All Combinations of Six 2x4 Lego Bricks

#1

Post by mattmalin » Tue, Sep 08, 2015, 8:08 AM UTC

All Combinations of Six 2x4 Lego Bricks
c-mt.dk

Re: All Combinations of Six 2x4 Lego Bricks

#2

Post by mattmalin » Tue, Sep 08, 2015, 8:10 AM UTC

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

Re: All Combinations of Six 2x4 Lego Bricks

#3

Post by DiThi » Tue, Sep 08, 2015, 12:23 PM UTC

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

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

Re: All Combinations of Six 2x4 Lego Bricks

#4

Post by tgb » Tue, Sep 08, 2015, 1:55 PM UTC

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.

Re: All Combinations of Six 2x4 Lego Bricks

#5

Post by jgrahamc » Tue, Sep 08, 2015, 2:18 PM UTC

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-...

Re: All Combinations of Six 2x4 Lego Bricks

#6

Post by teraflop » Tue, Sep 08, 2015, 2:22 PM UTC

tgb wrote: post #4

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. Th…

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.

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

Re: All Combinations of Six 2x4 Lego Bricks

#7

Post by fogleman » Tue, Sep 08, 2015, 2:34 PM UTC

jgrahamc wrote: post #5

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-...

The Ikea one seems more practical.

Re: All Combinations of Six 2x4 Lego Bricks

#8

Post by dublinben » Tue, Sep 08, 2015, 2:44 PM UTC

jgrahamc wrote: post #5

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-...

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

Re: All Combinations of Six 2x4 Lego Bricks

#9

Post by ggus » Tue, Sep 08, 2015, 2:47 PM UTC

jgrahamc wrote: post #5

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-...

and how many combinations if I buy two sets?

Re: All Combinations of Six 2x4 Lego Bricks

#10

Post by pavel_lishin » Tue, Sep 08, 2015, 2:48 PM UTC

dublinben wrote: post #8

jgrahamc wrote: post #5

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-...

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

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