71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
From the squares I can find the difference for a 2x2 grid.
(2x11)-(1x12) = 10
(25x34)-(24x35) = 10
(63x72)-(62x73) = 10
(58x67)-(57x68) = 10
(90x99)-(89x100)=10
For all of these you can that the difference is a constant 10, so anywhere you put a 2x2 square on a 10x10 grid you will always get the same difference of ten. This can also be shown algebraically.
Let ‘N’ equal the number in the top left of the square.
N
N+1
N+10
N+11
We can change this into the equation:
(N+1)(N+10) – N (N+11)
Multiply out the brackets:
N²+10N + N + 10 - N² - 11N
Simplify to
N²+11N+10-N²-11N
=10
We can simplify this further until we are left on with 10, which is equal to the constant.
3x3 Squares
We can now investigate using a larger square. I will again selected random 3x3 squares on this grid.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
(3x21)-(1x23) =40
(8x26)-(6x28) =40
(44x62)-(42x64)=40
(58x76)-(56x78)=40
For the 3x3 grids the constant difference is 40. Therefore if you put a 3x3 square anywhere on a 10x10 grid the difference will be equal to 40. We can again show the algebraic method of working out the difference.
Let ‘N’ equal the top left number in the square.
N
N+2
N+20
N+22
We then change this into the equation
(N+2)(N+20) – N (N+22)
Multiply out the brackets
N²+ 20N +2N+ 40-N²-22N
This can be simplified to so that you are only left with 40, which is the constant difference for a 3x3 grid.
4x4 Squares
I will now investigate for the last time with square boxes using a larger 4x4 square. I will place them randomly on the grid.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
(4x31)-(34x1) =90
(9x36)-(6x39) =90
(67x94)-(64x97)=90
The difference for a 4x4 grid is always going to be no matter where you place the square on a 10x10 grid. We can again prove that the difference will always be 90 using algebra.
Let ‘N’ be the number at the top left of the square.
N
N+3
N+30
N+30
We then change this into the equation
(N+3)(N+30)-N (N+30)
Multiply out the brackets
N²+30N+3N+90-N²-30N
If this equation is fully worked out the only number left will be 90, which is the constant of a 4x4 grid.
Quadratic Equations
I can now take the results I have and put them into a table
Square size Difference First Difference Second Difference
1x1 0
10
2x2 10 20
30
3x3 40 20 50
4x4 90
The constant difference is the third one when dealing with squares. Using this we can construct a quadratic formula that would be able to tell us the difference for any sort of square that would be put on a 10x10 number grid.
The quadratic formula is
UN= AN²+BN+C
We now need to work out what A, B and C are. The formulas for working these out are:
A= 2nd difference / 2
B= 1st difference-3A
C= 1st term - (a+b)
A=10
B= -20
C=10
So now the equation reads
UN=10N²-20N+10
We can test this by using the square, for example if we use the 3x3 grid, the nth term is 3, if we insert this into the formula:
UN=10x3²-20x3+10
UN=90-60+10
UN=40
This formula will find the difference of the to two opposing corners on a 10x10 grid for any square shape.
Investigating Further
Now that we have worked out a quadratic formula for squares on a 10x10 grid, we can investigate further to see if we can work out a formula for rectangles on this grid. We will have to use the same process as before, although it is slightly extended.
2xN Rectangle
I will now try four different shapes of rectangle, each with one side that is two numbers long.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
These rectangles are
2x3
2x4
2x5
2x6
I am doing the same, as when I was working out the differences of the squares, except I will just write down the differences instead of showing all of the working out. I am still multiplying the red numbers first then multiplying the yellow ones, after that I am subtracting the sum of the yellows from the reds.
The differences for these rectangles are:
M N Difference First Difference
2x3 20
10
2x4 30
10
2x5 40
10
2x6 50
If we look at this we can see that the difference is equal to 10 times the nth term subtract the first difference, which is 10. We can write this algebraically
D=10N-10
This can also be simplified to D=10(N-1)
We have now worked out the formula for a 2xN grid.
3xN Rectangle
I will now enlarge the rectangle. Changing the ‘M’ to a three, where it was a constant two in the 2xN Rectangle.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
These rectangles and their differences are
M N Difference First Difference
3x3 40
20
3x4 60
20
3x5 80
20
3x6 100
This time we can see that if we multiply the nth term by 20 this time, then if we subtract the first term again we get the difference. This can be written algebraically.
The formula is written as D=20N-20
But can be simplified to be written as
D=20(N-1)
From this we begin to see that the first difference is going up in tens, and so is the formula, so we can predict that the formula for a 4xN grid will be 30(N-1), but we will have to test this to see if it is true.
4xN Grid
I am doing this rectangle grid to see if my predictions are correct, and if they are I will be able to construct a formula for any grid where I know what ‘M’ is.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
These rectangles and their differences are:
M N Difference First Difference
4x3 60 30
4x4 90 30
4x5 120 30
4x6 150 30
My prediction was correct. We need to multiply the nth term by 30 this time then subtract the first difference which is 30. So we can construct the formula D=30N-30 which can be reduced to the formula D=30(N-1)
Now that my predictions have been proven correct we can now work out any rectangle as long as we know the Mth term. For example if we want a rectangle that had the constant M as 7 then we could make the formula D=60(N-1).
But now we want a formula so that we can be given an Nth and an Mth term and from that we can work out the difference of the opposing corners.
