From the series of these results, a formula can be developed, as shown below.
……. (i)
The above formula is of quadratic format,. It may be pointed out here that if the difference between D & D1 (Figure 1 & 2) is constant, then it would be linear. And if the difference between D1 & D2 (Figure 1 & 2) is constant, then it would be a quadratic. Here the second condition holds true and thus the formula is quadratic.
Figure 1 Figure 2
Difference
-
I substituted the values for the quadratic,
D1
D2
-
Subtracted D1 by D2 to find a constant
Substitute 1
Testing
Both my testing were correct, therefore the formula for finding the difference between products of the diagonally opposite numbers for any square box in a 10 by 10 grid is:-
Where:-
As I have found the formula for any square box in a 10 by 10 grid, I will not try and attempt to find a general formula for any size square box for any size square grid. I will do the same to 12 by 12 grids as I did for 10 by 10 grids.
12 by 12 Grids
Substitute 1
Testing
-
This shows that the formula for any size squares in a 12 by 12 grid is to be: - .
I can observe from the above that a pattern is forming between the two formulas. In , a and c is always equal to the grid-length and b is always equal to the grid-length multiplied by 2.
Hence, I can deduce a general formula for any size square in any grid size. The formula is: -
This is when: -
-
n = grid-length
-
x = square length/height
Proof Testing
8 by 8 Grids
11 by 11 Grids
13 by 13 Grids
14 by 14 Grids
Rectangles
Hypothesis
Example
Analysis
10 by 10 Grids
12 by 12 Grids
8 by 8 Grids
13 by 13 Grids
14 by 14 Grids
11 by 11 Grids
Conclusion
Parallelograms
Hypothesis
When using horizontal parallelograms, the difference of products of the diagonally opposite numbers will be constant throughout the grid.
Example
This is a 3 by 2 parallelogram taken from a 10 by 10 grid.
Analysis
10 by 10 Grids
8 by 8 Grids
11 by 11 Grids
12 by 12 Grids
13 by 13 Grids
14 by 14 Grids
Conclusion
From the above analysis, I observe that the difference of products of the diagonally opposite numbers is constant throughout the grid. This proves the hypothesis as true.
From the set of these results and the pattern, we can derive a formula to determine the difference of products of the diagonally opposite numbers. The formula is: -
…… (iii)
Where: -
-
x = base length of parallelogram
-
y = vertical height of parallelogram
-
n = grid-length
The formula states that ‘to find the difference of products of the diagonally opposite numbers, we need to multiply the grid-length minus one by the base length minus one by the vertical length minus one’. I have based the formula around the properties of rectangles. This is because the parallelogram is basically a rectangle when the angle between the adjacent sides is not 90°. So what I have done is use the formula (ii) in modified format. That is, by changing the slope of the vertical sides.
Testing
8 by 8 Grids
10 by 10 Grids
11 by 11 Grids
12 by 12 Grids
13 by 13 Grids
14 by 14 Grids