Number Grid Product Differences Investigation

Above is a 10x10 matrix with 3x3 matrices inside it. To work out the product difference I do:-

The top left x the bottom right – the top right x the bottom left.

I took an example from the grid to find out the product difference.

So the calculation that I did was (75x97) – (77x95) = 40. With this calculation I figured out that the product for a 3x3 was 40.

I will do another 2 examples to prove this

I did (13x35) – (15x33) = 40. This shows that the product difference is the same anywhere in the grid.

I did (18x40) – (20x38) = 40. This shows that the product difference is the same anywhere in the grid.

The examples prove that this does work anywhere on the grid.

4x4

For the 4x4 I did the same as before.

The above is a 10x10 matrix with 4x4 matrices in it.

I took the same calculation to work this out which was: -

The top left x the bottom right – the top right x the bottom left.

This is an example from the grid

I did the calculation of  (45x78) – (48x75) = 90. This is the product difference of 4x4 matrices in a 10x10 matrix.

I will do another 2 examples to prove this

I did the calculation of  (7x40) – (10x37) = 90. This is the product difference of 4x4 matrices in a 10x10 matrix.

I did the calculation of  (61x94) – (64x91) = 90. This is the product difference of 4x4 matrices in a 10x10 matrix.

The examples prove that it works anywhere on the grid.

5x5

For the 5x5 matrices I did the same as before

The above is a 10x10 matrix with 5x5 matrices in it.

I took the same calculation to work this out which was: -

The top left x the bottom right – the top right x the bottom left.

This is an example from the grid

I did the calculation of (33x77) – (37x73) = 160. I had figured out that the product difference was 160 in 5x5 matrices in a 10x10 matrix.

I will do another 2 examples to prove this

I did the calculation of (23x67) – (27x63) = 160.

I did the calculation of (53x97) – (57x93) = 160.

The examples prove that it works anywhere on the grid.

6x6

I did exactly the same as I have done several times.

The above is a 10x10 matrix with 6x6 matrices in it.

I took the same calculation to work this out which was: -

The top left x the bottom right – the top right x the bottom left.

This is an example from the grid

I did the calculation of  (35x90) – (40x85) = 250. This was the product difference for 6x6 matrices in a 10x10 matrix.

I did 2 more examples to prove this.

I did the calculation of  (25x80) – (30x75) = 250. This was the product difference for 6x6 matrices in a 10x10 matrix.

I did the calculation of  (43x98) – (48x93) = 250.

This proves that it works anywhere on the grid.

I then decided to see if I could find a formula.

N.b The Nth term is 1 less than the size of the squares

The second difference is halved and put into the formula. This is because

Now if I wanted to work out the product difference of a 7x7 matrices in a 10x10 matrix I would use the formula of 10n2.

So my calculation would be 10x62 which would = 10x 36 which would be 360. And if I know work out the long way this is what the answer would be.

The above is a 10x10 matrix with 7x7 matrices in it.

I took the same calculation to work this out which was: -

The top left x the bottom right – the top right x the bottom left.

This is an example from the grid

I did the calculation of (34x100) – (40x94) = 360. This product difference is the same as the other answer that we used with the formula.

Another example of this is below to prove that this works.

 

I did the calculation of (2x68) – (8x62) = 360. This product difference is the same as the other answer that we used with the formula.

This proves it works anywhere on the grid.

 So the formula for figuring out product differences in 10x10 matrices is 10n2.

To see if this formula would work in an any size matrix I would use the formula from before and see if this worked with a different sized matrix such as a 10x8 matrix with a 2x2 matrices in it also a 8x8 with a 3x3 matrices in it and also a 6x6 matrix with a 4x4 matrix this will determine whether or not the formula works for an any sized grid.

Above is a 10x8 matrix with 2x2 matrices in it.

Below is an example that I will use to workout the product difference using the formula 10n2. This is to show that this formula does work in any size matrix using any size matrices.

I did (32x43) – (33x42) = 10. This shows that the product difference is the same anywhere in the grid.

I do the calculation of 10x12 =10. This concludes that it works in a 10x 8 matrixes.

I tried it 2 more times just to make sure it worked.

I did (63x74) – (64x73) = 10.

I did (27x38) – (28x37) = 10.

This shows that the product difference is the same anywhere in the grid.

8x8

Above is an 8x8 matrix with 3x3 matrices in it.

Below is an example that I will use to workout the product difference using the formula 10n2.

I did (36x54) – (38x52) = 40. This shows that the product difference is the same anywhere in the grid.

I do the calculation of 10x22 = 40. This shows that the formula works in an 8x8 matrix

I did 2 more examples to prove this

I did (28x46) – (30x44) = 40

I did (17x35) – (19x33) = 40.

This shows that the product difference is the same anywhere in the grid.

6x6

Below is a 6x6 matrix with 4x4 matrices in it.

Below is an example that I will use to workout the product difference using the formula 10n2.

I did (8x29) – (11x26) = 90. This shows that the product difference is the same anywhere in the grid.

I do the calculation of 10x32 = 90. This shows that the product difference is the same in a 6x6 matrix with a 4x4 matrix inside it.

I did 2 more examples to prove this.

I did (15x36) – (18x33) = 90.

I did (7x28) – (10x25) = 90.

This shows that the product difference is the same anywhere in the grid.

The above concludes that the formula 10n2 works in any matrix.

Then I decided to try if could find a rule applied which would work for 3x2 matrices in a 10x10 matrix.

Here is a 10x10 grid. We are trying to investigate product differences within the 10x10 grid. As you can see I have drawn a 3x2 grid inside the 10x10 matrix. To find out the product differences I do: -

The top left number x the bottom right – the top right x the bottom right.

Below is an example to work out

        

I did the calculation of (27x39) – (29x37) = 20

I tried this 2 more times to see if we had the product.

 

I did the calculation of (77x89) – (79x87) = 20

I did the calculation of (77x89) – (79x87) = 20

This proved that the formula worked.

N.b The Nth term is 1 less than the size of the squares

I have done this to help me work out the formula.

With the information above put into a table I figured out the general formula using my knowledge gained from the previous work that the general formula is 10(L-1)(W-1). I came to this by firstly using the 2nd product difference and halving it this is because                                                                                        

This would for different sized matrices in different sized matrix. Examples of this would be like a 3x2 in an 8x8 grid etc.