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 February 14th, 2012, 11:22 AM #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, 11:33 AM #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, 11:55 AM   #3
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Re: Square and integers

Quote:
 Originally Posted by CRGreathouse 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?
8 units a side.
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, 12:37 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 two-dimensional 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, 12:57 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*(k-1)/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, 01:00 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, 01:30 PM   #7
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Re: Square and integers

Quote:
 Originally Posted by Bogauss How can you put 3 points in square 1x1 ?
One point at (0, 0), one point at (0, 1), one point at (sqrt(3/4), 1/2).

Quote:
 Originally Posted by Bogauss I know the URL indicated by you but it has nothing to do with our problem. It has to do with circles in circle.
The page certainly deals with more than that! The connection is clear to me but I'll not explain it if you don't see it; it's an incidental point.

February 14th, 2012, 01:36 PM   #8
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Re: Square and integers

Quote:
 Originally Posted by Bogauss Here is an example of 4 [attachment=1:25x7oe16]connections.GIF[/attachment:25x7oe16] You can place it in any 4 units x 4 units board
I already showed how to get 6 points on a 4x4 board.
Attached Images
 4x4.png (484 Bytes, 355 views)

 February 14th, 2012, 01:55 PM #9 Senior Member   Joined: Jan 2011 Posts: 560 Thanks: 1 Re: Square and integers
February 14th, 2012, 05:22 PM   #10
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Re: Square and integers

Hello, Bogauss!

A fascinating question . . .

Quote:
 Let a board be 8-by-8 inches. How many points can you place inside a board (the perimeter included) such that [color=beige]. . [/color]the distance between [color=blue]any two[/color] of those points can be expressed as integer? Now the same question with a board being n-by-n.

Most of what I have discovered has already been posted by others.

$\text{We cannot use a triangular grid.}$

Code:
          A
*
/ \
/   \
/     \
D * - - - * B
\     /
\   /
\ /
*
C
$\text{Although {AB,\,BC,\,CD,\,DA,\:\text{ and }BD\:\text{ are integral lengths,}$
[color=beige]. . [/color]$AC\text{ is not.}$

$\text{We cannot have any squares.}
\text{The sides of a square may be integers,}
\text{\;\;\;\;\;but its diagonals are not.}$

 Tags integers, square

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