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perfredlund

Post subject: Re: Patterns Posted: Thu Nov 10, 2011 7:53 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

Nice contribution Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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robin

Post subject: Re: Patterns Posted: Fri Dec 02, 2011 10:53 am 

Joined: Fri Oct 23, 2009 8:01 am 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 (4Ñ…4Ñ…4, 6Ñ…6Ñ…6 and Ñ‚. Ð´.) the chess pattern is going to always without 4 corners: Somebody tried to collect such pattern on a cube 4Ñ…4Ñ…4? Actually it not so is simple, as it seems at first. P. S. Who has cube 6Ñ…6Ñ…6, 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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pytlivyj_1

Post subject: Patterns Posted: Sun Dec 04, 2011 1:29 pm 

Joined: Tue Mar 24, 2009 12:52 pm

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 â„– 2; step 3.3. and step 3.4.  we carry out mirror analogue of algorithm â„– 2. 1. (l' U2)*2 F2 l' F2 r U2 r' U2 l2 = A2. 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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glazik

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

Joined: Wed Oct 12, 2011 12:54 pm

Hi, Simple algorithms of 4face Chessboards can already be found by clicking the links below: 4x4x4 cube http://www.randelshofer.ch/rubik/revenge/A840.01.html6x6x6 cube http://www.randelshofer.ch/rubik/vcube6/A840.01.htmlAlgorithms of 6face 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 4face chessboards and add the missing edge + center algorithms. Two twocycles on opposed faces may give short algorithms. These are computed using an online orbit solver: http://www.randelshofer.ch/rubik/apps/7x_super_cube_solver/index.htmlAlgorithm notation details can also be found at the same website: http://www.randelshofer.ch/rubik/patterns/doc/supersetENG_3x3.htmlHere 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 [ 94.98 KiB  Viewed 6011 times ]

Chessboard_6x.jpg [ 111.78 KiB  Viewed 6011 times ]



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pytlivyj_1

Post subject: Patterns Posted: Mon Dec 05, 2011 11:58 am 

Joined: Tue Mar 24, 2009 12:52 pm

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) [(23r' U2)*4 23r' z2 (23r' U2)*4 23r' y]*2  36 moves; 4) 34r2 B2 34r2 34f2 L2 34f2  6 moves; 5) (B2 L2)*2  4 moves. * 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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GoombaGeek

Post subject: Re: Patterns Posted: Mon Dec 05, 2011 4:45 pm 

Joined: Wed Jun 22, 2011 4:57 pm Location: The land of dreams, coincedentally located in Alberta

glazik wrote: Hi, Simple algorithms of 4face Chessboards can already be found by clicking the links below: 4x4x4 cube snip.png 6x6x6 cube snip.png Algorithms of 6face 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 4face chessboards and add the missing edge + center algorithms. Two twocycles on opposed faces may give short algorithms. These are computed using an online orbit solver: http://www.randelshofer.ch/rubik/apps/7x_super_cube_solver/index.htmlAlgorithm notation details can also be found at the same website: http://www.randelshofer.ch/rubik/patterns/doc/supersetENG_3x3.htmlHere 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!
_________________ 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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perfredlund

Post subject: Re: Patterns Posted: Wed Dec 07, 2011 3:06 pm 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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 8cycle. 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
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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perfredlund

Post subject: Re: Patterns Posted: Thu Dec 08, 2011 6:40 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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) [(23r' U2)*4 23r' z2 (23r' U2)*4 23r' y]*2  36 moves; 4) 34r2 B2 34r2 34f2 L2 34f2  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
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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glazik

Post subject: Re: Patterns Posted: Fri Dec 09, 2011 7:17 am 

Joined: Wed Oct 12, 2011 12:54 pm

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) [(23r' U2)*4 23r' z2 (23r' U2)*4 23r' y]*2  36 moves; 4) 34r2 B2 34r2 34f2 L2 34f2  6 moves; 5) (B2 L2)*2  4 moves.
* The decision is found without application of the computer! Good work! Your 84move solution is not that far from my (unoptimized) 88move 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 2cycles. 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 18move alg instead of the 40move initial alg! 2 Optimizing Edges The 2 algorithms are first separated into four 2 2cycles. 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 2cycles: 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 26move alg after move cancellation, to be compared to the initial 36move 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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pytlivyj_1

Post subject: Patterns Posted: Thu Dec 15, 2011 12:42 pm 

Joined: Tue Mar 24, 2009 12:52 pm

2 glazik
Chessboard  6x6x6 Cube  6 faces  46 moves (!!!):
Green  F; White  U.
(l2 3r2 R2) (b2 3f2 F2) (l2 3r2 R2) (d2 3u2) â€“ 11 moves; (3l2 r2 3b2 f2)*2 U2 (3l2 r2 3b2 f2 3l2 r2)  15 moves; 25f' 25u' R2 25u (24f' 3f2 b') 25u' R2 B2 25u (24r' 3r2 l') 25u' B2 25u (24r 3r2 l) â€“ 20 moves.


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glazik

Post subject: Re: Patterns Posted: Fri Dec 16, 2011 3:34 pm 

Joined: Wed Oct 12, 2011 12:54 pm

To pylivyj_1: You 46move alg is quite short indeed!
This 47move 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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perfredlund

Post subject: Re: Patterns Posted: Fri Dec 16, 2011 4:04 pm 

Joined: Fri May 06, 2005 10:13 am Location: Norway

glazik wrote: To pylivyj_1: You 46move alg is quite short indeed!
This 47move 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
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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glazik

Post subject: Re: Patterns Posted: Sat Dec 17, 2011 12:48 am 

Joined: Wed Oct 12, 2011 12:54 pm

perfredlund wrote: glazik wrote: To pylivyj_1: You 46move alg is quite short indeed!
This 47move 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/ ... UR2SUMRMLBtw, it's certainly a good idea to give links to animated cubes...


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perfredlund

Post subject: Re: Patterns Posted: Sun Dec 18, 2011 1:09 pm 

Joined: Fri May 06, 2005 10:13 am Location: Norway

glazik wrote: perfredlund wrote: glazik wrote: To pylivyj_1: You 46move alg is quite short indeed!
This 47move 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/ ... UR2SUMRMLBtw, 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 ... Per Breakdown of last 31 turns: Setup edges: U D N2R N4R D U First block 3cycle: T3F U VR U L2 U VR U L2 T3F Second block 3cycle: 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
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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perfredlund

Post subject: Re: Patterns Posted: Mon Dec 19, 2011 7:09 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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 !! Very good brain exercise Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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glazik

Post subject: Re: Patterns Posted: Mon Dec 19, 2011 12:29 pm 

Joined: Wed Oct 12, 2011 12:54 pm

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 !! Very good brain exercise Per Good work, indeed! Your edge permutation algorithm is now 24move 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 rewritten as a 38move 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 reusing my 4(2c) already existing alg, which is 12move 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 builtin 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 [ 61.76 KiB  Viewed 5475 times ]

Zhongguo_17x.jpg [ 68.97 KiB  Viewed 5475 times ]

Last edited by glazik on Thu Dec 22, 2011 10:41 am, edited 1 time in total.


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pytlivyj_1

Post subject: Patterns Posted: Mon Dec 19, 2011 1:28 pm 

Joined: Tue Mar 24, 2009 12:52 pm

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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glazik

Post subject: Re: Patterns Posted: Mon Dec 19, 2011 4:08 pm 

Joined: Wed Oct 12, 2011 12:54 pm

Ð‘Ð»Ð°Ð³Ð¾Ð´Ð°Ñ€ÑŽ!
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.
Ð”Ð¾ ÑÐ²Ð¸Ð´Ð°Ð½Ð¸Ñ!


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perfredlund

Post subject: Re: Patterns Posted: Tue Dec 20, 2011 4:58 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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! Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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perfredlund

Post subject: Re: Patterns Posted: Wed Dec 21, 2011 4:14 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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 !! Exploring shorter setups Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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glazik

Post subject: Re: Patterns Posted: Sat Dec 24, 2011 1:59 pm 

Joined: Wed Oct 12, 2011 12:54 pm

Quite nice pattern indeed! The idea of leaving corner stickers unpermuted for chessboardlike 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!
Attachments: 
PythagorianTiling_11x.jpg [ 110.41 KiB  Viewed 5293 times ]



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glazik

Post subject: Re: Patterns Posted: Tue Dec 27, 2011 9:21 am 

Joined: Wed Oct 12, 2011 12:54 pm

Here are some additional patterns, linked to animated cubes:  Pythagorean Tiling (6x6x6): link1 Assyrian Wall Painting (7x7x7): link2More details on algorithms are given in this document. Algorithms are mainly commutatorbased 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:
Attachments: 
PythagoreanTiling_16x.jpg [ 148.12 KiB  Viewed 5181 times ]

AssyrianWallPainting_19x.jpg [ 161.73 KiB  Viewed 5181 times ]

VasarelyComposition_22x.jpg [ 148.54 KiB  Viewed 5181 times ]



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perfredlund

Post subject: Re: Patterns Posted: Tue Dec 27, 2011 3:36 pm 

Joined: Fri May 06, 2005 10:13 am Location: Norway


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glazik

Post subject: Re: Patterns Posted: Wed Dec 28, 2011 4:16 pm 

Joined: Wed Oct 12, 2011 12:54 pm

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 3cycles, 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 (singlecolor pattern)6 Small Chessboards (2color pattern)3 Small Chessboards (singlecolor pattern)3 Small Chessboards (2color pattern)


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perfredlund

Post subject: Re: Patterns Posted: Thu Dec 29, 2011 6:43 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

Yet again i have a small contribution. I call it mixed dots 3 faces, 6x6x6. Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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glazik

Post subject: Re: Patterns Posted: Thu Dec 29, 2011 4:57 pm 

Joined: Wed Oct 12, 2011 12:54 pm

Hi Per, Your 'mixed dots 3 faces, 6x6x6 pattern' is the same as my '3 Small Checkerboards (2color pattern)'. Both algorithms have the same length: 20 moves Here are additional patterns on a 7x7x7 cube:  Boson diagram (see this link)  BasketweaveI 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 [ 156.58 KiB  Viewed 5027 times ]

Basketweave_13x.jpg [ 126.52 KiB  Viewed 5027 times ]



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perfredlund

Post subject: Re: Patterns Posted: Fri Dec 30, 2011 5:35 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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 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
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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glazik

Post subject: Re: Patterns Posted: Fri Dec 30, 2011 11:13 am 

Joined: Wed Oct 12, 2011 12:54 pm

After manual optimisation, I got a 28move 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 24move 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, 3face 2colorchessboards 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?
Attachments: 
Chessboards_RUF_12x.jpg [ 110.87 KiB  Viewed 4980 times ]

DominoTiling_8x.jpg [ 24.23 KiB  Viewed 4980 times ]



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KelvinS

Post subject: Re: Patterns Posted: Fri Dec 30, 2011 11:16 am 

Joined: Mon Mar 30, 2009 5:13 pm

That last design would make one really nasty bandaged cube!
_________________ If you want something youâ€™ve never had, youâ€™ve got to do something youâ€™ve never done  Thomas Jefferson


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perfredlund

Post subject: Re: Patterns Posted: Fri Dec 30, 2011 11:21 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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 Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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glazik

Post subject: Re: Patterns Posted: Fri Dec 30, 2011 5:14 pm 

Joined: Wed Oct 12, 2011 12:54 pm

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 3cycles for displaying 2 colors per face. The remaining centers are permuted using 2cycles on opposed faces, which leads to a fairly long algorithm, though. I guess Per may find a shorter one soon


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glazik

Post subject: Re: Patterns Posted: Sat Dec 31, 2011 3:44 am 

Joined: Wed Oct 12, 2011 12:54 pm

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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Waran

Post subject: Re: Patterns Posted: Sun Jan 01, 2012 9:24 am 

Joined: Sat Jan 28, 2006 6:04 am Location: Switzerland

I wish you all a Happy New Year! 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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Waran

Post subject: Re: Patterns Posted: Sun Jan 01, 2012 10:13 am 

Joined: Sat Jan 28, 2006 6:04 am Location: Switzerland

Maybe someone has a good idea how outlined hearts can be achieved more efficiently on a 7x7x7. A try to twist them 90Â° instead of 180Â° around the UD axis turned out to a real challenge. 4 Outlined Hearts (180Â° 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 (90Â° 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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glazik

Post subject: Re: Patterns Posted: Sun Jan 01, 2012 4:21 pm 

Joined: Wed Oct 12, 2011 12:54 pm

Hi Walter, Wow, 2012 augurs well for pretty patterns! Domino TilingNotice that a 19move 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 7move algs have been found this way (to be compared to your 12move 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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perfredlund

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 5:04 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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 On closer look V3U2 and N3D2 combined to V4U2, then it became 31 block turns!! Probably even better exists ... Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy
Last edited by perfredlund on Mon Jan 02, 2012 9:00 am, edited 1 time in total.


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perfredlund

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 5:29 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

And now an easy 6x pattern order 3 on the 7x7x7 cube. Not optimised. It can also be done similarly on any odd sized cube. Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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perfredlund

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 7:27 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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 3cycle and inserted spots with good cancellation. I didn't bother to transform B F (etc) to slice notation. Lazy today Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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DinoBen

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 10:32 am 

Joined: Sun Nov 20, 2011 7:16 pm

Has this been done before?
Attachments: 
DSCF5879.JPG [ 1.83 MiB  Viewed 4754 times ]

DSCF5878.JPG [ 1.71 MiB  Viewed 4754 times ]



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Waran

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 10:43 am 

Joined: Sat Jan 28, 2006 6:04 am Location: Switzerland

@ glazik Hi AndrÃ©, Awesome finding, the algorithm for the Domino Tiling became now quite handy! @ perfredlund Wow, what a massive reduction of the Outlined Hearts algorithm! Great job! 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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perfredlund

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 12:19 pm 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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perfredlund

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 12:51 pm 

Joined: Fri May 06, 2005 10:13 am Location: Norway

Nested U's inspired me to also post nested snake pattern!! Again too lazy to convert to proper slice notation Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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perfredlund

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 2:36 pm 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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perfredlund

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 3:10 pm 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy
Last edited by perfredlund on Mon Jan 02, 2012 3:35 pm, edited 1 time in total.


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perfredlund

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 3:34 pm 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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Waran

Post subject: Re: Patterns Posted: Mon Jan 02, 2012 6:30 pm 

Joined: Sat Jan 28, 2006 6:04 am Location: Switzerland

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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perfredlund

Post subject: Re: Patterns Posted: Tue Jan 03, 2012 4:01 am 

Joined: Fri May 06, 2005 10:13 am Location: Norway

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 Per
_________________ "Life is what happens to you while you are busy making other plans" John Lennon, Beautiful Boy


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glazik

Post subject: Re: Patterns Posted: Tue Jan 03, 2012 6:03 am 

Joined: Wed Oct 12, 2011 12:54 pm

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): MR2 MF2 D2 MR2 MF2 D2 WU' M3R2 WU WR2 M3D' MU SR2 MU' M3R2 M3U (16 btm) Filled Diamonds


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DinoBen

Post subject: Re: Patterns Posted: Tue Jan 03, 2012 12:50 pm 

Joined: Sun Nov 20, 2011 7:16 pm

Curvy copter checkerboard menzon? I put it together this way
Attachments: 
DSCF5885.JPG [ 1.84 MiB  Viewed 4613 times ]

DSCF5884.JPG [ 1.87 MiB  Viewed 4613 times ]



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glazik

Post subject: Re: Patterns Posted: Tue Jan 03, 2012 3:16 pm 

Joined: Wed Oct 12, 2011 12:54 pm


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