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# Pairing points

Best Answer k-man, 03 June 2014 - 07:07 PM

Spoiler for clarification with the drawing
Go to the full post

7 replies to this topic

### #1 bonanova

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Posted 01 June 2014 - 06:06 AM

Take a red pen and touch a sheet of paper with it at n randomly chosen points.

Now add n random points with a blue pen, for a total of 2n points, no three of

which lie on a straight line.

Is it possible in every case to pair the points so that no two of the n lines that

join each red point with its corresponding blue point will cross?

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #2 k-man

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Posted 02 June 2014 - 08:04 PM

It's easy to show with n=2 that it's not possible with lines, so I will assume that you meant line segments connecting a red and a blue points.

Spoiler for combination of two methods

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### #3 bonanova

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Posted 03 June 2014 - 05:00 AM

Sorry.

That was a nice discussion, and I wasn't very clear.

Begin with n red points and n blue points randomly intermingled.

Can n (straight) line (segments) each join a red point to a blue point?

Once drawn they remain in place.

They may not cross.

Spoiler for There is a simple answer if the connecting line (segments) can be curved:

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #4 k-man

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Posted 03 June 2014 - 03:48 PM

Maybe I wasn't clear, but my solution doesn't involve any curved lines. All straight line segments connecting a blue dot with a red dot and they don't cross.

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### #5 k-man

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Posted 03 June 2014 - 07:07 PM   Best Answer

Spoiler for clarification with the drawing

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### #6 bonanova

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Posted 03 June 2014 - 07:38 PM

Nice.

I misinterpreted "points contained by the hull" as "points within the hull."

You were saying perimeter; I thought interior.

BTW there is a very simple idea that does the job

Spoiler for Can you show that
.

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #7 k-man

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Posted 03 June 2014 - 08:31 PM

Spoiler for Can you show that
.

Spoiler for first thoughts...

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### #8 bonanova

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Posted 04 June 2014 - 06:11 AM

Spoiler for Can you show that
.

Spoiler for first thoughts...

That's it. I don't believe it's cyclic.

BTW your solution constructs a non-crossing pairing.

This observation proves that one exists, which is all that is asked, but does not construct it.

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

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