wu :: forums - Print Page


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

        riddles >> putnam exam (pure math) >> Diagonals of a polygon.
        
(Message started by: Grimbal on May 13^th, 2007, 3:11pm)

Title: Diagonals of a polygon.
Post by Grimbal on May 13^th, 2007, 3:11pm

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 N-sided 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 N-sided polygon are concurrent for N odd. Does anybody know a good argument for that?

Title: Re: Diagonals of a polygon.
Post by Barukh on May 13^th, 2007, 11:05pm

on 05/13/07 at 15:11:11, Grimbal wrote:

- the vertices lie on the vertices of a regular N-sided polygon.

Shouldn't this be "on the sides of a regular N-gon?

Title: Re: Diagonals of a polygon.
Post by towr on May 14^th, 2007, 12:46am

on 05/13/07 at 23:05:50, Barukh wrote:

Shouldn't this be "on the sides of a regular N-gon?

My guess would be no. You want the diagonals of the hexagon to coincide with those of the N-gon, so it has to be vertices in both cases.

Title: Re: Diagonals of a polygon.
Post by Grimbal on May 14^th, 2007, 2:10am

As towr said. The vertices of the hexagon are a subset of those of a regular N-gon.

Title: Re: Diagonals of a polygon.
Post by Barukh on May 14^th, 2007, 8:32am

I see.

Grimbal, your assumption is true. I don't know if there is a simple argument to prove it, though.

Maybe, the following (http://citeseer.ist.psu.edu/cache/papers/cs/9081/ftp:zSzzSzftp.msri.orgzSzpubzSzpublicationszSzpreprintszSz1995zSz1995-060zSz1995-060.pdf/poonen98number.pdf) is of some help.

Title: Re: Diagonals of a polygon.
Post by Grimbal on May 14^th, 2007, 9:56am

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 odd-gon has no 3 concurrent diagonals, using complex polynomials.

Title: Re: Diagonals of a polygon.
Post by balakrishnan on Aug 5^th, 2007, 5:37pm

I get [hide]8[/hide] as the smallest N.

Title: Re: Diagonals of a polygon.
Post by Obob on Aug 5^th, 2007, 10:55pm

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

Title: Re: Diagonals of a polygon.
Post by balakrishnan on Aug 6^th, 2007, 5:05am

Sorry I overlooked the problem.
I get N=[hide]24[/hide].

Title: Re: Diagonals of a polygon.
Post by Grimbal on Aug 8^th, 2007, 5:39am

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