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 2 by 2 box is 10, but I will try another 2 by 2 box just to check this
15 x 26 = 39
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 a 2 by 2 grid is 10, so I assume that if I did another square then I will get the answer of 10 because both grids have given me an answer of 10, but just in case I will do a final 2 by 2 box 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 2 by 2 boxes was 10 because all the 2 by 2 boxes gave me an answer of 10.
Now I am going to try a 3 by 3 square box
11 x 33 = 363
13 x 31 = 403
Diagonal difference: 40
This is different from a 2 by 2 box so I am going to do a few more.
42 x 64 = 2688
44 x 62 = 2728
Diagonal difference: 40
The diagonal difference for a 3 by 3 grid is 40, but I am going to try one more to see if all 3 by 3 boxes give the diagonal difference of 40.
51 x 73 = 3723
71 x 53 = 3763
Diagonal difference: 40
Now I am going to try out a 4 by 4 box. I predict that the diagonal difference will be 90.
55 x 88 = 4840
58 x 85 = 4930
Diagonal difference: 90
Now I am going to try a final box to make sure that all 4 by 4 boxes difference is 90!
4 x 37 = 148
34 x 7 = 238
Diagonal difference: 90
11 x 44 = 484
14 x 41 = 574
Diagonal difference: 90
Now I am going to try a 5 by 5 box
51 x 95 = 4845
55 x 91 = 5005
Diagonal difference: 160
15 x 59 = 885
19 x 55 = 1045
Diagonal difference: 160
The diagonal difference for any 5 by 5 box is 160. I predict that to get the 6 by 6 I will have to square 7 and then minus one and then times this by 10 (as shown below)
52 x 10
For a 6 by 6 box it should be
(6-1)2 x 10
52 x 10
25 x 10
250
The general formula for this is (n-1)2 x 10
So the grid for the general formula would look like the box shown below
Now I am going to try out the rectangular boxes for this I am going to use algebraic formula for each box.
I am going to start of by a 2 by 3 box.
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 another 2 by 3 box but this time I am going to try out my algebraic box first and then the number box.
(x+2) (x+10) – X (x+12)
= x2 + 2x + 10x + 20 -(x2 + 12)
= x2 + 12x + 20 – x2 -12x
= 20
Now I am going to do a number box for this formula
The difference for this should be 20; this is because it should be the same as the other 2 by 3 boxes that I have done above.
41 x 53 = 2173
43 x 51 = 2193
Diagonal difference: 20
Now I am going to try a 2 by 4 rectangular box. I predict that the difference would be 30.
55 x 68 = 3740
58 x 65 = 3770
Diagonal difference: 30
I was correct but to make sure I will do at least 2 more 2 by 4 boxes
14 x 27 = 378
24 x 17 = 408
Diagonal difference: 30
82 x 95 = 7790
85 x 92 = 7820
Diagonal difference: 30
Now I am going to do an algebraic formula for a 2 by 4 rectangular box
(x+3) (x+10) – X (x+13)
= x2 + 3x + 10x + 30 -(x2 + 13)
= x2 + 13x + 40 – x2 -13x
= 40
The answer matches my boxes above. This shows that my algebra is accurate!
Now I am going to try out a 2 by 5 rectangular box.
55 x 69 = 3795
59 x 65 = 3835
Diagonal difference: 40
I am going to try one more to verify that my answer is correct, but I think that my results should be accurate but just to make sure I am going still going to try this out on an algebraic box.
(x+3) (x+10) – X (x+13)
= x2 + 3x + 10x + 30 -(x2 + 13)
= x2 + 13x + 40 – x2 -13x
= 40
Now I am going to try out another algebraic and number box just to verify my working out and to make sure that everything is accurate.
32 x 46 = 1472
36 x 42 = 1512
Diagonal difference: 40
(x+3) (x+10) – X (x+13)
= x2 + 3x + 10x + 30 -(x2 + 13)
= x2 + 13x + 40 – x2 -13x
= 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 when multiplying the bottom left corner of the rectangle by the top right, the product is higher when you multiply the top left of the rectangle by the bottom right. I also 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 use the width as 3 and vary the length as I did previously.
First I am going to start of with a 3 by 4 box
65 x 88 = 5720
85 x 68 = 5780
Diagonal difference: 60
Now I am going to try a 3 by 5 rectangular box
63 x 87 = 5481
83 x 67 = 5561
Diagonal difference: 80
Now I am going to try a 3 by 6 rectangular box.
68 x 83 = 5644
63 x 88 = 5544
Diagonal difference: 100
Now I am going to try a 3 by 7 rectangular box.
22 x 48 = 1056
42 x 28 = 1176
Diagonal difference: 120
Now I am going to do a table to show my results.
I have noticed that this formula works for squares as well as rectangles. I have also noticed that the differences are always one less than the box size, so this made it easier for me to obtain a formula.
The formula for this is (W-1) (L-1) x 10
Now I am going to draw a table using the formula above to predict the boxes with the width of 4.
Now I am going to try and work out the algebraic formula for working out the diagonal differences for all squares.
So for a 6 by 6 box I predict that the diagonal difference would be 250
To show this I will do a number grid and also in algebra. The general differences formula that I predict is (n-1)2 x 10
Now to see if it works!
45 x 100 = 4500
50 x 95 = 4750
Diagonal difference: 250
Now to show a 6 by 6 box in algebra.
(x+50) (x+5)-x(x+55)
= x2+50x+5x+250-(x2+55x)
= x2+55x+250-x2-55x
= 250
I have noticed that my formula works for a 10 by 10 grid size. So now I am going to change the grid size to see if it has the same results or different. I am also trying to find out if this formula will work for any grid size!