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Path: utzoo!linus!decvax!dartvax!chuck
From: chuck@dartvax.UUCP (Chuck Simmons)
Newsgroups: net.puzzle
Subject: Re: High School Math problems:Answers and a toughie!
Message-ID: <2504@dartvax.UUCP>
Date: Tue, 23-Oct-84 04:01:07 EDT
Article-I.D.: dartvax.2504
Posted: Tue Oct 23 04:01:07 1984
Date-Received: Wed, 24-Oct-84 07:53:43 EDT
References: <1975@stolaf.UUCP> <1590@ucla-cs.ARPA> <1214@utah-gr.UUCP>
Organization: Dartmouth College, Hanover, NH
Lines: 34

<Someone should put a stop light at that intersection!>

>>    >Find the greatest number of intersections in an n-gon if all
>>    >vertices are connected. Ex.  If you draw lines connecting all four
>>    >vertices of a quadrilateral, you get one intersection.
>>
>>...every four distinct vertices give rise to one intersection
>>and conversely, each intersection determines four distinct vertices.  So
>>the the answer is the number of ways one can choose four vertices from
>>the given n. That is, it is "n choose 4"...
>
>I'm afraid it's more complicated than that.  For example, this formula
>gives 15 for a hexagon, but a simple picture shows only 13.  This is
>because of the three diagonals that intersect in a single point at the
>center.
>
>I don't have the answer, but it looks like it's back to the drawing
>board.

There are only 13 intersections if it is a regular hexagon.  It seems
reasonable that for arbitrary n-gons, we should be able to find some
skewed irregular version which has all "n choose 4" intersections.  But
it would be soothing to see a simple proof that this is so.  Unfortunately,
my math isn't that good.

-- Chuck
dartvax!chuck

               __
              /  \
             /    \
            /     /
            \____/    Now if this was a mac, I could draw the intersections.