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Number Grids - Algebra

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Introduction

Number Grid Coursework

I am doing this experiment to see if there are any patterns in squares on a 1 - 100 grid. I will then see if I can make a formula to express these patterns.

2 x 2 Squares

My 2 x 2 square is 14, 15, 24 and 25. The top left times by the bottom right is 14 x 25, this equals 350. The top right multiplied by the bottom left is 15 x 24 = 360. To finish I will take the smaller of the two numbers from the larger. This is 360 – 350 = 10.

My 2 x 2 square is 84, 85, 94 and 95. The top left times by the bottom right is 84 x 95, this equals 7980. The top right multiplied by the bottom left is 85 x 94 = 7990. To finish I will take the smaller of the two numbers from the larger. This is 7990 – 7980 = 10.

My 2 x 2 square is 27, 28, 37 and 38. The top left times by the bottom right is 27x 38, this equals 1026. The top right multiplied by the bottom left is 28 x 37 = 1036. To finish I will take the smaller of the two numbers from the larger. This is 1036 – 1026 = 10.

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Middle

X + 2

X + 10

X + 11

X + 12

X + 20

X + 21

X + 22

(X2 + 2X + 20X + 40) – (X2 +22X) = 40

FORMULA = (X2 + 22X + 40) - (X2 +22X) = 40

4 x 4 Squares

My 4 x 4 square is 17, 18, 19, 20, 27, 28, 29, 30, 37, 38, 39, 40, 47, 48, 49 and 50. The top left times by the bottom right is 17 x 50 = 850. The top right multiplied by the bottom left is 20 x 47 = 940. The difference between the smaller and bigger numbers is 940 – 850 = 90.

My 4 x 4 square is 1, 2, 3, 4, 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33 and 34. The top left times by the bottom right is 1 x 34 = 34. The top right multiplied by the bottom left is 4 x 31 = 124. The difference between the smaller and bigger numbers is 124 – 34 = 90.

My 4 x 4 square is 62, 63, 64, 65, 72, 73, 74, 75, 82, 83, 84, 85, 92, 93, 94 and 95. The top left times by the bottom right is 62 x 95 = 5890. The top right multiplied by the bottom left is 65 x 92 = 5980. The difference between the smaller and bigger numbers is 5980 – 5890 = 90.

My 4 x 4 square is67, 68, 69, 70, 77, 78, 79, 80,87, 88, 89, 90, 97, 98, 99 and 100. The top left times by the bottom right is 67 x 100 = 6700. The top right multiplied by the bottom left is 70 x 97 = 6790. The difference between the smaller and bigger numbers is 6790 – 6700 =90.

I have found that for any 4 x 4 square on a 1-100 grid the difference between the two numbers is 90.

Algebra – 4 x 4

X

X + 1

X + 2

X + 3

X + 10

X + 11

X + 12

X + 13

X + 20

X + 21

X + 22

X + 23

X + 30

X + 31

X +32

X + 33

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Conclusion

My 5 x 5 square is 51, 52, 53, 54, 55, 61, 62, 63, 64, 65, 71, 72, 73, 74, 75, 81, 82, 83, 84, 85, 91, 92, 93, 94 and 95. The top left times by the bottom right is 51 x 95 = 4845. The top right multiplied by the bottom left is 55 x 91 = 5005. The difference between the smaller and bigger numbers is 5005 – 4845 = 160.

My 5 x 5 square is 56, 57, 58, 59, 60, 66, 67, 68, 69, 70, 76, 77, 78, 79, 80, 86, 87, 88, 89, 90, 96, 97, 98, 99 and 100. The top left times by the bottom right is 56 x 100 = 5600. The top right multiplied by the bottom left is96 x 60 = 5760. The difference between the smaller and bigger numbers is 5760 – 5600 = 160.

I have found that they all end out that the difference between the two numbers of any 5 x 5 squares on a 1 – 100 grid is 160.

Algebra - 5 x 5

X

X + 1

X + 2

X + 3

X + 4

X + 10

X + 11

X + 12

X + 13

X + 14

X + 20

X + 21

X + 22

X + 23

X + 24

X + 30

X + 31

X +32

X + 33

X + 34

X + 40

X + 41

X + 42

X + 43

X + 44

(X2 + 40X + 4X + 160) – (X2 + 44X) = 160

FORMULA = (X2 + 44X + 160) – (X2 + 44X) = 160

Algebra for All

The algebra for all the equations is 10(n-1)2. This formula can be used to find out any size square on a 10 x 10 grid.

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