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number grid

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Introduction

NUMBER GRID

COURSE WORK

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To investigate the patterns generated from using rules in a square grid

The aim of the investigation is to find differences of small n x n squares in 10 x 10 square and then to see if there is any rule or pattern which connects the size of square chosen and the difference.

In order to find the difference of n x n square first step is to take nxn square and then multiply the corners diagonally. For example take 2x2 square and multiply its corners diagonally.

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25 x 36= 900             26 x 35= 910                                   910 - 900= 10        

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77 x 88 = 6776              78 x 87= 6786                     6786 - 6776= 10

After a few results I observed that

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Middle

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31 x 64 = 1984              34 x 61=2074       2074 – 1984 = 90

n th term

SIZED

DIFFERENCE

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2 x 2

10

2

3 x 3

40

3

4 x 4

90

I put my results into table and observed that the difference is square number because the box and the grid are square shaped.

I predict that for a 5 by 5 square the difference between the products will be 160.

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46 x 90 = 4140        50 x 86 = 4300              4300 - 4140 = 160

I will now attempt to prove my results algebraically.

After doing this I will check my answer using the nxn formula

2x2 square

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image00.png
 n            n+1   

n+10       n+11                                             (24x33)- (23x34)

                                                                                   792-782=10                                                                           

(n+1) (n+10)-n (n+11)

 n2 +10n + n+10 - n2 +11n                                                                    n2 +11n +10 - n2 +11n =10                              

In both ways the difference is

5x5 square

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image01.png

n                 n+4  

 n+40         n+44       

 n (n+44) - [(n+4) (n+40)]                                                   (5x41)-(1x45) 

  n2+44n - n2+40n + 4n +160                                                   205-45= 160                                                

  n2+44n +160 - n2+44n =160

In both ways the difference is 160

After doing this I will check if the nxn formula will work in a bigger grid size like 11x11  

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Conclusion

S-1 x G = D

Sized = 2 x 3   G = 10   D = 20

2-1 =1

3-1 =2

 1 x 2 =2

 2 x 10 =20

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37 x 60 =2220    40 x 57 = 2280 2280 - 2220 = 60

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

SIZED = 3 x 4       G = 10   D = 60

3-1 = 2

4-1 = 3

2 x 3 = 6    

 6 x 10 = 60

I could prove that this formula can work for both rectangle and square.

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53 x 64 = 3392       54 x 63 = 3402      3402 - 3392 = 10

 S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

S = 2 x 2       G = 10    D = 10

2-1 = 1

2-1 = 1

1 x 1 = 1

1 x 10 = 10

FINALLY

In this project I found that not all formula can work for rectangle and square.

But I found a formula can work for rectangle and square it does not matter the size of the box and I will proof that it does not matter for grid sizes too.

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28 x 54 = 1512    30 x 52 = 1560 1560 - 1512 = 48

S= SIZED G = GRID D = DIFFERENCE

S-1 x G = D

S = 3 x 3 G = 12    D = 48

3-1 = 2

3-1 = 2

2 x 2 = 4

4 x 12 = 48

If I were to extend this project further I would try and do cube in three dimensions.

ISRAA AL-RIFAEE                                                    -  -

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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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