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February 12th, 2012, 08:31 AM   #1
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Circle size and integers

Hi,

Here is a puzzle.
Can you find the size of a circle (diameter) such as the distances between the 6 points numbered on the diagram above CAN be expressed as INTEGERS

[attachment=0:37jaf8hc]sizecircle.GIF[/attachment:37jaf8hc]

Thank you for any comment.
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February 12th, 2012, 10:02 AM   #2
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Re: Circle size and integers

Hello, Bogauss!"]Hi,

Quote:
Can you find the size of a circle (diameter) such as the distances between
the 6 points numbered on the diagram above can be expressed as integers? [color=beige] . [/color] [color=blue] . . . no[/color]

First, re-label the vertices with
Let be the center of the circle.

Let the radius of the circle be , an integer.
Then the diameters are all integers
and the sides of the hexagon are all integers.


[color=beige]. . [/color]



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February 12th, 2012, 10:21 AM   #3
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Re: Circle size and integers

Hi,

Thank you for your comments.
Assume that you can place the numbers 1 to 6 in any point on the circumference is it still possible?
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February 12th, 2012, 01:15 PM   #4
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Re: Circle size and integers

Does the diameter need to be an integer?
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February 12th, 2012, 02:54 PM   #5
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Re: Circle size and integers

No.
What is required is that all the distances between the numbered points can be expressed as integers.
You can place the 6 distinct points where you want in the circumference.
Only at least one solution is required.
That's it.
I put the picture just to illustrate the case.
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February 17th, 2012, 05:50 PM   #6
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Re: Circle size and integers

I have found this on internet

http://www.contestcen.com/geom.htm

Quote

* 6 Points on a Circle #2
What is the smallest possible radius of a circle such that it is possible to place 6 points on the circumference with the 15 distances between the points being distinct integers?
Solved by: Jean Jacquelin


I did not find the solution yet.
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