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MultiSquares (Posted on 2008-08-28) Difficulty: 4 of 5
A plane surface is marked with points in a square gridded format.

How many squares,
using the points as their vertices,
can be drawn within a grid of:
1. 6 x 6 points,
2. 7 x 7 points
and
3. n x n points?

In this 3 x 3 array:
   .  .  .
   .  .  .
   .  .  .  
          there are 6.
The 5th envelopes them all and
the 6th uses the outer midpoints 
of the grid as its vertices.

See The Solution Submitted by brianjn    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solutions for 6x6 and 7x7 point | Comment 1 of 2
Where each point is orthogonally 1 unit distant on a grid, the number of squares that may be formed by connecting 4 points...
    ...on a 6x6 point grid, 105 squares:
  • 1 x 52 = [25] of side length 1
  • 1 x 42 = [32] of side length SQRT( 2)
  • 1 x 42 = [16] of side length 2
  • 2 x 32 = [18] of side length SQRT( 5)
  • 1 x 22 = [ 4] of side length SQRT( 8)
  • 1 x 32 = [ 9] of side length 3
  • 2 x 22 = [ 8] of side length SQRT(10)
  • 2 x 12 = [ 2] of side length SQRT(13)
  • 1 x 22 = [ 4] of side length 4
  • 2 x 12 = [ 2] of side length SQRT(17)
  • 1 x 12 = [ 1] of side length 5
    ...on a 7x7 point grid, 196 squares:
  • 1 x 62 = [36] of side length 1
  • 1 x 52 = [25] of side length SQRT( 2)
  • 1 x 52 = [25] of side length 2
  • 2 x 42 = [32] of side length SQRT( 5)
  • 1 x 32 = [ 9] of side length SQRT( 8)
  • 1 x 42 = [16] of side length 3
  • 2 x 32 = [18] of side length SQRT(10)
  • 2 x 22 = [ 8] of side length SQRT(13)
  • 1 x 32 = [ 9] of side length 4
  • 2 x 22 = [ 8] of side length SQRT(17)
  • 1 x 12 = [ 1] of side length SQRT(18)
  • 2 x 12 = [ 2] of side length SQRT(20)
  • 1 x 22 = [ 4] of side length 5
  • 2 x 12 = [ 2] of side length SQRT(26)
  • 1 x 12 = [ 1] of side length 6

The equation(s) for an nxn point grid, for me, becomes a little too complicated. As each additional row and column of points are added, so, too, are the possible side lengths. For example, in addition to increasing the number of existing side length squares to the next "square" (x1 or x2), an 8x8 point grid adds additional squares of side lengths of 5 [= SQRT(32+42)] and 5*SQRT(2) [= SQRT(12+72)]. 
  Posted by Dej Mar on 2008-08-28 12:16:55
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