# The problem is to investigate the differences of corner numbers on a multiplication grid.

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Introduction

## Number Grids

## The Problem

The problem is to investigate the differences of corner numbers on a multiplication grid.

## Introduction

To solve this problem I will have to choose several examples of squares on a grid:

E.g. 2x2, 3x3, 4x4

To work this out I will need to take the opposite corners of the square and subtract them from the other sum of the two corners. This is easier seen in the diagram shown below:

*The RED numbers are always multiplied then subtracted from the sum of the Blue numbers.

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

This is an example of a 2x2 square. I will then make an equation with this:

Difference = (2x11)-(1x12)

= 10

The difference for this example is ten.

2x2 Squares

I will use on grid to use several example of 2x2 squares by placing them randomly on the grid.

12 3 4 5 6 7 8 9 10

1112 13 14 15 16 17 18 19 20

21 22 23 2425 26 27 28 29 30

31 32 33 3435 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 5758 59 60

61 6263 64 65 66 6768 69 70

71 7273 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 8990

91 92 93 94 95 96 97 98 99100

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.

Middle

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

Conclusion

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.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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