Opposite Corners Investigation
This investigation is about finding the difference between the products of the opposite corner numbers in a number square. There are three variables which I can change whilst doing my investigation, they are the size of the grid, the shape of the grid and the numbers inside the grid. I am going to start by looking only at number squares with consecutive numbers
Consecutive Numbers
4 x 13 = 52
1 x 16 = 16
Difference = 36
0 x 15 = 0
3 x 12 = 36
Difference = 36
1 x 10 = 10
-2 x 13 = -26
Difference = 36
The difference seems to be the same, for these 3 the answer is 36 but this isn't proof.
Let X stand for the start number which can be any real number.
(X + 3) (X + 12) = X2 + 3X + 12X + 36
= X2 + 15X + 36
X (X + 15) = X2 + 15X
Difference = 36
So, the difference between the products of the opposite corner numbers in a 4x4 number square is 36. What about a 3x3 number square?
(X + 2) (X + 6) = X2 + 2X + 6X +12
= X2 + 8X +12
X (X + 8) = X2 + 8X
Difference = 12
So, the difference between the products of the opposite corner numbers in a 3x3 number square is 10. What about Other squares?
This investigation does not work with a square size of 1x1, as the square does not have four corners.
(X + 1) (X + 2) = X2 + X + 2X +2
= X2 + 3X +2
X (X + 3) = X2 + 3X
2
(X + 4) (X + 20) = X2 + 4X + 20X + 80
= X2 + 24X + 80
X (X + 24) = X2 + 24X
= 80
(X + 5) (X + 30) = X2 + 5X + 30X + 150
= X2 + 35X + 150
X (X + 35) = X2 + 35X
150
Predict + check
Looking at the patterns of numbers from my tables of results it appears for a grid size of NxN the difference is N(N - 1)2. I predict that for a 10x10 grid the difference will be 10 x 92 = 10 x 81 = 810.
I will check by drawing.
(X + 9) (X + 90) = X2 + 9X + 90X + 810
= X2 + 99X + 810
X(X + 99) = X2 + 99X
810
The check shows that the predicted formula is correct. But this is not proof.
(X + (N-1)) (X + N(N-1)) = X2 + X(N-1) + X(N(N-1)) + N(N-1)(N-1)
X (X + N(N-1) + N-1)) = X2 + X(N-1) + X(N(N-1))
N(N-1)(N-1)
=N(N-1)2
This formula is the same as before so I have proved my prediction.
Grid within a grid
The formula that I have figured out works for any sized square with a consecutive number grid but what about a grid within a grid?
I'm now going to see whether the corners have any algebraic relation to each other.