wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> hard >> Matchsticks Arrangement
        
(Message started by: Barukh on Jan 31^st, 2006, 11:10am)

Title: Matchsticks Arrangement
Post by Barukh on Jan 31^st, 2006, 11:10am

It is easy to arrange 3 matchsticks in the plane so that exactly 2 matchsticks meet at each end.

How many matchsticks are required to build a configuration where at each end exactly 3 matchsticks meet? 4 matchsticks? 5 matchsticks?

Note: Matchsticks cannot cross each other!

Title: Re: Matchsticks Arrangement
Post by Grimbal on Jan 31^st, 2006, 4:11pm

All in 2 dimensions?

In 3 dimensions I would ask [hide]Plato[/hide] for advice. But I don't guarantee optimality.

Given enough dimensions, [hide]n(n+1)/2[/hide] would be enough and optimal.

Title: Re: Matchsticks Arrangement
Post by ChunkTug on Jan 31^st, 2006, 8:09pm

Assuming 2D, for 3 matchsticks at each meeting I get this [hide]18[/hide] match configuration. (Can't seem to attach a image atm).

[hideb]3 of the diamonds (formed from two eqilateral triangles) connected at their acute ends by three matchsticks.
/\
\/

End result is a nonagon with angles 150^o,120^o,150^o... with the diamonds constructed on the interior of the 120 degree angles. [/hideb]

Title: Re: Matchsticks Arrangement
Post by Barukh on Feb 1^st, 2006, 12:36am

on 01/31/06 at 16:11:48, Grimbal wrote:

All in 2 dimensions?

Yes.

on 01/31/06 at 20:09:01, ChunkTug wrote:

Assuming 2D, for 3 matchsticks at each meeting I get this [hide]18[/hide] match configuration. (Can't seem to attach a image atm).

Nice solution, ChunkTug! But it can be improved.

Title: Re: Matchsticks Arrangement
Post by KeypadSDM on Feb 1^st, 2006, 2:21am

I can get [hide]15[/hide] using a modified argument for that one:

[hide]/_\
\ /

2 of them Connected together in the centre with a pattern like:

\/
|
/\[/hide]

Actually, if you think about it you can get down to [hide]12[/hide] by:
[hide]Arranging 2 of these together:
/_\
\ /
Now connect their tips with a matchstic, and slightly offset them. [/hide]

Title: Re: Matchsticks Arrangement
Post by Barukh on Feb 1^st, 2006, 3:45am

Well done, KeypadSDM! :D

Now, for the hard part: what about 4-matchsticks configuration?

Title: Re: Matchsticks Arrangement
Post by Grimbal on Feb 1^st, 2006, 4:26pm

120?
http://www.florian.net/puzzle/pic/connect4.gif

Title: Re: Matchsticks Arrangement
Post by Barukh on Feb 2^nd, 2006, 1:10am

on 02/01/06 at 16:26:25, Grimbal wrote:

120?

Good! But may be better.