
All Combinations of Six 2x4 Lego Bricks - mattmalin
http://c-mt.dk/counting/?view=paper
======
jgrahamc
Well that makes my "how many ways can I arrange an Ikea train set" look pretty
lame.

[http://blog.jgc.org/2010/01/more-fun-with-toys-ikea-
lillabo-...](http://blog.jgc.org/2010/01/more-fun-with-toys-ikea-lillabo-
train.html)

~~~
dublinben
You can create more arrangements if you treat the bridge like two pieces,
which it is.

~~~
pavel_lishin
How so? If your goal is to create a loop, the two pieces _must_ be joined
together.

~~~
lfowles
You could lengthen the bridge (theoretically, the pieces might not fit that
snug) with any combination of the pieces all the way down to the case of a
reverse bridge connected at the bottom.

~~~
JoshTriplett
Unless they fit very snugly, I don't think anything would hold up the extra
pieces in the middle, especially not well enough to support a train.

~~~
dublinben
The original creator of this type of train track offers just such a support
piece.[0] You can also 3D print your own supports![1]

[0]
[http://i.ebayimg.com/00/s/MTIwMFgxNjAw/z/DAUAAOSwGvhT3ny3/$_...](http://i.ebayimg.com/00/s/MTIwMFgxNjAw/z/DAUAAOSwGvhT3ny3/$_35.JPG)

[1]
[http://www.thingiverse.com/thing:110221](http://www.thingiverse.com/thing:110221)

~~~
pavel_lishin
Oh no, we're not combining Lego bricks with this, that's how we got here in
the first place!

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DiThi
It bugged me that the 2-brick possibilities pic in [1] wasn't ordered so I
made this [http://i.imgur.com/ShWZnwl.png](http://i.imgur.com/ShWZnwl.png)

[1]
[http://www.math.ku.dk/~eilers/lego.html](http://www.math.ku.dk/~eilers/lego.html)

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tgb
Won't their definition conflict with the rectilinear definition? Specifically,
there are distinct rectilinear configurations which are homotopic. With two
bricks connected by at corner, there are two rectilinear configurations but
both are homotopic. So their method used to count "homotopy equivalence
classes which contain at least one rectilinear configuration" would yield a
lower number than the previous result. This seems to contradict when the paper
says that their definition extends the previous one.

~~~
teraflop
Where are you getting two from? By the definitions on this page, it seems to
me like there are four corner-connected rectilinear configurations of two
bricks, and also four homotopy equivalence classes, so there's no
inconsistency.

    
    
        +--+       +--+
        |  |       |  |
        |  |       |  |
        |  |-+   +-|  |
        +--+ |   | +--+
          |  |   |  |
          |  |   |  |
          +--+   +--+
        
        +-----+       +-----+
        |     |-+   +-|     |
        +-----+ |   | +-----+
             |  |   |  |
             |  |   |  |
          	 +--+   +--+

~~~
tgb
Your ascii art has convinced me! I imagined you could rotate the left two into
each other but that makes no sense.

------
mattmalin
Note: this is not my work, just something I stumbled across.

GitHub project is here, including examples of all angles considered:
[https://github.com/LasseD/BrickCounting](https://github.com/LasseD/BrickCounting)

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fit2rule
I've always wanted to make a sorting box that, given any quantity of random
Lego Mindstorms parts, sorts the parts into shape/size/purpose bins. A static
box, no moving parts but the raw feed.

I suppose it will have to be a big box, mathematically.

~~~
DanBC
[http://thetechnicgear.com/2014/02/best-lego-sorting-
machines...](http://thetechnicgear.com/2014/02/best-lego-sorting-machines-
will-see/)

Some of these are interesting.

I'd be interested to see how good something that sorted purely by weight could
be. And then with a vision stage?

------
randomanybody
A minor typo in this paper yields the quote "Our result is that using sic
[sic] bricks, one can combine them into different models."

