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 February 12th, 2012, 08:31 AM #1 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 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.
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 $A,\,B,\,C,\,D,\,E,\,F.$
Let $O$ be the center of the circle.

Let the radius of the circle be $r$, an integer.
Then the diameters $AD,\,BE,\,CF$ are all integers
and the sides of the hexagon $AB,\,BC,\,CD,\,DE,\,EF,\,FA$ are all integers.

$\text{In }\Delta AOC\text{, the Law of Cosines:}$
[color=beige]. . [/color]$AC^2 \:=\:r^2\,+\,r^2\,-\,2r^2\cos(120^o) \:=\:3r^2
AC \:=\:r\sqrt{3},\,\text{ an irrational number.}$

$\text{Hence, the diagonals }\{AC,\,BD,\,CE,\,DF,\,EA,\,FB\}\text{ cannot be integers.}$

 February 12th, 2012, 10:21 AM #3 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 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?
 February 12th, 2012, 01:15 PM #4 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Circle size and integers Does the diameter need to be an integer?
 February 12th, 2012, 02:54 PM #5 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 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.
 February 17th, 2012, 05:50 PM #6 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 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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