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## #1 2014-10-17 07:35:23

EdV56
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From: Minneapolis
Registered: 2014-10-17
Posts: 20
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### Are There Graph Theoretic Methods for Polycube Puzzles?

Hi,
I was hoping start a little discussion at Math Stack Exchange about tree graphs and recording/displaying solutions to the SOMA cube puzzle.  you can see some of what I have been doing here:

http://math.stackexchange.com/questions/954037/can-i-record-soma-puzzle-solutions-with-tree-graphs

Not much activity other than me.

It seems reasonable to me that the solutions can be shown on a tree graph but:
1. How best to do it?  Start with tables and then sort the tables before graphing?
2. How much information is needed?  I think the piece color occupying  which of 8 vertices may suffice.
3. Are other graph methods of use to:
a. determine a bound on the number of solutions?
b. determine allowable piece positions?

This is part of a Leonardo's Basement project to make hands on math activities for elementary schools.  Some previous work can be seen here:
http://www.fam-bundgaard.dk/SOMA/NEWS/N060808.HTM

I realize this is quite a number of questions but I thought I would give this forum a try.  If I need to refine my query please let me know.

Thanks much!

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## #2 2014-10-17 12:23:27

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
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### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

What is a SOMA cube?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #3 2014-10-17 14:29:51

EdV56
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From: Minneapolis
Registered: 2014-10-17
Posts: 20
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### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

http://en.wikipedia.org/wiki/Soma_cube

http://www.mathematische-basteleien.de/soma25.jpg

Last edited by EdV56 (2014-10-17 14:32:01)

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## #4 2014-10-17 17:35:55

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

How would you want to represent it with a graph?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #5 2014-10-18 01:22:43

EdV56
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From: Minneapolis
Registered: 2014-10-17
Posts: 20
Website

### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

1. A tree graph would describe the 240 solutions and reflect left and right due to the similar helix pieces.  There are also as many 16 solutions that begin with the same three pieces in the same position which would become major branches off of the trunk.
reference - http://www.fam-bundgaard.dk/SOMA/NEWS/N030518.HTM

2. A weighted graph of the pieces "touching" in all 240 solutions would be a way to illustrate the constancy of some of the pieces.
reference - http://www.fam-bundgaard.dk/SOMA/NEWS/N990201.HTM

How do I post pictures?  Rather than links to pictures . . . .

Thanks much.

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## #6 2014-10-18 01:30:54

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

Right click the picture and click "Copy Image Location". Take that link and put it between the img tags.

`` [img]Put link here.[/img] ``

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #7 2014-10-18 01:35:57

EdV56
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From: Minneapolis
Registered: 2014-10-17
Posts: 20
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### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

thanks!

Last edited by EdV56 (2014-10-18 01:36:34)

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## #8 2014-11-02 20:29:20

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

Pretty interesting

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #9 2014-11-03 15:37:44

EdV56
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From: Minneapolis
Registered: 2014-10-17
Posts: 20
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### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

There is very interesting approach here to graph theoretic methods for tiling trominoes which I think could shed some light but get lost in their notation around page three:
Graphs of Tilings (specifically a simple trominoes puzzle)
http://web.calstatela.edu/faculty/sheubac/papers/Graphs%20of%20Tilings.pdf

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## #10 2014-11-03 17:01:54

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

If not, I think this thread should be moved to 'This is Cool' or 'Euler\'s Avenue'

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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## #11 2014-11-04 03:29:06

EdV56
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From: Minneapolis
Registered: 2014-10-17
Posts: 20
Website

### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

I am fine with it moving if that seems a more appropriate venue for discussions.

Thanks much

I will also refine my questions regarding
Graphs of Tilings (specifically a simple trominoes puzzle)
http://web.calstatela.edu/faculty/sheubac/papers/Graphs%20of%20Tilings.pdf

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## #12 2014-11-04 03:44:54

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
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### Re: Are There Graph Theoretic Methods for Polycube Puzzles?

@bobbym: Kindly move this to 'This is Cool' or 'Euler\'s Avenue' wherever appropriate.

Do you know any resource that can explain the structures of the Soma cube graphs from scratch? I never knew anything about them before this thread.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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