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# Between six towns

Best Answer bushindo, 12 April 2013 - 02:43 AM

Spoiler for PROOF pt.3

I believe there is a subtle error here

Spoiler for
Go to the full post

20 replies to this topic

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Posted 06 April 2013 - 03:30 AM

The smallest distance between any two of six towns is m miles. The largest distance between any two of the towns is M miles. Show that M/m . Assume the land is flat.

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### #2 dark_magician_92

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Posted 06 April 2013 - 12:01 PM

Can the arrangment be 3-D? i guess not.

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### #3 bonanova

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Posted 07 April 2013 - 09:33 AM

Spoiler for Close

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### #4 bonanova

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Posted 08 April 2013 - 05:47 AM

I would be interested to see the configuration that makes M = m * sqrt(3)

Spoiler for Graphical lower bound

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Posted 09 April 2013 - 05:56 PM

Show that, unless the towns are all collinear, it is always possible to choose three of them so that they form a triangle with maximum angle of at least 120°.

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Posted 09 April 2013 - 06:09 PM

I marked it solved originally but just realized you didn't show equality yet.  Sorry.

I would be interested to see the configuration that makes M = m * sqrt(3)

Spoiler for Graphical lower bound

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### #7 k-man

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Posted 10 April 2013 - 07:21 PM

I think you should mark this as solved. There is no way to place 6 points on a surface in a way that the minimum distance between any 2 points was m and the maximum distance between any 2 points was m*sqrt(3).

Spoiler for Informal proof that you can do at most 5

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### #8 bonanova

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Posted 10 April 2013 - 07:47 PM

I am exploring one more possibility.

Spoiler for In my previous post

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### #9 k-man

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Posted 10 April 2013 - 08:58 PM

I am exploring one more possibility.

Spoiler for In my previous post

Spoiler for that radius is...

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### #10 bonanova

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Posted 10 April 2013 - 09:48 PM

I won't spoiler this because it's not a solution but a refutation.

Consider a regular pentagon with side t and points at center and vertices.

The radius r of the circumcircle is the minimum distance among the six points.

The diagonal d is the maximum distance among the six points.

t = 2r sin 36o = 1.1756 r

d = PHI t where PHI is the golden ratio = 1.618...

M/n = d/r = 2 PHI sin 36o = 1.902 = sqrt (3.618)  > sqrt (3).

This is the answer from my post 3 where I conjectured a more dense arrangement of points.

I don't believe there is a more dense configuration of six points.

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