The difference between these two equations is
90. In order to test the rule I tried a few more
4 x 4 grids.
5 by 5
After testing a lot of grids I have discovered that the rule for a square of any size is
(n-1)2x10. To prove this I am going to test it for a 6 x 6 grid.
Prediction
I predict that for a 6 x 6 grid the difference will always be 250.
For a 6 x 6 grid the algebraic formula is:
n(n + 55) = n2 + 55n
(n + 5)(n + 50) = n2 + 55n + 250
Difference = 250
Testing
By predicting what the outcome would be with a 6 x 6 grid I have now proved that the rule of (n-1)2x10 does work for a square grid of any size.
Changing the Shape of the Grid
I am now going to change the shape of the grid to see what happens if I use a rectangle grid instead of a square.
the first rectangle I am going to test is a 2 x 3:
Since a 2 x 3 grid has a difference of 20. I will now turn the grid around and try a 3 x 2 grid to see if it has the same outcome.
By testing a 2 x 3 grid and then a 3 x 2 grid I have just proved that a rectangular grid of either size has a difference of 20.
After testing a couple of rectangular grids I have discovered that the rule for a rectangular grid of any size is (L-1)(W-1) x 10. where L = length and W = width
To prove this I am going to test it for a 3 x 4 grid.
For a 3 x 4 grid the algebraic formula is:
y (y + 32) = y2 + 32y
(y + 2) (y + 30) = y2 + 32y + 60
Difference = 60
By predicting what the outcome would be with a 3 x 4 grid I have now proved that the rule of (L-1)(W-1) x 10 does work for a rectangular grid of any size.
Conclusion
All of the grids that I have tested so far have been from a 10 x 10 grid, therefore the differences have all been multiples of 10. I predict that if I change the size of the entire grid from a 10 x 10 to a possible 6 x 6 grid then all of the differences will then be multiples of 6. If I am right in saying this then no matter what size the grid is changed to whether it be 7 x 7 or 8 x 8 the differences will always be multiples of its size. To test my prediction I am going to randomly choose a few square grids from a 6 x 6 grid and see whether my prediction is correct.
6 x 6 grid
After testing a few square grids from a 6 x 6 grid I found the differences to be 6 and 24, these are both multiples of 6. Therefore I have now proved that my prediction is correct.
