(a+2)(a+20) = a² + 22a + 40
(a² + 22a + 40) – (a² + 22a) = 40
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 3 X 3 grid inside a 10 X 10 grid will always be 40.
Now I will draw a 3 X 3 grid to test if my theory is correct.
41 x 63 = 2583
43 x 61 = 2623
2623 – 2583 = 40
I have tested my theory by using numbers. The difference is 40 so therefore it looks like my theory is right.
The Difference for any 3 X 3 grid is always 40 inside a 10 X 10 grid.
4 X 4 Grid
Now I will look at 4 X 4 grids within a 10 X 10 grid. Again I will work this out algebraically.
Here is the algebraic grid for a 4 X 4 grid within a 10 X 10 grid.
In my investigation I have to find the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number. So therefore I will multiply ‘a’ by ‘a+33’ and also I will multiply ‘a+3’ by ‘a+30’ and find the difference.
Therefore:
a(a + 33) = a² + 33a
(a + 3)(a + 30) = a² + 33a + 90
(a² + 33a + 90) – (a² + 33a) = 90
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 4 X 4 grid inside a 10 X 10 grid will always be 90.
Now I will draw a 4 X 4 grid to test if my theory is correct.
16 x 49 = 784
19 x 46 = 874
874 – 784 = 90
I have tested my theory by using numbers. The difference is 90 so therefore it looks like my theory is right.
The Difference for any 4 X 4 grid is always 90 inside a 10 X 10 grid.
5 X 5 Grid
Now I will look at 5 X 5 grids within a 10 X 10 grid. Again I will work this out algebraically.
Here is the algebraic grid for a 5 X 5 grid within a 10 X 10 grid.
Therefore:
a(a + 44) = a² + 44a
(a + 4)(a + 40) = a² + 44a + 160
(a² + 44a + 160) – (a² + 44a) = 160
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 5 X 5 grid inside a 10 X 10 grid will always be 160.
Now I will draw a 5 X 5 grid to test to see if my theory is correct.
21 x 65 = 1365
25 x 61 = 1525
1525 – 1365 = 160
I have tested my theory by using numbers. The difference is 160 so therefore it looks like my theory is right.
The Difference for any 5 X 5 grid is always 160 inside a 10 X 10 grid.
Now I am going to try and find out the formula that finds the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any size square.
Results
After looking at my table I have found out that if I subtract 1 from the square size and square it, the difference is always 10 times the number that I get. Therefore if I multiply the number by 10 I will get the difference. For example 4 –1 = 3, 3² = 9, 9 x 10 = 90 and the difference of the 4 X 4 square is 90.
Therefore the formula to work out the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any size square in a 10 X 10 grid is:
d = 10(s – 1)²
Now I will test my formula to see if it works. I am going to use an 8 X 8 grid to test to see if my formula works.
d = 10(s – 1)²
d = 10(8 – 1)²
d = 10 x 7²
d = 10 x 49
d = 490
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an 8 X 8 grid should be 490
23 x 100 = 2300
30 x 93 = 2790
2790 – 2300 = 490
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an 8 X 8 grid is 490. This proves that my formula is correct. But my formula wont work for a 1 X 1 grid or any grid that goes out of the 10 X 10 grid.
Rectangles
Now I a going to try and find the formula for the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in a rectangle. I will do this by finding the formula for changing one side, then I will find the formula for changing the other side and combine the two together.
First I will look at 2 X 3 grids within a 10 X 10 square. Also I will work out the differences algebraically.
Here is an algebraic grid for a 2 X 3 grid within a 10 X 10 grid.
Therefore:
a(a + 12) = a² + 12a
(a + 2)(a + 10) = a² + 12a + 20
(a² + 12a + 20) – (a² + 12a) = 20
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 2 X 3 grid inside a 10 X 10 grid will always be 20.
Now I will draw a 2 X 3 grid to test if my theory is correct.
53 x 65 = 3445
55 x 63 = 3465
3465 – 3445 = 20
I have tested my theory by using numbers. The difference is 20 so therefore it looks like my theory is right.
The Difference for any 2 X 3 grid is always 20 within a 10 X 10 grid.
2 X 4 Grid
Here is an algebraic grid for a 2 X 4 grid within a 10 X 10 grid.
Therefore:
a(a + 13) = a² + 13a
(a + 3)(a + 10) = a² + 13a + 30
(a² + 13a + 30) – (a² + 13a) = 30
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 2 X 4 grid inside a 10 X 10 grid will always be 30.
Now I will draw a 2 X 4 grid to test if my theory is correct.
71 x 84 = 5964
74 x 81 = 5994
5994 – 5964 = 30
I have tested my theory by using numbers. The difference is 30 so therefore it looks like my theory is right.
The Difference for any 2 X 4 grid is always 30 within a 10 X 10 grid.
2 X 5 Grid
Here is an algebraic grid for a 2 X 5 grid within a 10 X 10 grid.
Therefore:
a(a + 14) = a² 14a
(a + 4)(a + 10) = a² + 14a + 40
(a² + 14a + 40) – (a² 14a) = 40
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 2 X 5 grid inside a 10 X 10 grid will always be 40.
Now I will draw a 2 X 5 grid to test if my theory is correct.
86 x 100 = 8600
90 x 96 = 8640
8640 – 8600 = 40
I have tested my theory by using numbers. The difference is 40 so therefore it looks like my theory is right.
The Difference for any 2 X 5 grid is always 40 within a 10 X 10 grid.
2 X 6 Grid
Here is an algebraic grid for a 2 X 5 grid within a 10 X 10 grid.
Therefore:
a(a + 15) = a² + 15a
(a + 5)(a + 10) = a² + 15a + 50
(a² + 15a + 50) – (a² + 15a) = 50
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 2 X 6 grid inside a 10 X 10 grid will always be 50.
Now I will draw a 2 X 6 grid to test if my theory is correct.
13 x 28 = 364
18 x 23 = 414
414 – 364 = 50
I have tested my theory by using numbers. The difference is 50 so therefore it looks like my theory is right.
The Difference for any 2 X 6 grid is always 50 within a 10 X 10 grid.
Now I am going to try and find out the formula that finds the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any rectangle were I change the length but keep the width the same.
Results
After looking at my table I have found out that the difference is 10 subtracted off 10 multiplied the length of the rectangle. For example 5 x 10 = 50, 50 - 10 = 40 and the difference when the length of the rectangle is 5 is 40.
Therefore the formula to work out the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any rectangle were I change the length but keep the width the same is:
10n – 10 simplified to;
d = 10(L – 1)
Now I will test my formula to see if it works. I am going to use an 2 X 7 grid to test to see if my formula works.
d = 10(l – 1)
d = 10(7 – 1)
d = 10 x 6
d = 60
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an 2 X 7 grid should be 60
51 x 67 = 3417
57 x 61 = 3477
3477 – 3417 = 60
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in a 2 X 7 grid is 60. This proves that my formula is correct. But my formula wont work for a 1 X 2, a 2 X 1 grid or any grid that goes out of the 10 X 10 grid.
Now I will find the formula for rectangles where I change the width and keep the length the same.
3 X 2 Grid
Here is an algebraic grid for a 3 X 2 grid within a 10 X 10 grid.
Therefore:
a(a + 21) = a² + 21a
(a +1)(a + 20) = a² + 21a + 20
(a² + 21a + 20) – (a² + 21a) = 20
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 3 X 2 grid inside a 10 X 10 grid will always be 20.
Now I will draw a 3 X 2 grid to test if my theory is correct.
22 x 43 = 946
23 x 42 = 966
966 – 946 = 20
I have tested my theory by using numbers. The difference is 20 so therefore it looks like my theory is right.
The Difference for any 3 X 2 grid is always 20 within a 10 X 10 grid.
4 X 2 Grid
Therefore:
a(a + 31) = a² + 31a
(a + 1)(a + 30) = a² + 31a + 30
(a² + 31a + 30) – (a² + 31a) = 30
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 4 X 2 grid inside a 10 X 10 grid will always be 30.
Now I will draw a 4 X 2 grid to test if my theory is correct.
6 x 37 = 222
7 x 36 = 252
253 – 222 = 30
I have tested my theory by using numbers. The difference is 30 so therefore it looks like my theory is right.
The Difference for any 4 X 2 grid is always 30 within a 10 X 10 grid.
5 x 2 grid
Therefore:
a(a + 41) = a² + 41a
(a + 1)(a + 40) = a² + 41a + 40
(a² + 41a + 40) – (a² + 41a) = 40
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 5 X 2 grid inside a 10 X 10 grid will always be 40.
Now I will draw a 5 X 2 grid to test if my theory is correct.
47 x 88 = 4136
48 x 87 = 4176
4176 – 4136 = 40
I have tested my theory by using numbers. The difference is 40 so therefore it looks like my theory is right.
The Difference for any 5 X 2 grid is always 40 within a 10 X 10 grid.
6 X 2 Grid
Therefore:
a(a + 51) = a² + 51a
(a + 1)(a + 50) = a² + 51a + 50
(a² + 51a + 50) – (a² + 51a) = 50
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 5 X 2 grid inside a 10 X 10 grid will always be 40.
Now I will draw a 5 X 2 grid to test if my theory is correct.
42 x 93 = 3906
43 x 92 = 3956
3956 – 3906 = 50
I have tested my theory by using numbers. The difference is 50 so therefore it looks like my theory is right.
The Difference for any 6 X 2 grid is always 50 within a 10 X 10 grid.
Now I am going to try and find out the formula that finds the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any rectangle were I change the width but keep the length the same.
Results
After looking at the table I have noticed that the results are exactly the same as the ‘length of rectangle’ table so therefore the formula should be the same, apart from the fact that you have to change the letter ‘l’ to the letter ‘w’.
Therefore the formula to work out the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any rectangle were I change the width but keep the length the same is:
d = 10(w-1)
Now I will test my formula to see if it works. I am going to use an 8 X 2 grid to test to see if my formula works.
d = 10(w – 1)
d =10(8 – 1)
d = 10 x 7
d = 70
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an 8 X 2 grid should be 70
3 x 74 = 222
4 x 73 = 296
296 – 222 = 70
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an 8 X 2 grid is 70. This proves that my formula is correct. But my formula wont work for a 1 X 2, a 2 X 1 grid or any grid that goes out of the 10 X 10 grid.
Now I will combine the ‘length of rectangles’ formula with the ‘width of rectangles’ formula to get a formula that finds any rectangle.
Any Rectangle
To get the formula to find any rectangle I need to combine, the formula that finds the difference when you change length of the rectangle and the formula that finds the difference when you change the width of the rectangle.
Formula for length of rectangle; d = 10(L – 1)
Formula for width of rectangle; d = 10(w – 1)
If I multiply these two formulae together I should get the formula for any rectangle.
Therefore the formula should be:
d = 10(L – 1)(w – 1)
Now I will test my formula to see if it works. I am going to use an 4 X 5 grid to test to see if my formula works.
d = 10(L – 1)(w – 1)
d = 10(5 – 1)(4 – 1)
d = 10 x 4 x 3
d = 120
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an 4 X 5 grid should be 120
16 x 50 = 800
20 x 46 = 920
920 – 800 = 120
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an 4 X 5 grid is 120. This proves that my formula is correct. But my formula wont work for a 1 X 2, a 2 X 1 grid or any grid that goes out of the 10 X 10 grid.
The formula to get any rectangle inside a 10 X 10 grid is:
d = 10(L – 1)(w – 1)
Formulas
Formula for any square: d = 10(n – 1)² or d = 10(n – 1)(n – 1)
Formula for any rectangle: d = 10(L – 1)(w – 1)
After looking at these two formulae I have noticed that they are virtually the same, ‘L’ and ‘w’ are both lengths of sides and so is ‘n’, but (n – 1) is squared because both the sides are the same in any square. Also if you expand the formula for any square a little bit it is the same formula as the formula for any rectangle but using ‘n’ instead of ‘L’ and ‘w’ and they all stand for the lengths of the sides except ‘L’ and ‘w’ stand for the length of one side and ‘n’ stands for the length of both sides. This is easier to see when the formula for any square is expanded to ‘d = 10(n – 1)(n – 1).
Grid Size
Now I a going to try and find the formula for the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any square or rectangle in any size grid. To find this I will use the formula that I found to get any rectangle, as it will also work for squares.
First I will look at different size grids within a 12 X 12 square. Also I will work out the differences algebraically.
Here is the 12 X 12 grid that I will be using.
First I will look at 3 X 3 grids within a 12 X 12 square. Also I will work out the differences algebraically.
Here is an algebraic grid for a 3 X 3 grid within a 12 X 12 grid.
a(a + 26) = a² + 26a
(a + 2)(a + 24) = a² + 26a + 48
(a² + 26a + 48) – (a² + 26a) = 48
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 3 X 3 grid inside a 12 X 12 grid will always be 48.
Now I will draw a 3 X 3 grid to test if my theory is correct.
4 x 30 = 120
6 x 28 = 168
168 – 120 = 48
I have tested my theory by using numbers. The difference is 48 so therefore it looks like my theory is right.
The Difference for any 3 X 3 grid is always 48 within a 12 X 12 grid.
4 x 4 Grid
a(a + 39) = a² + 39a
(a + 3)(a + 36) = a² + 39a + 108
(a² + 39a + 108) – (a² + 39a) = 108
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 4 X 4 grid inside a 12 X 12 grid will always be 108.
Now I will draw a 4 X 4 grid to test if my theory is correct.
67 x 106 = 7102
70 x 103 = 7210
7210 – 7102 = 108
I have tested my theory by using numbers. The difference is 108 so therefore it looks like my theory is right.
The Difference for any 4 X 4 grid is always 108 within a 12 X 12 grid.
5 X 5 Grid
a(a + 52) = a² + 52a
(a + 4)(a + 48) = a² + 52a + 192
(a² + 52a + 192) – (a² + 52a) = 192
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 5 X 5 grid inside a 12 X 12 grid will always be 192.
Now I will draw a 5 X 5 grid to test if my theory is correct.
84 x 32 = 2688
36 x 80 = 2880
2880 – 2688 = 192
I have tested my theory by using numbers. The difference is 192 so therefore it looks like my theory is right.
The Difference for any 5 X 5 grid is always 192 within a 12 X 12 grid.
Results
Now I am going to try and find out the formula that finds the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any size square, then I will use that to work out the formula for any size rectangle.
In my table I have used ‘Length Of Square’ and ‘Width Of Square’ instead of the general size of the square because it will allow me to work out the formula for rectangles where the size is not the same as the width.
After looking at my table I have found out that if I subtract 1 from the length and width and multiply them by 12 I always get the difference. For example 3 –1 = 2, 3 - 1 = 2, (2 x 12) + (2 x 12) = 48 and the difference of the 3 X 3 square is 48.
Therefore the formula to work out the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any size square in a 10 X 10 grid is:
d = (12(L – 1)) + (12(w – 1)), simplified to;
d = 12(L – 1)(w – 1)
Now I will test my formula to see if it works. I am going to use an 8 X 8 grid to test to see if my formula works. Also because this formula includes the length and width it should also be able to work out the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number for any size rectangle in a 12 X 12 grid. So therefore I am going to use a 7 X 10 grid to test to see if it works for rectangles as well.
8 X 8 grid;
d = 12(L – 1)(w – 1)
d = 12(8 – 1)(8 – 1)
d = 12 x 7 x 7
d = 588
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an 8 X 8 grid should be 588.
17 x 108 = 1836
24 x 101 = 2424
2424 – 1836 = 588
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an 8 X 8 grid is 588. Now I will see if my formula will work for rectangles inside a 12 X 12 grid.
7 X 10 grid;
d = 12(L –1)(w – 1)
d = 12(10 – 1)(7 – 1)
d = 12 x 9 x 6
d = 648
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in an 7 X 10 grid should be 648.
50 x 131 = 6550
59 x 122 = 7198
7198 – 6550 = 648
The difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in a 7 X 10 grid is 648. This proves that my formula is correct. But my formula wont work for a grid that is less than a 2 X 2 grid or any grid that goes out of the 12 X 12 grid.
Therefore the formula to get the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number of any size square or rectangle in a 12 X 12 grid is:
d = 12(L – 1)(w – 1)
8 X 8 Grid
Now I am going to try and find the formula to get the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number of any size square or rectangle in an 8 X 8 grid. This will help me to find the formula for the difference between, the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any square or rectangle in any size grid.
Here is the 8 X 8 grid that I will be using.
First I will look at 3 X 3 grids within a 8 X 8 square. Also I will work out the differences algebraically.
Here is an algebraic grid for a 3 X 3 grid within an 8 X 8 grid.
a(a + 18) = a² 18a
(a + 2)(a + 16) = a² + 18a + 32
(a² + 18a + 32) – (a² 18a) = 32
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 3 X 3 grid inside an 8 X 8 grid will always be 32.
Now I will draw a 3 X 3 grid to test if my theory is correct.
21 x 39 = 819
23 x 37 = 851
851 – 819 = 32
I have tested my theory by using numbers. The difference is 32 so therefore it looks like my theory is right.
The Difference for any 3 X 3 grid is always 32 within an 8 X 8 grid.
4 X 4 Grid
a(a + 27) = a² + 27a
(a + 3)(a + 24) = a² + 27a + 72
(a² + 27a + 72) – (a² + 27a) = 72
According to this, the difference between the product of the top left number and the bottom right number, and the product of the top right number and the bottom left number in any 4 X 4 grid inside an 8 X 8 grid will always be 96.
Now I will draw a 4 X 4 grid to test if my theory is correct.
4 x 31 = 124
7 x 28 = 196
196 – 124 = 72
I have tested my theory by using numbers. The difference is 72 so therefore it looks like my theory is right.
The Difference for any 4 X 4 grid is always 72 within an 8 X 8 grid.
5 X 5 Grid