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.