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 Post subject: Another way for solving the Ultimate Skewb...
PostPosted: Sun Dec 18, 2005 6:13 am 
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Well, as usual, I avoid looking at other's solutions, and I analyse the puzzles using the trusty GAP program.

Here is the story of how I solved the Ultimate Skewb last night. Sorry for being lazy to type, but the image tells the whole story! :wink:

Image

It clearly shows all the orbits, while it also contains (the usual finger blister but easy to memorize) repeating sequences, that interchange and/or rotate individual tiles to eventually solve the puzzle.

GAP generator code:

a:=(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24);
b:=(1,20,25)(26,23,27)(13,11,28)(22,29,30)(12,15,31)(9,18,6)(32,21,3)(33,24,34);
c:=(10,35,15)(36,27,24)(19,37,3)(22,38,34)(11,14,39)(29,23,40)(41,20,2)(8,17,5);
d:=(37,32,20)(42,9,17)(28,15,39)(43,18,5)(11,35,31)(8,44,6)(41,25,3)(34,29,27);
G:=Group(a,b,c,d);
Order(G);


In fact the page has *more* routines than necessary to solve it. :)





Peter

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 Post subject:
PostPosted: Sun Dec 18, 2005 4:17 pm 
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Uh... OK? Any chance of a translation for those of us who don't speak GAP? :? :scrambled: Thanks in advance. L8r.


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 Post subject:
PostPosted: Sun Dec 18, 2005 10:30 pm 
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Hi Tim!

Sorry for the messy diagrams/numbers.

No GAP knowledge is required to solve the puzzle. You just need the figure and the routines.

The symbols Rh, Rl, Lh, Ll on the up right, correspond to the clockwise moves of the puzzle "Right-high", "Right-low", "Left-high" and "Left-low".
Those are based by keeping the shaded part stabilised and on the top.

The routines (spread all over the paper) indicate what is is changed while keeping the rest invariant.

For example, on the left bottom corner, we have the "rotation of the upleft and downright corners".
If you apply the moves
Lh* Rl* Lh* Rh* Ll* ten times, then the tile 4 will go to tile 44, 5 to 16, 7 to 42 etc. keeping all the other numbers (which are not in the routine) invariant.


All this is explained in "permuation orbit language", which itself is based on all the numbers/labels on the figure.


I am planning to give more details and better figures at some point in the future! :)


Peter

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 Post subject:
PostPosted: Mon Dec 19, 2005 9:41 am 
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You say "No GAP knowledge is required". Also, no knowledge how to solve the puzzle is required, right? Hmm... I think you're the first script kiddie in twisty puzzling :D


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 Post subject:
PostPosted: Mon Dec 19, 2005 11:00 am 
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correct! no need to know anything about GAP. :P

What I actually do, is test a few sequences, and if they are small and change only a few tiles, then they are accepted...!

It will be interesting to generate *all* such routines, and pick up the best ones, but the elements of each group are way too many and the program will eventually crash.... :(


The things needed to do are:

1) Write down the puzzle generators that correspond to a puzzle move/twist/rotation.
2) Assess the number of "efficient" sequences that are needed to solve the puzzle.
3) Test to find "efficient" sequences.


I did something similar for the Rubik's UFO, and it seems that two such easy to remember seqeunces (plus some basic knowledge of the non-twisting 96 combinations) are more than enough! :wink:

Most certainly, this can be applied on the 3x3x3 cube, but I am not sure if I will really be able to find more efficient sequences than the ones already in existence. But for new puzzles, yes, I think GAP can provide a lot of interesting information! :)



Peter



PS As you requested, I did not use GAP for the Cmetricks. They were too easy for that...! LOL :D :D :D :D :D

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 Post subject:
PostPosted: Mon Dec 19, 2005 12:14 pm 
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kastellorizo wrote:
PS As you requested, I did not use GAP for the Cmetricks.


Good to hear that :-). I was already thinking about asking you about this.

Currently I'm trying to solve a puzzle that I can't solve yet. I'm wondering whether you can solve it with GAP. I'll let you know which one it is once I can solve it...


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 Post subject:
PostPosted: Mon Dec 19, 2005 12:30 pm 
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Well, any puzzle is possible to be decoded. Of course, the more complex it is, the more time it will take! :wink:

By the way, I am sure you must have tried Tony Fisher's Cube...
(reference: http://twistypuzzles.com/cgi-bin/puzzle.cgi?pid=624)

I noticed that when scrambling it, the centres actually *rotate*, something that does not happen with the normal 3x3x3 cube.
So I was just wondering whether I should include this within the GAP code.
(that is, "replace" a single tile by a "quartiple" tile whose information includes orientation..!)
And if I do, there will be *more* tile-numbers!!! :?

Or maybe I should treat all tiles as single tiles? (perhaps solving it, will have the orientation automatically solved too... or maybe not?)

Peter

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 Post subject:
PostPosted: Mon Dec 19, 2005 3:30 pm 
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kastellorizo wrote:
Well, any puzzle is possible to be decoded.

With GAP? Try Square-1. ;)
Quote:
I noticed that when scrambling it, the centres actually *rotate*, something that does not happen with the normal 3x3x3 cube.
<...> (perhaps solving it, will have the orientation automatically solved too... or maybe not?)

Ah, yes... GAP could certainly help you there.

But -- just for a bit of "retro" fun -- why not take an ordinary 3x3x3 and mark the centres so that you can see their orientations? Scramble it, then try to solve it without the help of GAP.

If that's not enjoyable enough, you could always imagine the brain/Cube combination to be a real, bio-logical PC with an ABS monitor!


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 Post subject:
PostPosted: Mon Dec 19, 2005 5:35 pm 
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StefanPochmann wrote:
Currently I'm trying to solve a puzzle that I can't solve yet. I'm wondering whether you can solve it with GAP. I'll let you know which one it is once I can solve it...


Alright, wasn't trivial but wasn't too hard, either. I meant the "Rack'em up" puzzle. I have a very simple and very very inefficient solution method now :D

http://www.twistypuzzles.com/cgi-bin/pu ... i?pid=1445


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 Post subject:
PostPosted: Mon Dec 19, 2005 11:15 pm 
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StefanPochmann wrote:
I have a very simple and very very inefficient solution method now :D

http://www.twistypuzzles.com/cgi-bin/pu ... i?pid=1445



LOL sounds exactly like my method above! :P

Thanks to both of you for the suggestions regarding Tony Fisher's cube. I am glad it is verified that I need to code in some extra parameters! :)


Peter

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