Stewart Coffin Auction

[Canuck]

Hi all, I thought I'd let you all in on a very special auction to be hosted by Nick Baxter sometime in July, I'll post a firm date once it's set.  In talking to many long time puzzle craftsman all have been gracious enough to take part and donate puzzles for this auction with all proceeds for Stewart Coffin.  If anyone would like to participate in any way send me an email or PM.
Here's a direct link to Nick's auction page: http://puzzles.baxterweb.com/ 

Now here's what Rolly has submitted;  a very unique 'VertIgon' puzzle designed and handcrafted by him, thank you so very much Rolly!!!   This puzzle is an absolute work of art


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[rolly_wood]

 
you make me blush... also because I am not yet mathematically sure that there are no more than two independent solutions one (that shown) with a single ring (there are rightwise and leftwise sub-solutions, of course), another one with three. Seems there are not others with two ... but I have no proofs.... it is a puzzle for its designer too 
Is there any mathematician out there? 

[Canuck]

Is there any mathematician out there? 

There must be!

[gibell]

Ha!  I am no woodworker or puzzle solver, but I do answer to Mathematician.

You know, I have a puzzle which looks similar to this which uses wooden pieces with very strong magnets in the ends.

So what is the question?  Looks the the pieces build a dodecahedron, but in how many ways?  Are there magnets in it?

[rolly_wood]

No, there are 1 pin on the "head" and two holes on the "tail". For this reason pieces must trace out only closed circuits. The other hole left free hosts a dowel. The soln for which it has been thought is the well known hamiltonian cycle, but it may be assembled also with 3 rings 5,10,5 pieces on the top, equator and bottom. The question is: how many soln are there (independent - not to distinguish between left and rightwise), are there solutions with two closed rings? It seems to me the answer is no but I am not sure...
thank you George

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[gibell]

You're right what you are asking reduces to a pure math problem!

As I understand it, you are asking for the number of ways to decompose the graph of the dodecahedron into Hamiltonian cycles.  That is, a set of cycles that together go through every vertex once.  In the true spirit of the lazy mathematician I used Google and found the following post, which I believe answers the question:

http://sci.tech-archive.net/Archive/sci.math/2006-08/msg03382.html

It is a bit hard to understand, I'm not sure exactly what the notation Hx means, but it should be the number of such decomposing cycles of x cycles.  You will note that for the Dodecahedron the answer they come up with is 1 path (with a unique solution, up to symmetries) or 3 paths (2 on opposite faces and the other of length 10 around the equator).  Of course, being on the internet, it MUST be true, right?

I came up with a partial argument for why 2 cycles is not possible.  The lengths of cycles on this graph can only be of length 5, 8, 9, 10, ?, ?, 14, 17, 20.  Now if you must cover all 20 nodes in 2 cycles the options are: 5+15, 8+12, 9+11, 10+10.  I don't think a path of length 15 or 12 is possible, and the other two cases don't work out.  Of course, this is only a partial argument, it's a bit tedious figuring out all the path lengths that are possible and be certain that you have not missed anything.  The argument in the above link is similar to this.

So I do not think there are any other solutions that you have missed.  It would be nice to find a very simple argument for this ...

[rolly_wood]

I am sorry that I have not read your profile before posting my stupid question: the mathematician was already in and on-line. I apologise for that and thank you so much for having clarified me the matter. I printed many 12hedron planar schemes on paper and tried to find a solution with the pen! Then I arrived to the same conclusion empirically. Happy that you joined the forum George. I use simple math to do VRML models of puzzle (e.g. hamiltonian cycle on a truncated icosahedron http://www.youtube.com/watch?v=eIjcTZGTqQ8)  I had so many things to ask a mathematician but I do not want to bore you (and the community too) with such problems.
I have another question about vertIgon: I claimed the design was mine but of course not in the sense of the problem, but concerning the way I thought a puzzle. If you had not the dowels, you need to have 10 left and 10 right pieces or, depending on how they are cut, 6 left, 8 central and 6 right... In the way I chose instead all 20 are identical.
The question is: you said you have a similar puzzle, may you post a pic please? Or, do you think vertigon is different enough to claim it is "original"? I searched the web before but I did not find anything...

[rolly_wood]

To explain me better this is a prototype which needs three piece types, but does not need dowels (note that it is glued!)

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[gibell]

Your puzzle reminded me of something I bought, but they are not by any means identical.  It is a bunch of building blocks that have the same basic shape.  Take rectangular stock and saw them at 45 degrees so that the face is square.  There are 4 strong magnets embedded in each square face.  Alternate the direction of cuts so that the side view is a trapezoid.  Is this some fundamental puzzle shape in Coffins book?

You can then link together these puzzle pieces to make various shapes: squares and hexagons mainly.  You can make 3D closed paths as well, including one which looks vaguely like a dedecahedron, but looking at it closely it is not a dedecahedron.  Basically, you can't use these pieces to make any face with 5 sides, I believe.

It goes by the name of "math maker", and is made in the Netherlands.  See

http://www.mathmaker.nl/

and look under "building blocks".

There are more photos on this web site:

http://philos.quasarshop.de/Detail.asp?ARNR=5550&AG1=&AG=&ID=&ALID=&Lang=E

[Canuck]

Wow, thanks for that link George!  I had a look at the different 'recipes' pdf...over 900 pages 

Very interesting set of pieces, they'd be great for modeling tons of shapes

[rolly_wood]

I cannot download that pdf now and the video on u-tube is very slow. However I understood, very interesting!, it seems the shape you referred, if it is that one, is composed by hexagons and squares and is similar to a truncated octahedron..
Ok happy that nothing siimilar was in circulation, or at least, as far as I presently know... thank you 
Regarding Coffin I do not remember... but he does not like too much platonic 12hedron, he prefers the rhombic one for obvious reasons. Instead other puzzlemakers such as Philippe Dubois took inspiration from it.

EDIT I read again carefully your post and you did not refer to 12hedron regarding Coffin, it was due to my bad english

[rolly_wood]

Is it involved RD here also? cut of pieces is 45 and you can join them rotated 45 also. It seems that in the plane of the pieces this obtain the dihedral angle of RD ie 60 degrees. Thats why hexagons come out, am I wrong?

[gibell]

Exactly correct, that is how you get hexagons.  The ball-shape that can be formed uses 24 pieces (a full set).

My two kids really like these pieces.  They are really fun to play around with.  I bought them last year from a Belgian fellow at G4G8 (Gathering for Gardner).

[rolly_wood]

they loves magnets I know... however my son (11) get bored quite soon and accepted to allow me to destroy them (not wooden but plastic like this http://www.pburch.net/Pictures/toys/icosahedron-geomag.jpg) to remove magnets...
I reused them to make this http://img91.imageshack.us/img91/9492/dscf4275go3.gif and the metallic pins which, in the toy, connected two magnets, were used to do the vertigon: sustainable recycling... 

[rolly_wood]

John I realized that probably you wanted to maintain a rather clean topic to post the donations for the auction... Now I switched the discussion to the puzzle which is OFF topic respect to the title...  Please if it is so, move my messages in the topic of the vertIgon "call for a name" ... sorry

I was excited for having found a Mathematician in flesh and bones.