD'
MB D2 MB' WD MB D2 MB' WD'
MU' MR' MU L' MU' MR MU
R F MD TF' R' MU R TF MD' F' R' MU' L (78 btm)


P.S. If someone likes to fill in the centers, here is an algorithm that just does the edges:

Double Python (Edges Only)
D MF' D F D' MF D F' U D2 ML' U L U' MR2 ML U L' U' MR2 SU' R' D' MR D R D' MR'
MF2 U2 MB' U2 MB L2 MF' L2 U2 MF' U2 MF' (40 btm)

I solved the second part in 39 turns.

Double python (Edges only).

I will maybe work on the complete pattern later

Per

I improved the start and also achieved some cancellation.

Double Python (Edges Only).

Next i will work on the centres ... tomorrow

Per

Hi Per,

Here is a solution to the initial pattern with 2 moves less
I haven't been able to find anything shorter up to now...

Edges 1: (2c)
(U2 R2) (MD2 B2 MU' B2 MU R2 MD' R2 B2 MD' B2 MD') (R2 U2)

Edges 2: 4(3c)
U [U R U' F L2 F', MR ML] U'

Complete alg (30 moves)
U2 R2 MD2 B2 MU' B2 MU R2 MD' R2 B2 MD' B2 MD' R' U' F L2 F' MR ML F L2 F' U R' U' ML' MR' U'

Double Python (Edges Only)

Exploiting some flexibility of even sized cubes:

Single Ring (120°).

No optimization attempted yet. I simply did centers then edges!

Per

And then there is also:

Nested Rings (120°).

Small cancellations achieved.

Per

I couldn't resist to optimizing the Single Ring pattern

1- Edges
F' [R M2D2 R', U'] F

2- Centers
[MF R2 MF' MB' R2 MB, M2D]

3- Optimization: insertion of the alg of edges into that of centers (21 moves)
MF R2 MF' R B M2R B' R' B M2R' TB' R2 MB M2D MB' R2 MB MF R2 MF' M2D'

Single Ring (120°)

Some improvements:

Single Ring (120°). Edit: I did not see the Glazik's improvement !!

Better center start (2 swaps).

Nested rings (120°).

Further improvements are surely possible to find.

Per

Rings (Order 2). Again possible only on even sized cubes. Highly non-optimized. More a porof of concept.

Come on Glazik bring the metric down

Per

These algorithms are further improving the posted solutions above:

Single Ring (120°)
F2 R MD R2 MU' MD' R2 MU R' F2
R2 U MF R2 MF' MB' R2 MB U' R2 (20 btm)

Nested Rings (120°)
R F' R' (TL' U2 TL MU TL' U2 TL MU') R F R'
TR (TU TF' MD2 TF TU' TF' MD2 F U2 MF MD MF' U2 MF MD') TR' (31 btm)


@ glazik
Wow, great improvement for the 'Double Python (Edges Only)' pattern!

@ Per

Here is an attempt at optimizing your Rings pattern algorithm.
The algorithm order is 4 instead of 2, but it has been shortened to a mere 15 moves
Part 1
U2 B2 L M2B2 L2 M2B2 L B2 D2

Part 2
D2 ML2 B2 MR2 TB2 ML2 MB2 D2

Complete algorithm (15 moves)
U2 B2 L M2B2 L2 M2B2 L B2 ML2 B2 MR2 TB2 ML2 MB2 D2

Rings (Order 4)


By inverting and transforming Part 1, a wide loop pattern can be obtained:
Part 1
U2 F2 R M2F2 R2 M2F2 R F2 D2

Part 2
D2 ML2 B2 MR2 TB2 ML2 MB2 D2

Complete algorithm (15 moves)
U2 F2 R M2F2 R2 M2F2 R F2 ML2 B2 MR2 TB2 ML2 MB2 D2

Wide Loop (Order 4)

@ Walter

I tried a number of ways to lower the move count of the Single Ring and Nested Rings algorithms, but found none shorter than 20 and 30 moves, respectively. So, I guess that they are probably not that far from optimum, unless Per could find a breakthrough...

For centers of the Double Python pattern, I will simply wait for a solution from Per and try to optimize it afterwards

I improved my previous contribution by 5 turns. Now it's only 17 ...

Rings (Order 2).

I like the fact that the sequence only requires 180° turns

Per

Why are these order 4? If you apply the sequences twice the cube is solved. What am I missing??

Per

I found another way to do Rings (Order 2).

Not an improvement to the 15-mover already posted by Glazik, but it has only half turns.

1. R2 U2* F2 U2 B2 D2 B2 R2 M2U2

2. N2R2 T3B2 N2R2 T2B2 N2R2 N2F2 (a modified parity alg)

Insert 2. into 1 at *.

Per

Here is a possible explanation:

The order of Part 1 is 4, whereas that of Part 2 is 2, so the order of Part 1 + Part 2 is 4.

Executing the alg of Part 1 two times solves centers but not edges. Edges of Part 1 need to be executed 4 times to be fully solved.

This solution is not as 'clean' as yours, btw...

Well OK. But add (some) physical turns to your sequence and it becomes a sequence of order2. So essentially it has order 2 anyway

Per

The algorithm for the edges is already incredibly short, so that's an excellent starting point to beat the 78 moves for the entire 'Double Python'.


Here comes the order 2 version of the 'Wide Loop':

Wide Loop (Order 2)
L2 D2 MR2 F2 ML2 TF2 MR2 TF2 (D2 F2)2 L2 WD2 (14 btm)

The Rings (Order 2) actually looks nicer on the 6x6x6 cube. But then it takes 16 turns. Foe some strange reason the applet shows 17 block turns. This is a bizzarre bug. I inserted 7 turns into 9 turns. How can that become 17???

Edit:

The error lies within this sequence: N3R2 T2B2 N3R2 T3B2 N3L2 N3B2 M2R2 - counted as 8 turns in the applet. This requires some investigation

I found the culprit. T2B2 is counted as 2 block turns. T3B2 is correctly counted as 1. The error is the same on any size cube, from 3 and up to 7. Fix ASAP please !!!!

Per

Cube in Cube in Cube (Order2).

Cube in Cube (Order 2).

Not optimized yet!!

Per

Patched rings (Order 6).

The idea was nicer than the final outcome

Per

I already noticed this bug yesterday when I analyzed your 'Rings (Order 2)' algorithm. This algorithm is actually a 14-mover and not a 15-mover.

In the current applet version T2B2 is counted as 2 block moves instead of 1 move. This was caused due to an incorrect entry in the notation file and also because of a problem concerning repetitive moves in special cases.

This bug has been corrected right away. Since we had to lay hand on a very sensitive part of the parser, we like to run some more tests before releasing a bug fix. We will update the applets and CubeTwister as soon as possible.

The move T2B2 is actually an alternative tolken which is accepted along with the shorter TB2. Both tokens stand for the same move, but TB2 compared to T2B2 is counted correctly.

So, as a workaround you can use TB2 instead to get correct move counts.

Why do you need additional moves when you are scaling an algorithm to another cube model?

6x6x6 Rings (Order 2)
TR2 (TU2 (MR2 TF2 ML2 T3F2 MR2 T3F2) TU2 TB2 TD2 TB2) TR2 M2D2 (14 btm)

4x4x4 Rings (Order 2)
R2 (U2 (MR2 F2 ML2 TF2 MR2 TF2) U2 B2 D2 B2) R2 WD2 (14 btm)

I found two 22-mover to build 'Cube in a Cube in a Cube (Order 2)' patterns:

Cube in a Cube in a Cube (Order 2)
F SU L' B2 L SD F U SR2 U' L2 F2
TF TU TR' TU2 TR TU' TF TU TR2 TF2 (22 btm)

Cube in a Cube in a Cube (Order 2)
R WD' R' B2 R SD' F U' WR2 D R2 B2
TF TD TF' TR2 TB TU' TB TU TB2 TR2 (22 btm)


The 'Cube in a Cube (Order 2)' can be done in 20 moves:

Cube in a Cube (Order 2)
F2 L2 D F' TB2 R2 MB2 TR2 MB2 MR2 U' R B2 R' SD' F SR2 B' U B (20 btm)

Cube in a Cube (Order 2)
F2 L2 D F' TB2 R2 MB2 TR2 MB2 MR2 U' R B2 R' WD R WF2 R' D R (20 btm)

When scaling the original 6 mover became 7 turns. Not sure why. And the slight cancellation also vanished ... My mistake

Per

I further optimized the Cube in a Cube (Order 2).

Now it takes only 17 block turns:

R2 F2 U- L F- D F2 D- F L- U2 T2R2 U2 N2R2 T2U2 N2R2 N2U2

Or even better Cube in a Cube (Order 2).



Windowed Cube in a Cube (Order 6). Trivial variety ...

Per

Are the interactive applets down for maintenace? Not working here on either of my browsers. Restart does nothing

Per

Awesome, what a great improvement! This is probably an optimal or nearly optimal solution!

Sorry for the inconvenience during the update, the applets should work now.

So let's test them with some nice spiral patterns:

7x7x7 Triskelion
L D2 B D2 F' R' D R2 F R B2 D' R2 D2 R' B
(V4U N3R' N3U' N3L MD MR' N3D NL NU' V4L') F B L2 U2 (30 btm)

6x6x6 Triskelion
L D2 B D2 F' R' D R2 F R B2 D' R2 D2 R' B
(V3U MR' MU' ML MD NL NU' V3L') F B L2 U2 (28 btm)

5x5x5 Triskelion
L D2 B D2 F' R' D R2 F R B2 D' R2 D2 R' B
VU MR' MD NL NU' VL' F B L2 U2 (26 btm)

4x4x4 Triskelion
L D2 B D2 F' R' D R2 F R B2 D' R2 D2 R' B
(MU ML MU' ML') F B L2 U2 (24 btm)

3x3x3 Triskelion
L' B' U D R U' R' D2 R2 D L D' R' L' F U (16 btm)

2x2x2 Triskelion (or actually 'Cube in a Cube')
U F' R U F2 R2 U' F' R F' (10 btm)

I tried to optimize your 'Patched Rings' a little.

Patched Rings
R L B2 (U' D' MF2 SD' MB2 ML2 MB2 D2) B2 R' L'
ML' B MD2 B' TU' B MD2 B' TU ML
MR F' MU2 F TD F' MU2 F TD' MR' (34 btm, or 33 btm actually)

These are two more spirals for the 5x5x5. The algorithms are not optimized yet:

Interlaced Spirals
F' R2 D' SR F R L F D F SD' L' U'
U R' (NF2 R F R' VF2 R F' R' MF2) R U'
D' L (NB2 L' B' L VB2 L' B L MB2) L' D
D2 F2 R' L' (ND WF MU' ND' WF' MU) R L F2 D2
U2 F' B' L2 (NL ND' NL' ND) L2 B F U2 (65 btm, or 61 btm actually)

Interlaced Spirals With a Detour
WR WF' WR' NF NB' MR MF MR'
F' R2 D' SR F R L F D F SD' L' U'
TU' TF MR2 TF' R' TF MR2 TF' R TU
TD TB' MR2 TB L TB' MR2 TB L' TD'
SF L2 B2 (U' NR2 U SR U' NR2 U SR') B2 L2 SF'
U2 B2 R L (NB' NU' NB NU) R' L' B2 U2 (67 btm)

Nice bug fix and nice new contributions.

Anyway, i have an easy contribution myself:

Block 4-flip.

Per

This is an expanded Block 4-flip:

Kind of 12-flip.

Per

I took the idea from the block 4-flip further. First i modified the start and got

T3F2 N2R N2L- T3F2 N2R2 N2L2

Then i commuted that with D SR2 D- to get a "8-flip".

T3F2 N2R N2L- T3F2 N2R2 N2L2 D SR2 D- N2R2 N2L2 T3F2 N2R- N2L T3F2 D SR2 D-

Then conjugated with F- to get a sequence that could be repeated 3 times around the cube.

Super-flip on small edges.

I guess shorter is still possible ...

Actually it can be brute forced. Start with SL F R- B U- L2 U B- L- R2 U2 F2 U- and commute with N2R2 N2L2 N2F2 N2B2 N2U2 N2D2.

Super-flip on small edges.

I still find the first solution cuter and more "pure"

Per

I combined my latest pattern with the 6-sieves pattern.

8 Escher rings.

Per

@Per
Superflip
Here is an improvement to the Superflip pattern algorithm.

1- Basic form of 4 2-cycles of Edges, flipped on 2 opposed faces R/L:
NF' [L2 U2, NF NB] [NF NB, U2 L2] NF

This basic form can be easily extended to all opposed faces by inserting only 1 face move F2 to L2 U2, 4 slice moves NR NL NU ND to NF NB and 2 setup moves ND' NR' to NF', to give a 30-mover, as follows:

2- Edges 12 2-cycles, flipped on all opposed faces F/B, R/L, U/D
ND' NR' NF' [L2 U2 F2, NF NB NR NL NU ND][NF NB NR NL NU ND, F2 U2 L2] NF NR ND

Superflip on small edges

@Walter
CubeTwister
It seems that CubeTwister 2.0alpha143 doesn't work either on PC or iMAC: all I got is a blank window. Maybe you could check the version again ?

Here are further improvements of the Superflip pattern:

Superflip on Small Edges:
F D R D2 B U B L B2 D
NR2 NL2 NU2 ND2 NF2 NB2
D' B2 L' B' U' B' D2 R' D' F'
NR2 NL2 NU2 ND2 NF2 NB2 (32 btm)


glazik's algorithm offers even more possibilities for optimization:

Superflip on Small Edges:
ND' NR' NF'
(L2 U2 F2 NF NB NR NL NU ND F2 U2 L2
F2 U2 L2 NU' ND' NR' NL' NF' NB' L2 U2 F2)
NF NR ND (30 btm)



Hmm, yes, your right.
I think I know what the blank window causes. I will fix this as soon as possible.

Great superflip improvements. Here's a few new small flips:

4 small flips (version 1).

[R- F D- R F-,N2U2 N2D2]

4 small flips (version 2).

[R- F D- R F-,N2U N2D-]

For some reason i could not paste proper notation into the URL's ... Parser could not handle it!!

Per

Let me add more small flips.

F-B small flips. Easy commutator. Once again this site does not handle the simplified notation ... grrr

Per

Some previous algorithms have been slightly improved :

Patched Rings (30 moves)
R L B2 (U' D' MF2 SD' MB2 ML2 MB2 D2) B2 L' R'
TR [U, TF' MD2 TF] TR'
TR [F' MU2 F, TD] TR'
Patched Rings

4 Small Flips (Version 1) (11 moves)
NU SL U2 SL' NU ND SL U2 SL' ND' NU2
4 Small Flips (Version 1)

4 Small Flips (Version 2) (13 moves)
F2 ND SL U2 SL' NU ND SL U2 SL' ND2 NU' F2
4 Small Flips (Version 2)

By adding permuted centers to flipped edges, we got (yet unoptimized) deep flips:
4 Deep Flips (Version 1) (23 moves)
NU SL U2 SL' NU ND SL U2 SL' ND' NU2
B2 (L2 NU' NR2 NU TR2 L2 NU NR2 NU' TR2) B2
4 deep flips (version 1)

4 Deep Flips (Version 2) (24 moves)
F2 ND SL U2 SL' NU ND SL U2 SL' ND2 NU' F2
NF2 (L2 NU NR2 NU' TR2 L2 NU' NR2 NU TR2) NF2
4 deep flips (version 2)

2 Deep Flips (26 moves)
ND2 F2 NU' F2 NU L2 ND' L2 F2 ND' F2 ND'
NL NF' NL' F' NL NF D NL' NU NL D' NL' NU' F
2 deep flips

@ André (a.k.a. glazik)
The bug has been fixed in the current version of CubeTwister 2.0alpha144.



I guess the TwistyPuzzles.com editor doesn't like Left and Right Square Bracket characters within tags like 'URL'.

But it may work if you replace these characters with the corresponding URL Encodings. In this case %5B stands for the Left Square Bracket and %5D for the Right Square Bracket:

4 Small Flips (Version 1)
[R' F D' R F', NU2 ND2] (14 btm)

4 Small Flips (Version 2)
[R' F D' R F', NU ND'] (14 btm)

I honestly thought the 2 deep flips was well known from speedsolving bigger cubes. This sequence is only 12 block turns

Per

This commutator gives you easily both versions of 4 deep flips:

[T2R- N2D- T2R2 N2D2 T2R2 N2D T2R,T2U2]

Per

One more: 2 block flips.

Thanks to Glazik

Per

I got inspired to make a rings pattern. I started off with the following:

L D L D- U2 L D2 B2 D U2 L D L- R2 D L2 F2 R2

It twists all but 1 corner clockwise. I reversed it with double layer turns to achieve 8 rings.

1 of the rings needed adjustment to also be clockwise:

R F- R- T2L- U2 T2L N2U T2L- U2 T2L N2U- R F R-

8 small clockwise rings.

I guess must be near optimal (49 turns).

Per

I managed to squeeze out 2 more moves from your '8 Small Clockwise Rings'.
BTW, this pattern also looks nice on a 5x5x5:

8 Small Clockwise Rings
U R2 U2 L' (U L2)2 WF2 L' U' WR2 WF2 (D L')2
(TL TD')2 MF2 MR2 TU TL MF2 (TL2 TU')2 TL TU2 TR2 TU'
B L' B' (TF' D2 TF ND TF' D2 TF ND') B L B' (48 btm or 47 btm actually)