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 Post subject: The potential of Magics with square tiles
PostPosted: Mon Apr 21, 2008 2:16 pm 
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Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
Firstly, I want to thank Mr Rubik for discovering those awesome folding puzzles (magics), as their depth
was more than I could imagine. In this post, I will try and summarise how I envisioned to classify
all closed loop types for this mechanism.

I will answer specific questions, and even in person this August, but for more details,
you might think of subscribing to the awesome magazine CFF (Cubism For Fun), because there is
where I intend to send the complete final PDF file when it becomes ready (with all the details).
(and that reminds me, I need to renew my own subscription!!!)

And before I start, let me mention that during the process, I had to change my notation five times,
as I was constantly discovering that some of the categories were already being a subclass of
some other existing category. Such is the beauty of those puzzles that they were full of surprises.

Some definitions:

Image


And now it's time to list Equivalence cases to make sure that we don't bump to the same cases.
In other words, if you have a closed loop magic puzzle which can take the shape of one
figure of a specific Equivalence class, it is possible to also take all the other shapes shown in the
figures of the same Equivalence class.


Image

Image

Image



And here's an extra note of a case which sometimes is not obvious (and VERY tricky).

Image




And now, I proudly present you the three general classes, by using the simplest (to date?) notation.
(the notation becomes much more simplified for the case where n=odd, as stated later).

Image




And here is the key explanation regarding the puzzles which have an odd number of tiles.
It can be apparent that their complexity, is also the reason it simplifies their structure.

Image



Some simple examples related to the notation used:

Rubik's Magic: M(8,0,0)
Secret of Atlantis: M(9,0,0)
Flexible Tetragon: M(8,1,0)
Magic Auto: M(12,2,0)
Mobius Line: M(8,1/2)

The only cases where a magic can be completely flat, is when the puzzle belongs to the first Class
(i.e. planar magics), n=even, t2=0, and n is greater or equal than 4(t1+1).

From all the above, it is easy to understand that there is still a lot of potential for making magics.
I hope you have enjoyed this ride to the inside world of magics.

:)


Pantazis

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 Post subject: Re: The potential of Magics with square tiles
PostPosted: Mon Apr 21, 2008 3:01 pm 
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Joined: Sun Jun 13, 2004 12:45 pm
Location: Rochester, MN
How do I subscribe to CFF? I google'd and couldn't find any references from this millenium.

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 Post subject: Re: The potential of Magics with square tiles
PostPosted: Mon Apr 21, 2008 3:10 pm 
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Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
Here is a link including a small discussion. Georges hosts some links which will answer your question.

I just hope that my results will be vigorously challenged and verified. Most of what I show above,
are about magics which I had built temporarily, just to validate my theoretical results.

:)


Pantazis

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 Post subject: Re: The potential of Magics with square tiles
PostPosted: Mon Apr 21, 2008 11:12 pm 
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Joined: Sat Apr 21, 2007 11:21 pm
Location: Marin, CA
I love this forum.

This is amazing, Pantazis. Thanks. I'm still trying to comprehend it.

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 Post subject: Re: The potential of Magics with square tiles
PostPosted: Tue Apr 22, 2008 12:11 am 
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Joined: Sun Jun 04, 2006 10:05 am
Location: Minneapolis, Minnesota, USA
io wrote:
I love this forum.

This is amazing, Pantazis. Thanks. I'm still trying to comprehend it.

You're not the only one.

*Puts on dunce cap*

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 Post subject: Re: The potential of Magics with square tiles
PostPosted: Tue Apr 22, 2008 12:42 am 
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Joined: Mon Sep 09, 2002 2:19 pm
Location: Yaroslavl, Russia and Maryland, USA
I am too old and my brain is too rusty for all this stuff.
But I have a gut feeling that you're right on point, pal! 8-)

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 Post subject: Re: The potential of Magics with square tiles
PostPosted: Tue Apr 22, 2008 2:22 am 
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Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
I apologise if I presented the general stuff first. :oops:

The results above are just the summary of years of testing.
As stated, the notation had changed five times, but the last one I used
seemed to last a lot more than the others, and I have decided to use it from now on.

Using loops and with full and half twists was the most logical approach,
as it simply covers all cases. Why? Because by using "exhaustion method" there were
no other alternative ways! The next step was to find out which of the cases
fall under the same class and simplify the notation. This is where the Equivalence cases helped a lot.
Of course, the corner effect of the puzzles with odd number of tiles, was a key part,
so that a universal notation for both even and odd cases would be established.

During this Odyssey, there were many interesting outcomes. For example, have a look
at the Equivalence types A, figure 6. I think it is one of the most intriguing cases! :)
It looks really amazing and unbelievable to have that figure equivalent with tiles in a straight line,
but it is a very true property. And I admit, that was one of the totally unexpected results.

Regarding magics with odd number of tiles, there is still one case I have not tested,
but it won't affect at all the definitions above, this is why I decided to present the results early.

Maybe some simple examples will help, but I am too busy now (until June)
so there won't be any spectacular update until then.

The simplest cases of all, which can be regarded as some nice examples, are six (which are actually five, because two
happen to be connected by an Equivalence type). All those six cases fall inside the three General Classes mentioned.

Some extra points to be mentioned:
(a) every half or full twist is considered to be a positive one (i.e. [+] or [+/-], and NOT [-] or [-/+]), unless stated otherwise.
(b) A cyan tile is only connected to it two gray neigbours and no other tiles. The same goes for a gray tile between two cyan ones.

And I know some people have said they are not twisty puzzles and even compared it with the Rubik's snake toy. :shock:
In fact, folding puzzles with strings are the ones which twist themselves *and* our mind more than any other puzzle!!!

:)


Pantazis

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 Post subject: Re: The potential of Magics with square tiles
PostPosted: Tue Apr 22, 2008 3:32 am 
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kastellorizo wrote:
Firstly, I want to thank Mr Rubik for discovering those awesome folding puzzles (magics)

Are you sure it was Mr Rubik? I somehow remember he invented the magics just like he invented the clock and the 4x4/5x5. I'm not 100% sure, but I think that's what I remember from talking with Janos Kovacs at Rubik's Studio in 2005.


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 Post subject: Re: The potential of Magics with square tiles
PostPosted: Tue Apr 22, 2008 5:55 am 
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Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
Stefan Pochmann wrote:
kastellorizo wrote:
Firstly, I want to thank Mr Rubik for discovering those awesome folding puzzles (magics)

Are you sure it was Mr Rubik? I somehow remember he invented the magics just like he invented the clock and the 4x4/5x5. I'm not 100% sure, but I think that's what I remember from talking with Janos Kovacs at Rubik's Studio in 2005.


Well, although the Rubik's Magic Patent only states Mr Rubik's name, I will not be surprised
if it was another person, or if it even required more than one person to come up with the idea.

Using the Jacob's Ladder concept to build something more solid and puzzling, was simply genius.
And whoever was behind it (most probably Mr Rubik), I thank him/them, as it certainly made
most of my spare time very challenging and enjoyable.

And I really hope with some new designs, based on the three classes mentioned above, I will take my... "revenge"!!!


:)

Pantazis

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