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Topic: A ten-sided polygon has the following property ...
Replies: 40   Last Post: Jun 12, 2014 12:44 AM

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 quasi Posts: 12,067 Registered: 7/15/05
Re: A ten-sided polygon has the following property ...
Posted: Jun 11, 2014 1:00 AM

quasi wrote:
>
>For each integer n >= 3, let f(n) be the greatest integer m
>such that there exists an m-gon in the plane whose boundary
>can be covered by n lines.
>
>Some instant observations:
>
>(1) f(n) >= n, for all n.
>
>(2) f is non-decreasing.
>
>An easy observation:
>
>(3) f(n) >= 2n, for all n >= 5.
>
>Some tentative data:
>
> f(3)=3
> f(4)=4
> f(5)=10
> f(6)=12
> f(7)=18
> f(8)=21
> f(9)=32
> f(10)=36
> f(11)=48
>
>Corrections and/or extensions to the above are welcome.
>
>A few conjectures:
>
>(1) f is strictly increasing.
>
>(2) f <= C(n,2), for all n.
>
>(3) As n -> oo, f/C(n,2) -> 1.

Correcting a few otypos ...

A few conjectures:

(1) f is strictly increasing.

(2) f(n) <= C(n,2), for all n.

(3) As n -> oo, f(n)/C(n,2) -> 1.

quasi

Date Subject Author
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6/11/14 quasi
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