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Introduction

Number Grids

 12 13 14 22 23 24 25 26 27

We were asked to investigate the difference in the products of sets of numbers on a number grid. These products are obtained by multiplying the top left number in a square box, by the bottom right number in the box. Then you find the product of the top right and bottom left and find the difference between the two products.

12 x 23 = 276

13 x 22 = 286

Difference (D) = 10

I will try this with other 2 x 2 boxes on the grid.

 27 28 29 37 38 39 47 48 49
 43 44 45 53 54 55 63 64 65

43 x 54 = 2322                        27 x 38 = 1926

44 x 53 = 2332                        28 x 37 = 1936

D            = 10                              D    = 10

On a 2 x 2 square box, the difference is always 10.

 n n+1 n+2 n+10 n+11 n+12 n+20 n+21 n+22

n x (n+11) = n2 +11n

(n+1)(n+10)= n2 +11n +10  These numbers produce the difference

I will now try this with squares larger than 2 x 2.

 3 4 5 13 14 15 23 24 25
 27 28 29 37 38 39 47 48 49

3 x 25 = 75                             27 x 49 = 1323

5 x 23 = 115                             29 x 47 = 1363

D   =  40                                 D     = 40

 n n+1 n+2 n+10 n+11 n+12 n+20 n+21 n+22

n(n+22) = n2 + 22n

(n+2)(n+20) = n2 + 22n + 40

Squares of equal sizes always have the same difference.

E.g. 2 x 2, D=10, 3 x 3, D=40.

From now on I will only use n grids in my diagrams as they show the numbers that produce the difference.

4 x 4

Middle

n+10

n+11

n+12

n+13

n+20

n+21

n+22

n+23

n+30

n+31

n+32

n+33

n(n+23) = n2 + 23n

(n+3)(n+20)= n2 + 23n + 60

D = 60

 n n+1 n+2 n+3 n+4 n+10 n+11 n+12 n+13 n+14 n+20 n+21 n+22 n+23 n+24 n+30 n+31 n+32 n+33 n+34

n(n+34)= n2 + 34n

(n+4)(n+30)=n2+34n+120

D = 120

This is a table of my results.

 X 2 3 4 D 20 60 120

2 x 10 = 20                From this I can tell that the formula for

3 x 20 = 60                finding X(X+1) rectangles is

4 x 30 = 120                10X(X-1)  OR  10X2 –10X This number is always 10(X-1)

Now I will look at rectangles where one pair of sides are two units longer than the other pair of sides.

X(X+2)

 n n+1 n+2 n+3 n+4 n+10 n+11 n+12 n+13 n+14 n+20 n+21 n+22 n+23 n+24 n+30 n+31 n+32 n+33 n+34

n(n+13)= n2 + 13n (n+3)(n+10)= n2 +13n+30

D = 30

 n n+1 n+2 n+3 n+4 n+10 n+11 n+12 n+13 n+14 n+20 n+21 n+22 n+23 n+24 n+30 n+31 n+32 n+33 n+34

n(n+24)=n2 +24n (n+4)(n+20)=n2 +24n+80

D = 80

 n n+1 n+2 n+3 n+4 n+5 n+10 n+11 n+12 n+13 n+14 n+15 n+20 n+21 n+22 n+23 n+24 n+25 n+30 n+31

Conclusion

 n n+1 n+2 n+9 n+10 n+11 n+18 n+19 n+20

n(n+10) = n2 + 10n

(n+1)(n+9) = n2 + 10n + 9

D = 9

 n n+1 n+2 n+9 n+10 n+11 n+18 n+19 n+20

n(n+20) = n2 + 20n

(n+2)(n+18) = n2 + 20n + 36

D = 36

 n n+1 n+2 n+3 n+9 n+10 n+11 n+12 n+18 n+19 n+20 n+21 n+27 n+28 n+29 n+30

n(n+30) = n2 + 30n

(n+3)(n+27) = n2 + 30n + 81

D = 81

 X 2 3 4 D 9 36 81

I will call the number of columns in a grid Z.

From this I can see that the formula is 9(X-1)2OR 9X2 –18X +9

The formula for a square on a 10x10 grid is 10X2 –20X +10

By comparing these two formulae I can tell that the formula for any square on any grid size is ZX2 –2ZX +Z

I predict that for the formula for rectangles I will have to substitute Z in to some of the values.

10(X2 +XY –Y –2X +1)                 Z(X2 +XY –Y –2X +1) For example if Z=9, X=4 and Y=3 the difference will be 162.

 20 26 47 53

20 x 53 = 1060

26 x 47 = 1222

D = 162

Now I have the formulae for any square or any rectangle on any size of grid. Given more time I would have found formulae for more irregular shapes like a T or a Pyramid.

Luke Ferngrove

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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