quasi
Posts:
10,450
Registered:
7/15/05


Re: A tensided polygon has the following property ...
Posted:
Jun 11, 2014 1:26 AM


quasi wrote: >quasi wrote: >> >>For each integer n >= 3, let f(n) be the greatest integer m >>such that there exists an mgon in the plane whose boundary >>can be covered by n lines.
Clarification:
The mgons are required to be piecewise linear simple closed curves with no pair of adjacent edges lying on the same line.
>>Some instant observations: >> >>(1) f(n) >= n, for all n. >> >>(2) f is nondecreasing. >> >>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(n) <= C(n,2), for all n. > >(3) As n > oo, f(n)/C(n,2) > 1.
quasi

