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        riddles >> hard >> Six points in the plane
        
(Message started by: NickH on Apr 20^th, 2006, 6:03am)

Title: Six points in the plane
Post by NickH on Apr 20^th, 2006, 6:03am

Let the minimum distance between any two of six points in the plane be m, and the maximum distance between any two of the six be M. Find the minimum possible value of M/m.

Title: Re: Six points in the plane
Post by SMQ on Apr 20^th, 2006, 6:35am

Well, as a quick upper limit, I see two obvious ways to achieve M/m = 2.

--SMQ

Title: Re: Six points in the plane
Post by towr on Apr 20^th, 2006, 7:02am

I think I can get ~1.9

Title: Re: Six points in the plane
Post by SMQ on Apr 20^th, 2006, 7:39am

Would that be [hide]1/(2sin([pi]/12))[/hide] ~= 1.932?

[hide]Start with three points in an equilateral triangle. With each point as a center, construct the short arc which has the other two points as endpoints. Place the remaining three points at the midpoints of the arcs.[/hide]

--SMQ

Title: Re: Six points in the plane
Post by NickH on Apr 20^th, 2006, 9:41am

Or is it [hide]2*sin(72°)[/hide] ~= 1.902?

[hide]The vertices of a regular pentagon, together with its center.[/hide]

Title: Re: Six points in the plane
Post by towr on Apr 20^th, 2006, 10:01am

The latter, ~1.902
It seemed the most obvious to me, after a hexagon

Title: Re: Six points in the plane
Post by Oyibo on Apr 21^st, 2006, 10:32am

How many points can be placed in a plane such that the ratio M/m does not exceed 3?

Title: Re: Six points in the plane
Post by Grimbal on Apr 27^th, 2006, 8:46am

I can do 12. Fore more, I'll have to think.

[hide](understand it on a regular triangular grid)
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