71x93=6603
73x91=6643
6643-6603=40
Here is a table for the difference for a ten by ten grid:
The difference is calculated by the (length of the square – 1)² x10, that is how I was able to calculate the other differences of the different length of square.
(L-1)² x 10= difference [algebraic form]
L = length of the square
The limitations with this formula are that the shape has to be a square, the length of the grid has to be 10 and the constant difference between each number has to be 1.
(n + a (L-1)) x (n + (L-1)) – (n x (n + a (L-1) + (L-1)= difference
I then simplified the above equation to make:
a( L² -2L + 1) = difference
a is the number of how many the grid goes across
The limitations to this formula are that the shape has to be a square and the constant difference between the numbers has to be one.
I then investigated whether there was a different formula for rectangles
(2x3 on 10x10 grid)
1x13=13
3x11=33
33-13=20
I then changed the length of the rectangle to 4 but kept the height and the size of the grid the same
1x14=14
4x11=44
44-14=30
Using the above figures I was able to come up with a formula:
n + a (h-1)) x (n + (L-1)) – (n x (n + a (h-1) + (L-1))) = difference
I then simplified the formula to:
a (Lh – h – L + 1) = difference
L is the length of the rectangle
h is the height of the rectangle
a is the number of how many the grid goes across
The limitations to this formula are that the constant difference between each number has to be one. The formula for the rectangle works for both the largest amount of numbers on the horizontal and vertical side.
I have found that the formulae for the square and the rectangle are very similar but in the rectangle formula the height of the rectangle is included but for a square the height is the same as the length so the height value is not needed in the square formula. So a formula for four sided shapes on a grid of numbers with a constant difference of 1 is a (Lh – h – L + 1), the a is how much the grid goes across, the h is the height of the shape and the L is the length of the shape.
Evaluation
Rather than simplifying the formulae step by step I have jumped from the starting formula to the ending one, in the future I shall simplify my formulae step by step.
