# Number grids. In this investigation I have been attempting to work out a formula that will find the difference between the products of the top left and bottom right of a number grid and the top right and bottom left of a number grid.

Extracts from this essay...

Introduction

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Middle

99 100 A 10x10 number grid If you choose any 2x2 box on a 10x10 number grid then the difference should equal 10... 35x46=1610 Difference=10 36x45=1620 The difference equals 10. So what would happen if you tried the same thing with a 3x3 box? 37x19=703 Difference=40 17x39=663 The difference equals 40. So if you try it with a 4x4 box... 61x94=5734 Difference=90 91x64=5824 The difference equals 90. Lets just try one more: 55x99=5445 Difference=160 95x59=5605 We now have enough results. A 9x9 number grid If you choose any 2x2 box on a 9x9 number grid then the difference should equal 9... 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

Conclusion

For a 3x3 box N(N+20)=N2+20N (N+2)(N+18)=N2+20N+36 As before the difference equals 20. From these results we can work out that, if S is the size of the square, then these are the results. For a SxS box N[N+10(S-1)]=N2+10(S-1)N [N+(S-1)][N+9(S-1)] The difference as an expression would be: =N2+9(S-1)N+(S-1)N+9(S-1)2 =N2+10(S-1)+9(S-1)2 Difference=9(S-1)2 N[N+(G+1)x(S-1)=N2+(G+1)x(S-1)N [N+(S-1)][N+G(S-1)] =N2+(S-1)G(S-1)+NG(S-1)+N(S-1) =N2+ (S-1)2 +N(S-1)(9+1) When G = the grid size and S = square size The formula is G(S-1)2 Rectangle This rectangle is of length L and of width W. For a grid of 10x10. N[N+(L-1)+10(W-1)]=N2+(L-1)N+10(W-1)N [N+(L-1)+10(W-1)] [N+(L-1)] N2+10(W-1)N+(L-1)N+ Difference=10(L-1)(W-1) Rectangular grid= LxW Difference =G(L-1)(W-1) Finally we can work out how the change of multiples in the grid would affect the formula. In this formula L=Length, W=width, M=multiple and G=grid size. N[N+M(L-1)+MG(W-1)] =N2+N(l-1)M+NMG(w-1)N [N+M(L-1)][N+GM(W-1)] N2+NGM(W-1)+NM(L-1)+ This formula is the final part of this investigation. I cannot think of another way to extend my investigation. This investigation has enlightened me to the real-life ways in which math's can be applied. Previously I was not aware of how complicated and interesting number grids can be!

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## Here's what a teacher thought of this essay

A well written piece of work with only a couple of minor errors. This piece of work shows an excellent application of multiplying double brackets. 4 stars ****

Marked by teacher Mick Macve 18/03/2012