February 14th, 2012, 12:22 PM  #1 
Senior Member Joined: Jan 2011 Posts: 560 Thanks: 1  Square and integers
Let a board 8x8. How many points can you place inside a board (the perimeter included) such as the distance between any 2 of those points can be expressed as integer? Now the same question with a board being nxn. Thank you for any comment 
February 14th, 2012, 12:33 PM  #2 
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: Square and integers
Where can the points be placed? In the centers of each of the 64 squares, on any of the 81 intersections of grid lines, or anywhere inside? Can the grid be of any size, or is it 8 units to a side? 
February 14th, 2012, 12:55 PM  #3  
Senior Member Joined: Jan 2011 Posts: 560 Thanks: 1  Re: Square and integers Quote:
You can place the points anywhere inside (included the perimeter). Let me give you a picture Anywhere on the green zone you can place the points. [attachment=0:mke3qqsv]8units.GIF[/attachment:mke3qqsv]  
February 14th, 2012, 01:37 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: Square and integers
You can fit three points on a 1x1 square  actually a square of side length at least (sqrt(2)+sqrt(6))/4 if I do my calculations correctly. You can't put in more in any twodimensional figure until the diameter is at least 2, which corresponds to side length sqrt(2). But that size doesn't actually get you more points  all you can do with the extra space is put three points down the diagonal. I guess that gives a (bad) lower bound: you can fit at least ceil(n * sqrt(2)) points into an n x n square. Hmm. The more I work on this the more it feels like http://www2.stetson.edu/~efriedma/packing.html and the less it feels like number theory. 
February 14th, 2012, 01:57 PM  #5 
Senior Member Joined: Jan 2011 Posts: 560 Thanks: 1  Re: Square and integers
How can you put 3 points in square 1x1 ? For any k points you k*(k1)/2 connections 2 to 2 3 points > 3 connections 4 points .....> 6 connections and so on ALL the connections have to be expressed as integers Here is an example of 4 [attachment=0:1vhv52p9]connections.GIF[/attachment:1vhv52p9] You can place it in any 4 units x 4 units board 
February 14th, 2012, 02:00 PM  #6 
Senior Member Joined: Jan 2011 Posts: 560 Thanks: 1  Re: Square and integers
I know the URL indicated by you but it has nothing to do with our problem. It has to do with circles in circle. 
February 14th, 2012, 02:30 PM  #7  
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: Square and integers Quote:
Quote:
 
February 14th, 2012, 02:36 PM  #8  
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: Square and integers Quote:
 
February 14th, 2012, 02:55 PM  #9 
Senior Member Joined: Jan 2011 Posts: 560 Thanks: 1  Re: Square and integers 
February 14th, 2012, 06:22 PM  #10  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 408  Re: Square and integers Hello, Bogauss! A fascinating question . . . Quote:
Most of what I have discovered has already been posted by others. Code: A * / \ / \ / \ D *    * B \ / \ / \ / * C [color=beige]. . [/color]  

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