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Topic: Matchsticks Arrangement (Read 712 times) |
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Barukh
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Matchsticks Arrangement
« on: Jan 31st, 2006, 11:10am » |
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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!
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Grimbal
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Re: Matchsticks Arrangement
« Reply #1 on: Jan 31st, 2006, 4:11pm » |
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All in 2 dimensions?
In 3 dimensions I would ask Plato for advice. But I don't guarantee optimality.
Given enough dimensions, n(n+1)/2 would be enough and optimal.
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ChunkTug
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Re: Matchsticks Arrangement
« Reply #2 on: Jan 31st, 2006, 8:09pm » |
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Assuming 2D, for 3 matchsticks at each meeting I get this 18 match configuration. (Can't seem to attach a image atm).
| hidden: | 3 of the diamonds (formed from two eqilateral triangles) connected at their acute ends by three matchsticks.
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End result is a nonagon with angles 150o,120o,150o... with the diamonds constructed on the interior of the 120 degree angles. |
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Barukh
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Re: Matchsticks Arrangement
« Reply #3 on: Feb 1st, 2006, 12:36am » |
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on Jan 31st, 2006, 4:11pm, Grimbal wrote:
Yes.
on Jan 31st, 2006, 8:09pm, ChunkTug wrote:| Assuming 2D, for 3 matchsticks at each meeting I get this 18 match configuration. (Can't seem to attach a image atm). |
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Nice solution, ChunkTug! But it can be improved.
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KeypadSDM
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Re: Matchsticks Arrangement
« Reply #4 on: Feb 1st, 2006, 2:21am » |
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I can get 15 using a modified argument for that one:
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2 of them Connected together in the centre with a pattern like:
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Actually, if you think about it you can get down to 12 by:
Arranging 2 of these together:
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Now connect their tips with a matchstic, and slightly offset them.
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| « Last Edit: Feb 1st, 2006, 2:31am by KeypadSDM » |
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Barukh
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Re: Matchsticks Arrangement
« Reply #5 on: Feb 1st, 2006, 3:45am » |
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Well done, KeypadSDM! 
Now, for the hard part: what about 4-matchsticks configuration?
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Barukh
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Re: Matchsticks Arrangement
« Reply #7 on: Feb 2nd, 2006, 1:10am » |
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on Feb 1st, 2006, 4:26pm, Grimbal wrote:
Good! But may be better.
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