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19gon and Pigeonhole principle 
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#1
Dec1612, 05:42 PM

P: 138

Hey!
I have got some question for you. Decide if you can choose seven tops of the regular 19gon and four of them are tops of trapezoid. (I think  Pigeonhole principle, but how?) 


#2
Dec1712, 01:39 PM

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P: 26,157

Hi Numeriprimi!
What do you mean by "tops"? 


#3
Dec1712, 03:18 PM

P: 824

I think he means the vertices. How I would try to prove it. Feel free to stop reading once you think that you have the answer...
1) A trapezoid is defined by having two parallel sides. So you want to construct a set of points and none of the connections between the points are to be parallel. 2) If we numerate the points we can start forming all the families of parallel lines. 3) If we enumerate in a circle one family is {(2,19),(3,18),(4,17),(5,16),..., (10,11)} You see that even one "length 1" pair is included. 4) There are 19 of these families, and they account for all the possible connections there are. 5) The possible connections between n points are (n^2n)/2 6) Pidgeonhole 


#4
Dec1712, 05:04 PM

P: 138

19gon and Pigeonhole principle
Yes, I mean vertices... Sorry for my English because is quite hard to choose right word with same meaning in my language when is a lot of words :)
So, I will read and understand your answer after school because I going to sleep. Then I will write when I won't understand you. For now... thanks very much :) 


#5
Jan213, 07:23 PM

P: 824

Numeriprimi. You PM me, but maybe others are interested in the answer as well. So I'll discuss the questions here. I hope that is ok.
So between 7 points there are (7*77)/2=21 connections. If two of them are in parallel you are done. Connections are in parallel if they are in the same family. There are only 19 families. 


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