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.

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