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        riddles >> medium >> 6 points in a unit circle
        
(Message started by: ecoist on Sep 17^th, 2007, 2:58pm)

Title: 6 points in a unit circle
Post by ecoist on Sep 17^th, 2007, 2:58pm

Show that, given any 6 points inside a circle of radius 1, some two of the given points are at most 1 unit apart.

Title: Re: 6 points in a unit circle
Post by towr on Sep 17^th, 2007, 3:22pm

[edit]misread the problem[/edit]
[hide]Evenly spaced they're all 1 unit apart. So there isn't enough room for them all to be further apart.
If any are further apart, the total distance going from one neighbour to the next around the circle back to the first even goes down

Hmmm, you probably want something more rigorous. I wonder if holy pigeons have anything to do with it..[/hide]

Title: Re: 6 points in a unit circle
Post by ecoist on Sep 17^th, 2007, 4:50pm

I never thought that this problem was obvious, yet many people believe that it is obvious. I have seen three different rigorous proofs.

Title: Re: 6 points in a unit circle
Post by Aryabhatta on Sep 17^th, 2007, 8:28pm

on 09/17/07 at 16:50:53, ecoist wrote:

I never thought that this problem was obvious, yet many people believe that it is obvious. I have seen three different rigorous proofs.

To be fair, I am pretty sure towr is not claiming that this is obvious.

And I agree with you, many people think things are obvious when they actually are not. For instance, the Jordan Curve Theorem.

omigod. Do i sound like srn347?

Title: Re: 6 points in a unit circle
Post by ecoist on Sep 17^th, 2007, 9:32pm

Sorry, left the wrong impression. towr clearly sees that the problem is not obvious. Others thought the problem was obvious, but I couldn't convince them otherwise. And no, Aryabhatta, I doubt if even the ventriloquist who won "America's Got Talent", could sound like srn347!

Title: Re: 6 points in a unit circle
Post by towr on Sep 18^th, 2007, 12:15am

Oh wait, it says in the circle (on the disk), not on the circle (circumference).
I need to read better.

Title: Re: 6 points in a unit circle
Post by mikedagr8 on Sep 18^th, 2007, 12:32am

Seems like I should be able to do this...

Title: Re: 6 points in a unit circle
Post by Grimbal on Sep 18^th, 2007, 6:47am

You could consider the radius (line from the center to the border) where each point is and prove that if the distance between 2 points is >=1, the angle between their radii is >=pi/3.

Title: Re: 6 points in a unit circle
Post by rmsgrey on Sep 18^th, 2007, 8:23am

How about:

[hideb]
If any of the points are the centre, then the rest of the circle is at most 1 away from it, so done.

Otherwise, pick any of the 6 points and draw the diameter through it, and the two diameters Pi/3 radians away from it, dividing the circle into 6 equal wedges.

Each wedge has the property that any two points in it are at most 1 apart, so if any wedge has two points in, we're done.

But our original point is on the boundary between two wedges (by construction) so counts as being "in" both of them, leaving only 4 other wedges and 5 other points, so at least one wedge must have two points in, so we're done.[/hideb]

Not sure how obvious that was.

Title: Re: 6 points in a unit circle
Post by ecoist on Sep 18^th, 2007, 8:44am

Don't know about others, but I would like a proof of the obvious(?)

[hide]Each wedge has the property that any two points in it are at most 1 apart.[/hide]

Omigod! Am I sounding like srn347?

Title: Re: 6 points in a unit circle
Post by Grimbal on Sep 18^th, 2007, 9:08am

[hide]A wedge fits in a Reuleaux Triangle (http://mathworld.wolfram.com/ReuleauxTriangle.html) which has constant width 1. Obviously it cannot contain a segment of length>1.[/hide]

Title: Re: 6 points in a unit circle
Post by joefendel on Sep 18^th, 2007, 9:10am

Strikes me that this is equivalent to the question of asking how many non-overlapping disks of radius 0.5 we can fit in the disk of radius 1.5.

Any mileage we can get from that formulation?

Title: Re: 6 points in a unit circle
Post by towr on Sep 18^th, 2007, 9:28am

on 09/18/07 at 09:10:13, joefendel wrote:

Strikes me that this is equivalent to the question of asking how many non-overlapping disks of radius 0.5 we can fit in the disk of radius 1.5.

Any mileage we can get from that formulation?

You could put 7 such .5 radius circles in the 1.5 radius disc. However, not without them touching (having a non-zero distance between them).

Title: Re: 6 points in a unit circle
Post by Michael_Dagg on Sep 18^th, 2007, 11:02am

Indeed so, otherwise you could place six discs of radius r > 1/2
inside a disc of radius 1+r; equivalently, you could place six unit discs
inside a disk of radius s = (1+r)/r < 3 . Well, you can indeed
place not just six but seven unit disks inside such a disk when s = 3 ;
that's the "kissing pennies" arrangement. But it's a very tight arrangement:
increase the radius just a little and not only do the six pennies around
the outside crash into each other, but they crash into the center one too!

Title: Re: 6 points in a unit circle
Post by JP05 on Sep 18^th, 2007, 6:26pm

Isn't this just a circle packing problem?