Investigate the diagonal difference of a 2 by 2 grid inside a 10 by 10 grid

Results

First I am going to try out all the squares then I will start with the rectangular.

I will first find the sequence of the diagonal difference in all of the different size grids.

 I am trying to calculate the differences between these products.

I am going to multiply the opposite corners and then find the difference.

11 x 22= 242

12 x 21= 252

The diagonal difference is 242– 252 = 10

I have found out that the diagonal difference for the two by two grid is 10, but I will try another two by two grid just to check this.

 15 x 26 = 390

 16 x 25 = 400

Diagonal difference: 400 – 390 = 10

The diagonal difference is 10 again.

I again found that the diagonal difference is 10 so I know that the diagonal difference of two by two grids is 10, so I assume that if I did another square then I will get the answer of 10 because both grids have gave me an answer of 10, but just in case I will do a final two by two grid to prove that the diagonal difference is 10.

27 x 38 = 1026

28 x 37= 1036

Diagonal difference: 1026– 1036 = 10

After doing this, I found out that the diagonal difference of two by two grids was 10 because all the two by two grids gave me an answer of 10.

Now I am going to try a 3 by 3 square grid.

11 x 33 = 363

13 x 31 = 403

Diagonal difference: 40

Now I am going to try another 3 by 3 grid to show if my diagonal difference is correct.

42 x 64 = 2688

44 x 62 = 2728

Diagonal difference: 40

The diagonal difference for a 3 by 3 grid is 40.

Now I am going to try out a 4 by 4 grid. My prediction is that the diagonal difference will be 90 because it is one below and then times by 10.

55 x 88 = 4840

58 x 85 = 4930

Diagonal difference: 90

My prediction was correct. Now I am going to try another to make sure that I didn’t do anything wrong.

11 x 44 = 484

14 x 41 = 574

Diagonal difference: 90

Now I am going to try a 5 by 5 grid.

51 x 95 = 4845

55 x 91 = 5005

Diagonal difference: 160

Now I am going to try out work out the algebraic formula for working out the diagonal differences for all squares.

So for a 6 by 6 grid I predict that the diagonal difference would be; 250.

To show this I will do a number grid and also in algebra.

The general difference formula that I predict is (n-1)2 x 10.

Now to show that it works!

45 x 100 = 4500

50 x 95 = 4750

Diagonal difference: 250

Now to show this in an algebraic form.

(x+50) (x+5)-x(x+55)

          = x2+50x+5x+250-(x2+55x)

          = x2+55x+250-x2-55x

          = 250

My formula works. So if I were to do a 7 by 7 I would do 62x10.

Now I am going to put my results in a table.

 

I have shown that for a 5 by 5 square grid you will have to 42x10 to get the answer, so if I wanted to do a 7 by 7 grid I would have to do; 62x10

7 x 7 it would be (7-1)2x10

                = 62 x 10

                =36 x 10

                =360

As I have said before that the general formula is:

(n-1)2x10

So the grid would look like this:

Now I a going to try out the rectangular grids. For this I will try to use an algebraic formula for each grid.

I am going to start of with a 2 by 3 rectangular grid.  

35 x 47 = 1645

45 x 37 = 1665

Diagonal difference: 20

Now in algebra:

(x+2) (x+10) - x(x+12)

                    = x2+2x+10x+20-(x2+12)

                    = x2+12x+20-x2-12x

                =        20

Now I am going to try a 2 by 4 rectangular grid. I predict that the difference would be 30.

 55 x 68= 3740

58 x 65= 3770

Diagonal difference: 30

I was correct.

Now I am going to do an algebraic formula for a 2 by 4 rectangular grid.

         (x+3) (x+10) – x(x+13)

          = x2+3x+10x+30-x2+13x

          = x2+13x+30-x2-13x

          = 30

This matches my grid above. This shows that my algebra is accurate.

Now I am going to try a 2 by 5 rectangular grid.

32 x 46= 1472

36 x 42= 1512

Diagonal difference: 40

(x+4) (x+10)-x(x+14)

                      = x2+14x+10x+40-(x2+14x)

                  = x2+14x+40-x2-14x

                  = 40

Now I am going to put my results in a table, for the working out that is shown above.

During the investigation I have discovered that my research is correct and when I observed my results using algebra the outcomes were the same as to when I used numbers.

Now I am going to change the width to 3 and keep the length the same.

Now I am going to try a 3 by 4 rectangular grid.

65 x 88 = 5720

85 x 68 = 5780

5780-5730 = 60

Now I am going to try a 3 by 5 rectangular grid

63 x 87=5481

83 x 67=5561

Diagonal difference: 80

Now I am going to try a 3 by 6 rectangular grid.

68 x 83 = 5644

63 x 88 = 5544

Diagonal difference: 100

Now I am going to do a 3 by 7 rectangular grid.

22 x 48 = 1056

42 x 28 = 1176

Diagonal difference: 120

Now I am going to do a table to show my results.