Not Too Many Points

Probability Level 5

What is the minimum number of points we need on a plane (no 3 points lie on a straight line) such that we are GUARANTEED to be able to form a convex 6-gon with 6 of these points as vertices?


The answer is 17.

Kartik Sharma by Kartik Sharma , 21, India

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1 solution

Kartik Sharma
Jul 7, 2014

This is known as Happy Ending Problem.

This has been found that 17 pints guarantee convex 6-gon, 5 points a 4-gon and 9 points a 5- gon.*

*I know they are not in any sequence.(6,4,5)

This is actually a conjecture - for a convex n-gon, 2^n-2 + 1 points are enough..

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I've edited your question for clarity. Can you review the statement for accuracy?

How would you show that 17 points is enough?

Calvin Lin Staff - 6 years, 11 months ago

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How can I and if I do that will make it easier...

I don't know how to show that 17 points are enough, it is a fact and I hope @Michael Mendrin comes to my 'rescue'.. BTW, it is a geometry problem, I suppose.

Kartik Sharma - 6 years, 11 months ago

The fact that 17 points are necessary was proved by Szekeres & Peters in 2006. They carried out a computer search which eliminated all possible configurations of 17 points without convex hexagons while examining only a tiny fraction of all configurations.

It is unlikely that we (currently) have the means to a solution which doesn't rely on computers.

Calvin Lin Staff - 6 years, 11 months ago

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We can use the conjecture which I think is likely to be true and will be proved, I suppose.The simplicity of the conjecture is the reason for my comment.

Kartik Sharma - 6 years, 11 months ago

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The conjecture has not been proved in general.

I am referring to the specific case of the conjecture when n = 6 n=6 , which has been proven in 2006.

Calvin Lin Staff - 6 years, 11 months ago

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@Calvin Lin – I know I know and I have also said that it 'will' be proved because of it simplicity.(But yes, I should not forget Fermat's Last Theorem;))

Kartik Sharma - 6 years, 11 months ago

Level 5!!!! Oh my god!!!!

Kartik Sharma - 6 years, 11 months ago

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