Author 
Topic: Diagonals of a polygon. (Read 3898 times) 

Grimbal
wu::riddles Moderator Uberpuzzler
Gender:
Posts: 7527


Diagonals of a polygon.
« on: May 13^{th}, 2007, 3:11pm » 
Quote Modify

I took part in a logical and mathematical games contest. One of the problem was (story removed): There is a convex hexagon such that  all sides have a different length,  all 3 diagonals are concurrent  the vertices lie on the vertices of a regular Nsided polygon. What is the minimum possible N? As a starter, the given answer was an odd number, but in my opinion it can not be because by experience (i.e. computer), no 3 digaonals of a regular Nsided polygon are concurrent for N odd. Does anybody know a good argument for that?


IP Logged 



Barukh
Uberpuzzler
Gender:
Posts: 2276


Re: Diagonals of a polygon.
« Reply #1 on: May 13^{th}, 2007, 11:05pm » 
Quote Modify

on May 13^{th}, 2007, 3:11pm, Grimbal wrote: the vertices lie on the vertices of a regular Nsided polygon. 
 Shouldn't this be "on the sides of a regular Ngon?


IP Logged 



towr
wu::riddles Moderator Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730


Re: Diagonals of a polygon.
« Reply #2 on: May 14^{th}, 2007, 12:46am » 
Quote Modify

on May 13^{th}, 2007, 11:05pm, Barukh wrote: Shouldn't this be "on the sides of a regular Ngon? 
 My guess would be no. You want the diagonals of the hexagon to coincide with those of the Ngon, so it has to be vertices in both cases.

« Last Edit: May 14^{th}, 2007, 12:47am by towr » 
IP Logged 
Wikipedia, Google, Mathworld, Integer sequence DB



Grimbal
wu::riddles Moderator Uberpuzzler
Gender:
Posts: 7527


Re: Diagonals of a polygon.
« Reply #3 on: May 14^{th}, 2007, 2:10am » 
Quote Modify

As towr said. The vertices of the hexagon are a subset of those of a regular Ngon.

« Last Edit: May 14^{th}, 2007, 2:43am by Grimbal » 
IP Logged 



Barukh
Uberpuzzler
Gender:
Posts: 2276


Re: Diagonals of a polygon.
« Reply #4 on: May 14^{th}, 2007, 8:32am » 
Quote Modify

I see. Grimbal, your assumption is true. I don't know if there is a simple argument to prove it, though. Maybe, the following is of some help.


IP Logged 



Grimbal
wu::riddles Moderator Uberpuzzler
Gender:
Posts: 7527


Re: Diagonals of a polygon.
« Reply #5 on: May 14^{th}, 2007, 9:56am » 
Quote Modify

Thanks. It shows that identifying concurrent diagonals is far from trivial and cannot be solved in the hour I had, not without prior knowledge of the problem. But in the introduction it refers to an earlier paper: Herman Heineken, "Regelmässige Vielecke und ihre Diagonalen", 1962. He proves that an oddgon has no 3 concurrent diagonals, using complex polynomials.

« Last Edit: May 14^{th}, 2007, 9:57am by Grimbal » 
IP Logged 



balakrishnan
Junior Member
Gender:
Posts: 92


Re: Diagonals of a polygon.
« Reply #6 on: Aug 5^{th}, 2007, 5:37pm » 
Quote Modify

I get 8 as the smallest N.


IP Logged 



Obob
Senior Riddler
Gender:
Posts: 489


Re: Diagonals of a polygon.
« Reply #7 on: Aug 5^{th}, 2007, 10:55pm » 
Quote Modify

You cannot choose six vertices of an octagon in such a way that the consecutive distances between adjacent vertices are all different.


IP Logged 



balakrishnan
Junior Member
Gender:
Posts: 92


Re: Diagonals of a polygon.
« Reply #8 on: Aug 6^{th}, 2007, 5:05am » 
Quote Modify

Sorry I overlooked the problem. I get N=24.


IP Logged 



Grimbal
wu::riddles Moderator Uberpuzzler
Gender:
Posts: 7527


Re: Diagonals of a polygon.
« Reply #9 on: Aug 8^{th}, 2007, 5:39am » 
Quote Modify

Yep. That's what I got later with a computer program.


IP Logged 



