# Number Grid Investigation

**Topics:**Harshad number, Multiplication, Triangular number

**Pages:**33 (3881 words)

**Published:**January 16, 2008

For this coursework I shall investigate and explain thoroughly the patterns, rules and formulae found in a number grid when placing a square at any point in the grid, multiplying the top left and bottom right corners and the top right and bottom left corners and finding the difference. I will look at all the variables and use systematic methods when doing this.

Suggested Variables:

Width of grid

Size of Square

Multiplication Tables

Width of Grid: Does the width of the grid make a difference to formulae? I shall keep the size of the square at a constant (2 x 2)

Width of Grid size 10)

12345678910

11121314151617181920

21222324252627282930

12 x 23 = 276

22 x 13 = 286

286 - 276=10

I shall now use letters to prove this correct

X

X+1

X+10

X+11

X(X+11) = X² + 11X

(X+1)(X+10) = X²+11X+10

(X²+11X+10) - (X²+11X) = 10

Width of Grid Size 9)

123456789

101112131415161718

192021222324252627

11 x 21 = 231

20 x 12 = 240

240 - 231 = 9

I shall now use letters to prove this correct

X

X+1

X+9

X+10

X(X+10) = X² + 10X

(X+9)(X+1)=X²+10X+9

(X²+10X+9) - (X²+10X) = 9

Width of Grid Size 8)

12345678

910111213141516

1718192021222324

10 x 19 = 190

18 x 11 = 198

198 - 190 = 8

I shall now use letters to prove this correct

X

X+1

X+8

X+9

X(X+9) = X² + 9X

(X+8)(X+1) = X²+9X+8

(X²+9X+8) - (X²+9X) = 8

Width of GridDifference

10 10

9 9

8 8

After doing these 3 widths I can see a definite pattern emerging, using a 2x2 square the difference is always equal to the grid width.

W=D

To prove this I shall use another grid width of 5. If my equation is correct the difference should be 5.

12345

678910

1112131415

7 x 13 = 91

12 x 8 = 96

96 - 91 = 5

I shall now use letters to prove this correct

X

X+1

X+5

X+6

X(X+6) = X²+6X

(X+5)(X+1) = X²+6X+5

(X²+6X+5) -(X²+6X) = 5

This proves my equation correct as the outcome is as I predicted.

I shall now work on changing the square size. To begin with I shall keep the width of the grid at a constant size (10) however I shall experiment with both to try and find a formula that works with both factors.

Size of Square:

Does the size of the square affect the formula. I shall keep the grid width the same size to begin with but I will change it when I find a 1st formula.

Size of square 2x2)

12345678910

11121314151617181920

21222324252627282930

12 x 23 = 276

22 x 13 = 286

286 - 276=10

I shall now use letters to prove this correct

X

X+1

X+10

X+11

X(X+11) = X² + 11X

(X+1)(X+10) = X²+11x+10

(X²+11x+10) - (X²+11X) = 10

Size of Square 3x3)

12345678910

11121314151617181920

21222324252627282930

31323334353637383940

12 x 34 = 408

32 x 14 = 448

448 - 408 = 40

I shall now use letters to prove this correct

X

X+2

X+20

X+22

X(X+22)=X²+22X

(X+20)(X+2)=X²+22X+40

(X²+22X+40) - (X²+22X) = 40

Size of Square 4x4)

12345678910

11121314151617181920

21222324252627282930

31323334353637383940

41424344454647484950

12 x 45 = 540

42 x 15 = 630

630 - 540 = 90

I shall now use letters to prove this correct

X

X+3

X+30

X+33

X(X+33)=X²+33X

(X+30)(X+3)=X²+33x+90

(X²+33X+90) - (X²+33X) = 90

Size of SquareDifference

2x2 10

3x3 40

4x4 90

After looking at this I can see that if you take 1 from the height or width of the square and square that number then multiply it by 10 you will get the difference.

(S-1)²x10=D

To prove this equation...

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