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 Post subject: Re: Patterns
PostPosted: Thu Nov 10, 2011 7:53 am 
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Nice contribution :)

Per

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 Post subject: Re: Patterns
PostPosted: Fri Dec 02, 2011 10:53 am 
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Location: Belgium (North)
pytlivyj_1 wrote:
As is known, on odd cubes to collect a pattern "a chess cube" it is rather easy from the assembled condition with the help of turns of external and internal layers on 180 degrees in axes x, y, z. And on any even cube (4х4х4, 6х6х6 and т. д.) the chess pattern is going to always without 4 corners:

Image

Somebody tried to collect such pattern on a cube 4х4х4? Actually it not so is simple, as it seems at first.

P. S. Who has cube 6х6х6, try from the assembled condition to collect on him a similar chess pattern without 4 corners. A task not from simple!


What's the algoritm of this patern?

Thanks


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 Post subject: Patterns
PostPosted: Sun Dec 04, 2011 1:29 pm 
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2 robin

* If the cuboid contains two even parties that a chess cuboid always will be without four corners!!!

Initial situation: Green - F; White - U.

Step 1. (l2 R2) (b2 F2) (l2 R2) (u2 D2),
Step 2. D (r2 U l2 U')*2 (l2 U r2 U')*2 D',
Step 3.
step 3.1. and step 3.2. - we carry out analogue of algorithm № 2;
step 3.3. and step 3.4. - we carry out mirror analogue of algorithm № 2.

Image
1. (l' U2)*2 F2 l' F2 r U2 r' U2 l2 = A
2. B' U R' U' (A) U R U' B
3. R' U R U' (A) U R' U' R
4. R B (A) B' R'

step 3.1.Green - F; White - U;
step 3.2. Dark blue - F; White - U;
step 3.3. Red - F; White - U;
step 3.4. Violet - F; White - U.


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 Post subject: Re: Patterns
PostPosted: Sun Dec 04, 2011 4:46 pm 
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Hi,

Simple algorithms of 4-face Chessboards can already be found by clicking the links below:

4x4x4 cube
http://www.randelshofer.ch/rubik/revenge/A840.01.html

6x6x6 cube
http://www.randelshofer.ch/rubik/vcube6/A840.01.html

Algorithms of 6-face chessboards are more complex though, and not widely available. Here are some details of possible solutions. Although shorter algorithms may exist, consider this is as an attempt to display the patterns and not necessarily to find the optimal solution.

We can start from already existing 4-face chessboards and add the missing edge + center algorithms.
Two two-cycles on opposed faces may give short algorithms. These are computed using an on-line orbit solver:

http://www.randelshofer.ch/rubik/apps/7x_super_cube_solver/index.html

Algorithm notation details can also be found at the same website:

http://www.randelshofer.ch/rubik/patterns/doc/supersetENG_3x3.html

Here are the results:

//
// Chessboard --- 4x4x4 Cube --- 4 faces --- 8 moves
U2 MD2 R2 ML2 F2 MB2 R2 ML2

// Edges --- Orbit 01 --- 18 moves
// (AC)(VX)(ET)(GR) --- 4(2c)
ML MR SU R2 SU' MR' ML' SU R2 F2 SU' MF' MB' SU F2 SU' MB MF

// Centers --- Orbit 03 --- 10 moves
// (FS)(HQ) --- 2(2c)
(MF2 D2 MR2 MF2 MR2)2

// New chessboard --- 4x4x4 Cube --- 6 faces --- 35 moves
U2 MD2 R2 ML2 F2 MB2 R2 ML' MR SU R2 SU' MR' ML' SU R2 F2 SU' MF' MB' SU F2 SU' MB MF MF2 D2 MR2 MF2 MR2 MF2 D2 MR2 MF2 MR2


//
// Chessboard --- 6x6x6 Cube --- 4 faces --- 12 moves
U2 MU2 ND2 R2 MR2 NL2 F2 MF2 NB2 R2 MR2 NL2

// Edges --- Orbit 01 --- 18 moves
// (AC)(VX)(ET)(GR) --- 4(2c)
N3L N3R SU R2 SU' N3R' N3L' SU R2 F2 SU' N3F' N3B' SU F2 SU' N3B N3F

// Edges --- Orbit 06 --- 18 moves
// (BD)(UW)(FQ)(HS) --- 4(2c)
NB NF SU F2 SU' NF' NB' SU F2 R2 SU' NR' NL' SU R2 SU' NL NR

// Centers --- Orbit 04 --- 10 moves
// (FS)(HQ) --- 2(2c)
(NF2 D2 N3R2 NF2 N3R2)2

// Centers --- Orbit 08 --- 10 moves
// (FS)(HQ) --- 2(2c)
(N3F2 D2 NR2 N3F2)2

// Centers --- Orbit 05 --- 10 moves
// (ER)(GT) --- 2(2c)
(NB2 U2 NL2 NB2 NL2)2

// Centers --- Orbit 07 --- 10 moves
// (ER)(GT) --- 2(2c)
(N3B2 U2 N3L2 N3B2 N3L2)2

// New chessboard --- 6x6x6 Cube --- 6 faces --- 88 moves
U2 MU2 ND2 R2 MR2 NL2 F2 MF2 NB2 R2 MR2 NL2
N3L N3R SU R2 SU' N3R' N3L' SU R2 F2 SU' N3F' N3B' SU F2 SU' N3B N3F
NB NF SU F2 SU' NF' NB' SU F2 R2 SU' NR' NL' SU R2 SU' NL NR
NF2 D2 N3R2 NF2 N3R2 NF2 D2 N3R2 NF2 N3R2
N3F2 D2 NR2 N3F2 NR2 N3F2 D2 NR2 N3F2 NR2
NB2 U2 NL2 NB2 NL2 NB2 U2 NL2 NB2 NL2
N3B2 U2 N3L2 N3B2 N3L2 N3B2 U2 N3L2 N3B2 N3L2


Attachments:
Chessboard_4x.jpg
Chessboard_4x.jpg [ 94.98 KiB | Viewed 6857 times ]
Chessboard_6x.jpg
Chessboard_6x.jpg [ 111.78 KiB | Viewed 6857 times ]
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 Post subject: Patterns
PostPosted: Mon Dec 05, 2011 11:58 am 
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Chessboard --- 6x6x6 Cube --- 6 faces --- 84 moves (my method):

Green - F; White - U.

1) (R2 3r2 l2) (F2 3f2 b2) (R2 3r2 l2) (D2 3d2 u2) --- 12 moves;
2) D (r2 3l2 U 3r2 l2 U')*2 (3r2 l2 U r2 3l2 U')*2 D' --- 26 moves;
3) [(2-3r' U2)*4 2-3r' z2 (2-3r' U2)*4 2-3r' y]*2 --- 36 moves;
4) 3-4r2 B2 3-4r2 3-4f2 L2 3-4f2 --- 6 moves;
5) (B2 L2)*2 --- 4 moves.

Image

* The decision is found without application of the computer!


Last edited by pytlivyj_1 on Fri Dec 09, 2011 7:41 am, edited 1 time in total.

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 Post subject: Re: Patterns
PostPosted: Mon Dec 05, 2011 4:45 pm 
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Location: The land of dreams, coincedentally located in Alberta
glazik wrote:
Hi,

Simple algorithms of 4-face Chessboards can already be found by clicking the links below:

4x4x4 cube
snip.png

6x6x6 cube
snip.png
Algorithms of 6-face chessboards are more complex though, and not widely available. Here are some details of possible solutions. Although shorter algorithms may exist, consider this is as an attempt to display the patterns and not necessarily to find the optimal solution.

We can start from already existing 4-face chessboards and add the missing edge + center algorithms.
Two two-cycles on opposed faces may give short algorithms. These are computed using an on-line orbit solver:

http://www.randelshofer.ch/rubik/apps/7x_super_cube_solver/index.html

Algorithm notation details can also be found at the same website:

http://www.randelshofer.ch/rubik/patterns/doc/supersetENG_3x3.html

Here are the results:

//snip.alg


Very, very nice! I could hardly see how you did it, but then I saw the trick: the blue/green corners are actually a bit off. I almost missed that! :)

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3x3x3 PB: 38.9 seconds
Well, I accumulated puzzles without even trying this Christmas. Whoops.
(Bermuda 8 planets, Rex Cube, Master Skewb, London Natural History Museum keychain 2x2x2, Impossiball)


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 Post subject: Re: Patterns
PostPosted: Wed Dec 07, 2011 3:06 pm 
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An easy contribution this time. A pattern on the shengshou crazy magic 2x2x2. Available from lightake for instance.

It emulates a single turn on a 2x2x2 gliding cube. But only from solved state. It is not possible to achieve a TRUE 8-cycle. I use Upper case for normal layer turns and lower case for "face disc" turns.

Then the pattern is simply:

R B D R u R' D' B' R' (9)

Picture later.

Per

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 Post subject: Re: Patterns
PostPosted: Thu Dec 08, 2011 6:40 am 
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pytlivyj_1 wrote:
Chessboard --- 6x6x6 Cube --- 6 faces --- 84 moves (my method):

Green - F; White - U.

1) (R2 3r2 l2) (F2 3f2 b2) (R2 3r2 l2) (D2 3d2 u2) --- 12 moves;
2) D (r2 3l2 U 3r2 l2 U')*2 (3r2 l2 U r2 3l2 U')*2 D' --- 26 moves;
3) [(2-3r' U2)*4 2-3r' z2 (2-3r' U2)*4 2-3r' y]*2 --- 36 moves;
4) 3-4r2 B2 3-4r2 3-4f2 L2 3-4f2 --- 6 moves;
5) (B2 L2)*2 --- 4 moves.

* The decision is found without application of the computer!


Computer applets are "allowed" for verification purposes. For instance turn cancellations between steps, and the final sequence !! Also it's nice to see a live demo :)

Per

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 Post subject: Re: Patterns
PostPosted: Fri Dec 09, 2011 7:17 am 
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pytlivyj_1 wrote:
Chessboard --- 6x6x6 Cube --- 6 faces --- 84 moves (my method):

Green - F; White - U.

1) (R2 3r2 l2) (F2 3f2 b2) (R2 3r2 l2) (D2 3d2 u2) --- 12 moves;
2) D (r2 3l2 U 3r2 l2 U')*2 (3r2 l2 U r2 3l2 U')*2 D' --- 26 moves;
3) [(2-3r' U2)*4 2-3r' z2 (2-3r' U2)*4 2-3r' y]*2 --- 36 moves;
4) 3-4r2 B2 3-4r2 3-4f2 L2 3-4f2 --- 6 moves;
5) (B2 L2)*2 --- 4 moves.

* The decision is found without application of the computer!


Good work!
Your 84-move solution is not that far from my (unoptimized) 88-move algorithm, btw ;-)

As was pointed out by Per, it is not so uncommon to use a solver in competition for checking solutions against computer generated sequences. But I think that all we are concerned with in this thread is how to try to find the shortest posible algorithm, by optimization, using all tools at our disposal, even including pencil and paper :-)

Here is a way of optimizing the already found algorithms. I note that other solutions may exist as well, but mine is based on optimization of 2-cycles.

1- Optimizing Centers

Noting that that all 4 algorithms show the same structure, we can try to merge them into a single sequence. From the algorithm of centers of orbit 05, we got a structure template:
(U2 NL2 NB2 NL2 NB2)2
Merging the 4 algorithms can now easily be done, based on the template, by inserting slice moves:
(U2 N3L2 NR2 N3B2 NF2 N3L2 NR2 N3B2 NF2)2
This gives a 18-move alg instead of the 40-move initial alg!

2- Optimizing Edges

The 2 algorithms are first separated into four 2 2-cycles. Cycles of similar structure are then grouped 2 by 2:
NB NF SU F2 SU' NF' NB' SU F2 SU'
N3F' N3B' SU F2 SU' N3B N3F SU F2 SU'

The two algs are now merged into a single alg:
NB NF N3F' N3B' SU F2 SU' NF' NB' N3B N3F SU F2 SU'

The same process is applied to the remaining 2-cycles:
SU R2 SU' NR' NL' SU R2 SU' NL NR
SU R2 SU' N3L N3R SU R2 SU' N3R' N3L'
SU R2 SU' NR' NL' N3L N3R SU R2 SU' NL NR N3R' N3L'

Combining the 2 algs gives:
NB NF N3F' N3B' SU F2 SU' NF' NB' N3B N3F SU F2 SU' SU R2 SU' NR' NL' N3L N3R SU R2 SU' NL NR N3R' N3L'

which gives a 26-move alg after move cancellation, to be compared to the initial 36-move alg:
NB NF N3F' N3B' SU F2 SU' NF' NB' N3B N3F SU F2 R2 SU' NR' NL' N3L N3R SU R2 SU' NL NR N3R' N3L'

So, the overall algorithm is now 56 moves long:
U2 MU2 ND2 R2 MR2 NL2 F2 MF2 NB2 R2 MR2 NL2
NB NF N3F' N3B' SU F2 SU' NF' NB' N3B N3F SU F2 R2 SU' NR' NL' N3L N3R SU R2 SU' NL NR N3R' N3L'
(U2 N3L2 NR2 N3B2 NF2 N3L2 NR2 N3B2 NF2)2

Any idea of further optimization?


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 Post subject: Patterns
PostPosted: Thu Dec 15, 2011 12:42 pm 
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2 glazik

Chessboard --- 6x6x6 Cube --- 6 faces --- 46 moves (!!!):

Green - F; White - U.

(l2 3r2 R2) (b2 3f2 F2) (l2 3r2 R2) (d2 3u2) – 11 moves;
(3l2 r2 3b2 f2)*2 U2 (3l2 r2 3b2 f2 3l2 r2) - 15 moves;
2-5f' 2-5u' R2 2-5u (2-4f' 3f2 b') 2-5u' R2 B2 2-5u (2-4r' 3r2 l') 2-5u' B2 2-5u (2-4r 3r2 l) – 20 moves.


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 Post subject: Re: Patterns
PostPosted: Fri Dec 16, 2011 3:34 pm 
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To pylivyj_1:
You 46-move alg is quite short indeed!

This 47-move alg is not too far off:
(MU2 ND2) (R2 NL2 MR2) (F2 MF2 NB2) R2
(MF2 NB2) (NL2 MR2 MF2 NB2) U2
(NR2 ML2 NF2 MB2) (NR2 ML2 NF2)
MB' MF SU F2 SU' VB' VF' SU F2 SU' NB NF
NL NR SU R2 SU' VL' VR' SU R2 SU' MR ML

4 chessboards + centers:
(MU2 ND2) (R2 NL2 MR2) (F2 MF2 NB2) R2
(MF2 NB2) (NL2 MR2 MF2 NB2) U2
(NR2 ML2 NF2 MB2) (NR2 ML2 NF2) MB2

16 permuted edges:
MB MF SU F2 SU' VB' VF' SU F2 SU' NB NF
NL NR SU R2 SU' VL' VR' SU R2 SU' MR ML

The (4 chessboards + centers) alg is indeed short: only 24 moves, but I think the alg of the 16 permuted edges may still be too long...


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 Post subject: Re: Patterns
PostPosted: Fri Dec 16, 2011 4:04 pm 
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glazik wrote:
To pylivyj_1:
You 46-move alg is quite short indeed!

This 47-move alg is not too far off:
(MU2 ND2) (R2 NL2 MR2) (F2 MF2 NB2) R2
(MF2 NB2) (NL2 MR2 MF2 NB2) U2
(NR2 ML2 NF2 MB2) (NR2 ML2 NF2)
MB' MF SU F2 SU' VB' VF' SU F2 SU' NB NF
NL NR SU R2 SU' VL' VR' SU R2 SU' MR ML


Is this really the pattern?? It's not a checkerboard pattern ...

Per

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 Post subject: Re: Patterns
PostPosted: Sat Dec 17, 2011 12:48 am 
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perfredlund wrote:
glazik wrote:
To pylivyj_1:
You 46-move alg is quite short indeed!

This 47-move alg is not too far off:
(MU2 ND2) (R2 NL2 MR2) (F2 MF2 NB2) R2
(MF2 NB2) (NL2 MR2 MF2 NB2) U2
(NR2 ML2 NF2 MB2) (NR2 ML2 NF2)
MB' MF SU F2 SU' VB' VF' SU F2 SU' NB NF
NL NR SU R2 SU' VL' VR' SU R2 SU' MR ML


Is this really the pattern?? It's not a checkerboard pattern ...

Per

It is actually;-)
You just have to replace NL- with NL in your expression for the algorithm.

Here is the corrected link:
http://www.randelshofer.ch/cube/vcube6/ ... UR2SU-MRML

Btw, it's certainly a good idea to give links to animated cubes...


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 Post subject: Re: Patterns
PostPosted: Sun Dec 18, 2011 1:09 pm 
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glazik wrote:
perfredlund wrote:
glazik wrote:
To pylivyj_1:
You 46-move alg is quite short indeed!

This 47-move alg is not too far off:
(MU2 ND2) (R2 NL2 MR2) (F2 MF2 NB2) R2
(MF2 NB2) (NL2 MR2 MF2 NB2) U2
(NR2 ML2 NF2 MB2) (NR2 ML2 NF2)
MB' MF SU F2 SU' VB' VF' SU F2 SU' NB NF
NL NR SU R2 SU' VL' VR' SU R2 SU' MR ML


Is this really the pattern?? It's not a checkerboard pattern ...

Per

It is actually;-)
You just have to replace NL- with NL in your expression for the algorithm.

Here is the corrected link:
http://www.randelshofer.ch/cube/vcube6/ ... UR2SU-MRML

Btw, it's certainly a good idea to give links to animated cubes...


So i messed about with this pattern for quite a while. I managed a slight improvement.
See it in action here.
The first 14 turns were easy. The last 8 edges (31 turns) were quite painful ... :oops:

Per

Breakdown of last 31 turns:

Setup edges: U- D- N2R N4R D U
First block 3-cycle: T3F U- VR- U L2 U- VR U L2 T3F-
Second block 3-cycle: T3F- D2 L- VD L D2 L- VD- L T3F
Undo setup: U- D- N3L N5L D

PS!
One can replace T3F with VF and T3F- with VF- to turn optimise it slightly. But i do think the T3F turns look nicer :D

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 Post subject: Re: Patterns
PostPosted: Mon Dec 19, 2011 7:09 am 
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I improved yet again the 6x6x6 checkerboard 6 faces pattern. I improved the rather inefficient last 31 turns like so:

setup edges: R VF VB M2F2 R- M2F2 VB- VF- L (9)
permute edges: VU2 L2 VU2 T3L2 VU2 VL2 (6)
undo setup: L- VF VB M2F2 R M2F2 VB- VF- R- (9)

The final result can be viewed here, and has a total of 38 block turns !! :mrgreen:

Very good brain exercise :lol:

Per

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 Post subject: Re: Patterns
PostPosted: Mon Dec 19, 2011 12:29 pm 
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perfredlund wrote:
I improved yet again the 6x6x6 checkerboard 6 faces pattern. I improved the rather inefficient last 31 turns like so:

setup edges: R VF VB M2F2 R- M2F2 VB- VF- L (9)
permute edges: VU2 L2 VU2 T3L2 VU2 VL2 (6)
undo setup: L- VF VB M2F2 R M2F2 VB- VF- R- (9)

The final result can be viewed here, and has a total of 38 block turns !! :mrgreen:

Very good brain exercise :lol:

Per


Good work, indeed!

Your edge permutation algorithm is now 24-move long:
(R VF VB M2F2 R' M2F2 VB' VF' L) (VU2 L2 VU2 T3L2 VU2 VL2) (L' VF VB M2F2 R M2F2 VB' VF' R')

It has the same length as mine:
N3B N3F (SU F2 SU') VB' VF' (SU F2 SU') NB NF NL NR (SU R2 SU') VL' VR' (SU R2 SU') N3R N3L

I haven't found a shorter one, even by using a (suboptimal) cube solver, not really a brainy day for me :-)

Your updated solution can be re-written as a 38-move algorithm:

(N2U2 N4U2 D2) R2 (N2F2 N4F2 B2) T3D2 (N2R2 N4R2 L2) (T3D2 R2 T3D2)
(R VF VB M2F2 R' M2F2 VB' VF' L) (VU2 L2 VU2 T3L2 VU2 VL2) (L' VF VB M2F2 R M2F2 VB' VF' R')

This algorithm is composed of 2 parts:
Part 1: all centers + nearly all edges (14 moves)
Part 2: the 8 remaining eges (24 moves)

Part 1 is already short (14 m), but Part 2 can be made much shorter by re-using my 4(2c) already existing alg, which is 12-move long:
(N3B N3F) (SU F2 SU') VB' VF' (SU F2 SU') (NB NF)

So by mixing our 2 algs, we now get a 26 mover for 6x6x6 cubes:
(N2U2 N4U2 D2) R2 (N2F2 N4F2 B2) T3D2 (N2R2 N4R2 L2) (T3D2 R2 T3D2)
(N3B N3F) (SU F2 SU') VB' VF' (SU F2 SU') (NB NF)

This seems to be a nice example of cooperative work ;-)


Notice that this algorithm can easily be scaled up/down, depending on cube size, and that only 5 moves are added per increase of 2 in size:

4 + 2 CHESSBOARDS
4x4x4 Cube
21 m
(N2U2 D2) R2 (N2F2 B2) TD2 (N2R2 L2) (TD2 R2 TD2)
(SU F2 SU') (NB' NF') (SU F2 SU') (NB NF)

6x6x6 Cube
26 m (+5)
(N2U2 N4U2 D2) R2 (N2F2 N4F2 B2) T3D2 (N2R2 N4R2 L2) (T3D2 R2 T3D2)
(N3B N3F) (SU F2 SU') VB' VF' (SU F2 SU') (NB NF)

8x8x8 Cube
31 m (+5)
(N2U2 N4U2 N6U2 D2) R2 (N2F2 N4F2 N6F2 B2) T4D2 (N2R2 N4R2 N6R2 L2) (T4D2 R2 T4D2)
(N3B N3F) (SU F2 SU') (V3B' V3F') (SU F2 SU') (NB N4B NF N4F)

10x10x10 Cube
36 m (+5)
(N2U2 N4U2 N6U2 N8U2 D2) R2 (N2F2 N4F2 N6F2 N8F2 B2) T5D2 (N2R2 N4R2 N6R2 N8R2 L2) (T5D2 R2 T5D2)
(N3B N5B N3F N5F) (SU F2 SU') (V4B' V4F') (SU F2 SU') (NB N4B NF N4F)

12x12x12 Cube
41 m (+5)
(N2U2 N4U2 N6U2 N8U2 N10U2 D2) R2 (N2F2 N4F2 N6F2 N8F2 N10F2 B2) T6D2 (N2R2 N4R2 N6R2 N8R2 N10R2 L2) T6D2 R2 T6D2
(N3B N5B N3F N5F) (SU F2 SU') (V5B' V5F') (SU F2 SU') (NB N4B N6B NF N4F N6F)

As there is no app for displaying cubes of size greater than 7 on the website of Walter and Werner, I developed a quick javascript Algorithm Viewer for cube sizes from 2 to 127, based on a built-in NxNxN engine and added to an already existing program.

By clicking this link and copying algs to the page, it is possible to display a planar cube model, showing the permuted stickers on all 6 faces. A planar cube model is then drawn up on an HTML5 canvas, showing the cube state on all 6 faces. Notice that canvas parameters have been optimized for viewing planar cubes of size 2 to 127, at a minimum screen horizontal resolution of 1920 pixels.

See examples:


Attachments:
Chessboards_6x.jpg
Chessboards_6x.jpg [ 61.76 KiB | Viewed 6321 times ]
Zhongguo_17x.jpg
Zhongguo_17x.jpg [ 68.97 KiB | Viewed 6321 times ]


Last edited by glazik on Thu Dec 22, 2011 10:41 am, edited 1 time in total.
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 Post subject: Patterns
PostPosted: Mon Dec 19, 2011 1:28 pm 
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Excellent job! I congratulate!
Chessboard --- 6x6x6 Cube --- 6 faces --- 26 moves!!!
The task was very interesting. The very original and simple decision!


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 Post subject: Re: Patterns
PostPosted: Mon Dec 19, 2011 4:08 pm 
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Благодарю!

You started the quest for algorithms of chessboards on 6x6x6 cubes and we ended up with short scalable algs for any NxNxN cube of even size! So, thanks for initiating the work.

До свидания!


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 Post subject: Re: Patterns
PostPosted: Tue Dec 20, 2011 4:58 am 
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So let me just add a link to a live demo of the final result.

View it here.

Now i must analyse how the 12 turns work! :lol:

Per

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 Post subject: Re: Patterns
PostPosted: Wed Dec 21, 2011 4:14 am 
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Hi :)

I added 12 extra turns at the end to the checkerboard 6 faces pattern. Then we get diagonal stripes on all 6 faces. Also quite nice !! :D

Exploring shorter setups :P

Per

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 Post subject: Re: Patterns
PostPosted: Sat Dec 24, 2011 1:59 pm 
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Quite nice pattern indeed!

The idea of leaving corner stickers unpermuted for chessboard-like patterns can be further extended to a Pythagorean Tiling on a cube, as shown on the demo 11x11x11 cube below (to display the alg, simply click this link and select the appropriate demo and page tab).

Merry Xmas!


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 Post subject: Re: Patterns
PostPosted: Tue Dec 27, 2011 9:21 am 
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Here are some additional patterns, linked to animated cubes:

- Pythagorean Tiling (6x6x6): link1
- Assyrian Wall Painting (7x7x7): link2

More details on algorithms are given in this document. Algorithms are mainly commutator-based and I guess shorter versions could certainly be found ;-)

These basic patterns can be applied to (virtual) cubes of bigger size to give interesting compositions:


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PythagoreanTiling_16x.jpg
PythagoreanTiling_16x.jpg [ 148.12 KiB | Viewed 6027 times ]
AssyrianWallPainting_19x.jpg
AssyrianWallPainting_19x.jpg [ 161.73 KiB | Viewed 6027 times ]
VasarelyComposition_22x.jpg
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 Post subject: Re: Patterns
PostPosted: Tue Dec 27, 2011 3:36 pm 
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Nice patterns!! Now i will contribute a simple yet not trivial pattern. Checkerboard 2 faces 6x6x6.

And a version with 2 smaller checkerboard faces 6x6x6.
Per

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 Post subject: Re: Patterns
PostPosted: Wed Dec 28, 2011 4:16 pm 
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Analyzing your algorithms gives the following partitions, which may help in understanding the moves, using CubeTwister for example.
I guess that finding such a short alg wasn't a trivial task. Congratulations!

// Checkerboard --- 2 Faces --- 6x6x6 Cube --- 20 moves
T3F2 N2R2 N4R2 N2U2 N4U2 V4L N2B2 N4B2 V4R T3F2 V4L N2B2 N4B2) V4R U2 R2 U2 R2 U2 R2

// Analysis: 3 Parts
(T3F2) (N2R2 N4R2 N2U2 N4U2) (T3F2) (T3F2 V4L N2B2 N4B2 V4R)2 (U2 R2)3

// Part 1: Edges
(T3F2) (N2R2 N4R2 N2U2 N4U2) (T3F2)

// Part 2: Centers
(T3F2 V4L N2B2 N4B2 V4R)2

// Part 3: AUF & ARF
(U2 R2)3


// Small Checkerboard --- 2 Faces --- 6x6x6 Cube --- 12 moves
V4L N2D2 N4D2 V4R U D V4L N2D2 N4D2 V4R U' D'

// Analysis: Commutator
[V4L N2D2 N4D2 V4R, U D]


The following patterns are variations of 'small' checkerboards, using 3-cycles, either on all 6 faces or constrained to faces R U F only, with single or 2 colors per face. Algorithms are based on products of 2 commutators each, for example:
[NF N3B', L N3D NU' L'] [N3F NB', L ND N3U' L']

6 Small Chessboards (single-color pattern)

6 Small Chessboards (2-color pattern)

3 Small Chessboards (single-color pattern)

3 Small Chessboards (2-color pattern)


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 Post subject: Re: Patterns
PostPosted: Thu Dec 29, 2011 6:43 am 
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Yet again i have a small contribution.

I call it mixed dots 3 faces, 6x6x6.

Per

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 Post subject: Re: Patterns
PostPosted: Thu Dec 29, 2011 4:57 pm 
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Hi Per,

Your 'mixed dots 3 faces, 6x6x6 pattern' is the same as my '3 Small Checkerboards (2-color pattern)'.
Both algorithms have the same length: 20 moves ;-)

Here are additional patterns on a 7x7x7 cube:

- Boson diagram (see this link)
- Basketweave
I think this last pattern could be displayed on a 5x5x5 cube, by including edges.

As usual, more complex patterns can be built from these, on bigger cubes, as shown below:


Attachments:
Bosons_19x.jpg
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 Post subject: Re: Patterns
PostPosted: Fri Dec 30, 2011 5:35 am 
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The basketweave pattern has a much shorter setup. It has 4 parts:

1. N3F N5F V5U- N3F- N5F- V5U
2. N3D N5D N2F N4F N6F N5U N3U N2B N4B N6B
3. N3F N5U- N3F- N5U
4. N5F N3U- N5F- N3U

The complete basketweave pattern animation!!

I am also very sure this is not optimal yet :wink:

Per

PS! I forgot to mention that the pattern can be extended to interweaving bands by including permuted edges. But then the setup becomes very long and awkward :|

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 Post subject: Re: Patterns
PostPosted: Fri Dec 30, 2011 11:13 am 
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After manual optimisation, I got a 28-move Basketweave algorithm, which shows a nice symmetric structure, with many move cancellations:

[N2F N4F N2B', N3D N3U']
[N3D, N3B'] [N3B', N3U'] [N3U', N3F] [N3F, N3D]
[N2D N4D N2U', N3F N3B']

But this is still far from your 24-move alg :(

Notice that your alg should be setupped as SU(...)SU' to get the same pattern as mine, that is, no bar should continue on an adjacent face with the same color, not really a strict condition...

Btw, 3-face 2-color-chessboards can also be displayed on bigger cubes, as shown below.

Next challenge:
Could it be possible to display a Domino Tiling on (centers of) a 4x/6x/8x?


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Chessboards_RUF_12x.jpg
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 Post subject: Re: Patterns
PostPosted: Fri Dec 30, 2011 11:16 am 
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That last design would make one really nasty bandaged cube! :shock:

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 Post subject: Re: Patterns
PostPosted: Fri Dec 30, 2011 11:21 am 
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It would be nice to see a domino tiling pattern for 6x6x6 with animated applet. Either 3 or 6 faces. If i get the time i will find a (hopefully) short setup for that :wink:

Per

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 Post subject: Re: Patterns
PostPosted: Fri Dec 30, 2011 5:14 pm 
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Domino tiling can be done on centers of a 6x6x6 cube :-)

A short sequence of 13 moves has already been found for the outer ring. It is based on 3-cycles for displaying 2 colors per face.

The remaining centers are permuted using 2-cycles on opposed faces, which leads to a fairly long algorithm, though.

I guess Per may find a shorter one soon :wink:


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 Post subject: Re: Patterns
PostPosted: Sat Dec 31, 2011 3:44 am 
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A sequence of 33 moves has just been found for the domino tiling.
Maybe a shorter sequence could still be found?


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 Post subject: Re: Patterns
PostPosted: Sun Jan 01, 2012 9:24 am 
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I wish you all a Happy New Year! :D

I found a shorter sequence for inner centers of the Domino Tiling:
VU' NF NB' VU VD NF' NB MD' M2F NU' ND' M2F' NU
SF2 D2 (TU TD MF2 S2D' MB2 ML2 MB2 TU2) D2 SF2 (25 btm)


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 Post subject: Re: Patterns
PostPosted: Sun Jan 01, 2012 10:13 am 
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Maybe someone has a good idea how outlined hearts can be achieved more efficiently on a 7x7x7. A try to twist them 90° instead of 180° around the U-D axis turned out to a real challenge.

4 Outlined Hearts (180° version):
MR2 D MR SD2 MR' D2 MR SD2 MR' D WR2 D2 VR2 VL2 U VR2 VL2 D2 VR2 VL2 U'
N3D' L2 NF2 NB2 L2 N3D L2 NF2 NB2 SR U2 R' L' V3U R L U2 R' L' M2U' L2 MF2 L2 NU' L2 M3F2 L2 ND' L2 N3F2 N3B2 L2 ND (54 btm)

4 Outlined Hearts (90° version):
R F (VD2 R2 VU' R2 VU F2 VD' F2 (R2 VD')2) F' R'
R' (U B' VD B U' B' VD' B) R
B' (U L' VU' L U' L' VU L) B
F D (VB' D' B' VR B D SR' D' B' VR' B D SR VB) D' F'
MF2 (ND' MR2 ND R' ND' R' NU MR2 NU' R ND R) MF2
B ND' B N3L2 B' ND B N3L2 B N3B2 N3D' B' L' NU' NR2 L B N3D B' NR2 L' NU L N3B2 B2
B N3D' B NL2 B' N3D B NL2 B NB2 ND' B' L' N3U' N3R2 L B ND B' N3R2 L' N3U L NB2 B2
B2 R2 B' MR B R2 B' MR' MF2 B' D F MF2 L U' R2 L U MF2 U' R2 L' U L' F' D' MF2 (145 btm or 141 btm actually)


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 Post subject: Re: Patterns
PostPosted: Sun Jan 01, 2012 4:21 pm 
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Hi Walter,

Wow, 2012 augurs well for pretty patterns! :D

Domino Tiling

Notice that a 19-move algorithm can be found by optimizing your alg further, using Algorithm Finder Lite.

The optimization process has been detailed below, as an aid to the interested readers who are not familiar with the Finder.

- Select a cube size of 6
- Copy algorithm SF2 D2 (TU TD MF2 S2D' MB2 ML2 MB2 TU2) D2 SF2 to windows 'Algorithm Template(s)' and 'Generator Algorithm'
- Set all these 4 settings to ON: 'Shift Template', 'Invert Template', 'Rotation Symmetry', 'Reflection Symmetry'
- Click button 'Search'

The Finder will then invert, cycle, transform each processed algorithm by symmetry, combine/cancel moves and compare results to the Gen Alg. Solutions shorter than the Gen Alg may eventually be found.

A number of 7-move algs have been found this way (to be compared to your 12-move initial alg), from which TU' TD MB2 MR2 MB2 S2U MF2 has been selected to give a final 19 mover, after composition with the alg of Part 1:
(VU' NF NB' VU VD NF' NB MD' M2F NU' ND' M2F') (U') (TD MB2 MR2 MB2 S2U MF2): Domino Tiling


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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 5:04 am 
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Hi :)

I solved the 4 outllined hearts (180 degrees) in 4 steps like so:

1. WR- V3U- WR V3U2 WR- V3U- WR V3U2
2. N3D2 N2F2 N2R2 N3D2 N2R2 N2F2
3. MR2 MF2 M3R- N2D M3R N2D2 M3R- N2D M3R
4. WR2 D WF2 D2 WF2 D WR2 MR2 MF2

I need to invert this to have a new setup algorithm. See the new outlined hearts algorithm in action. Reduced from 54 to 31 block turns :mrgreen:

On closer look V3U2 and N3D2 combined to V4U2, then it became 31 block turns!! Probably even better exists ...

Per

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Last edited by perfredlund on Mon Jan 02, 2012 9:00 am, edited 1 time in total.

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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 5:29 am 
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And now an easy 6-x pattern order 3 on the 7x7x7 cube. Not optimised. It can also be done similarly on any odd sized cube.

Per

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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 7:27 am 
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While i'm in the mood. Nested U's on the 7x7x7 cube.

Based on the simple 12 mover on the 3x3x3 cube. This one again is composed of an edge 3-cycle and inserted spots with good cancellation. I didn't bother to transform B- F (etc) to slice notation. Lazy today :lol:

Per

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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 10:32 am 
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Has this been done before?


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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 10:43 am 
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@ glazik
Hi André,
Awesome finding, the algorithm for the Domino Tiling became now quite handy! :D


@ perfredlund
Wow, what a massive reduction of the Outlined Hearts algorithm! Great job! :D

Inspired by the Outlined Heart, I also created a filled version:

4 Filled Hearts:
MF2 MR2 WR2 D WF2 D2 WF2 D WR2 MR2 MF2
V3U2
N3D' L2 WF2 L2 N3D L2 NF2 NB2 L2 ND' L2 M3F2 L2 ND (26 btm)


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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 12:19 pm 
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Hi Ben !!

A bit hard to see from the pictures the exact pattern. But i got the idea anyway. I would guess you are the first. Not a common puzzle. Maybe someone did on a virtual puzzle??? By the way, what is the algorithm for that??

Per

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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 12:51 pm 
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Nested U's inspired me to also post nested snake pattern!! Again too lazy to convert to proper slice notation :wink:

Per

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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 2:36 pm 
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Slight improvement to the

filled hearts pattern:

It has 3 parts applied in reverse:

1. V3U2 M3R- VD M3R VD2 M3R- VD M3R
2. N3D2 N2F2 N2R2 N3D2 N2R2 N2F2
3. MF2 MR2 WR2 D WF2 D2 WF2 D WR2 MR2 MF2

Maybe i will look into the 90 degrees version later (outline and filled).

Per

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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 3:10 pm 
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I'm on the run today:

Filled diamonds pattern:

Composed of 4 parts from outside to inside:
1. MR2 MF2 D2 MR2 MF2 D2 (edges)
2. M3R- N2D N2U- M3R N2D2 N2U2 M3R- N2D N2U- M3R (centers)
3. WR- N3D N3U- WR N3D2 N3U2 WR- N3D N3U- WR (more centers)
4. MU2 (trivial)

Per

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Last edited by perfredlund on Mon Jan 02, 2012 3:35 pm, edited 1 time in total.

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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 3:34 pm 
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I improved my previous pattern.

Filled diamonds:

The breakdown:

1. MR2 MF2 D2 MR2 MF2 D2 (edges)
2. F2 R2 MU2 R2 MU2 F2 (more edges)
3. M3R2 WU- M3R2 WU (centers)
4. N2R2 N2L2 M3U N2R2 N2L2 M3U- (more centers)

Per

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 Post subject: Re: Patterns
PostPosted: Mon Jan 02, 2012 6:30 pm 
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By combining part 3 and 4 the algorithm can be improved even more:

Filled Diamonds:
(MR2 MF2 D2)2
F2 R2 MU2 R2 MU2 F2
WD M3R2 WD- WR2 M3D- NR2 NL2 M3D (20 btm)


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 Post subject: Re: Patterns
PostPosted: Tue Jan 03, 2012 4:01 am 
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Waran wrote:
By combining part 3 and 4 the algorithm can be improved even more:

Filled Diamonds:
(MR2 MF2 D2)2
F2 R2 MU2 R2 MU2 F2
WD M3R2 WD- WR2 M3D- NR2 NL2 M3D (20 btm)


Yes, i saw that coming ... It was very late last night :wink:

Per

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 Post subject: Re: Patterns
PostPosted: Tue Jan 03, 2012 6:03 am 
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Notice that (F2 R2 MU2 R2 MU2) F2 could be replaced with (MU' SR2 MU SR2) to give 18 moves:
(MR2 MF2 D2)2
(MU' SR2 MU SR2)
(WD M3R2 WD- WR2 M3D- NR2 NL2 M3D) (18 btm)

By tinkering with this last expression, I even got 2 moves less (cube rotation CR2 not included): :D
MR2 MF2 D2 MR2 MF2 D2 WU' M3R2 WU WR2 M3D' MU SR2 MU' M3R2 M3U (16 btm)

Filled Diamonds


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 Post subject: Re: Patterns
PostPosted: Tue Jan 03, 2012 12:50 pm 
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Curvy copter checkerboard menzon?
I put it together this way


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 Post subject: Re: Patterns
PostPosted: Tue Jan 03, 2012 3:16 pm 
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Here are 2 new patterns on a 5x5x5 cube:

Four Fives (or Eight Ones)

Tangled Snakes

Algorithms have not been optimized yet, so shorter solutions may certainly exist.


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