# Number Grid Investigation

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The Patterns, Rules and Formulae Found in a Number Grid

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

\$#\$#1\$#\$# 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)

1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22
Now I shall try to answer the question "does this differ with a different size grid?"

I shall try to fill the table shown underneath to get a clear view on all the data and try to see a formula from this data.

Grid Width Difference

4x4 10

3x3 9

2x2 8

Square size 2x2 grid width 10

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

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

Square Size 2x2, Grid Width 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 21 26 27

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

Square size 2x2, Grid Width 8)

1 2 3 4 5 6 7 8
9 10 11 12 13