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  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Circle size and integers Hello, Bogauss!"]Hi, Quote:
First, relabel 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]  
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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