Number Grids.

2*8

70

.·. As the number of columns goes up a further 10 is added so I predict a 2*9 rectangle will be 80. So the equation for this is: -

R=rows

C = columns

(R-1)*(C-1)*10

So I tested it: -

For a 2*3 rectangle

(R-1)*(C-1)*10

(2-1)*(3-1)*10

=1*2*10

Diagonal difference=20

For a 2*4 rectangle

(R-1)*(C-1)*10

(2-1)*(4-1)*10

=1*3*10

Diagonal difference=30

For a 2*8 rectangle

(R-1)*(C-1)*10

(2-1)*(8-1)*10

=1*7*10

Diagonal difference=70

So my formula works.

Squares on a 10*10 grid

Here I will study the diagonal difference of squares on a 10*10 grid, try to find a formula and then test it.

66

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76

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86

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88

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32

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42

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44

So I conclude that all 3*3 squares on a 10*10 grid have a diagonal difference of 40.

Now I will try a 4*4 square to see if these have identical diagonal differences.

7*50=850

20*47=940

Diagonal difference =940 - 850 =90

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27

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50

65*98=6370

68*95=6460

Diagonal difference =6460 - 6370 =90

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98

So from this I conclude that all 4*4 squares on a 10*10 grid have a diagonal difference of 90.

So here I have drawn up a table of results to show the diagonal difference of squares on a 10*10 grid.

Square

Diagonal difference

Workings

2*2

0

*1=1(1²)

3*3

40

2*2=4(2²)

4*4

90

3*3=9(3²)

5*5

60

4*4=16(4²)

6*6

250

5*5=25(5²)

7*7

360

6*6=36(6²)

8*8

490

7*7=49(7²)

9*9

640

8*8=64(8²)

All of the green highlighted numbers are square numbers.

I predict that a 10*10 square will have a diagonal difference of

9*9=81 or (9²)

=810

So: - 1*100=100

0*91=910

=910 - 100=810

My prediction works now I will find a formula.

(X-1)*(X-1)*10

I will try this for a 3*3 square

(3-1)*(3-1)*10

=2*2*10

Diagonal difference=40

Also for a 7*7 square

(7-1)*(7-1)*10

=6*6*10

Diagonal difference = 360

You can also write the equation as

Diagonal difference = (X-1)*(X-1)*10

Or

(X-1) ²*10

A SMALLER GRID

Here I am going to look at a 9*9 grid. Because the grid is smaller the numbers will move around.

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I will begin by doing some 3*2 rectangles

21*32= 672

23*30=690

Diagonal difference=690 - 672 =18

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32

57*68= 3876

59*66=3894

Diagonal difference=3894 - 3876 =18

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59

66

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68

So I conclude that all 3*2 squares on a 9*9 grid have a diagonal difference of 18.

I have tested the other way by using a 2*3 rectangle.

*20=20

2*19=38

Diagonal difference = 38 - 20 = 18

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0

1

9

20

40*59=2360

41*58=2378

Diagonal difference = 2378 - 2360 = 18

40

41

49

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58

59

So all 3*2 and 2*3 squares have a diagonal difference of 18.

I will study more rectangles on a 9*9grid and then draw up a table.

Here is a 4 * 2 rectangle

40*52=2080

49*43=2107

Diagonal difference=2107 - 2080 =27

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52

69*81=5589

72*78=5616

Diagonal difference=5616 - 5589 =27

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78

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81

So I conclude that all 4*2 rectangles on a 9*9 grid have a diagonal difference of 27.

Now I will do a 5*2 rectangle from the 9*9 grid.

28*41=1148

32*37=1184

Diagonal difference = 1184 - 1148 =36

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41

64*77=4928

68*73=4964

Diagonal difference = 4964 - 4928 =36

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68

73

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77

So I conclude all 5*2 or 2*5rectangles on a 9*9 grid have a diagonal difference of 36.

Here is a table showing my results.

Rectangle

Rows * columns

Diagonal difference

3*2

8

4*2

27

All are multiples of 9

5*2

36

6*2

45

7*2

54

8*2

63

9*2

72

From this I have formed the formula: -

(R-1)*(C-1)*9

So I predict a 3*2 rectangle will be

(3-1)*(2-1)*9

=18

*12=12

3*10=30

Diagonal difference = 30 - 12 =18

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3

0

1

2

So my prediction and formula are right.

Squares on a 9*9 grid

I will now do the same for squares on a 9*9 grid.

2*2

21*31=651

22*30=660

Diagonal difference = 660 - 651 =9

21

22

30

31

3*3

41*61=2501

43*59=2337

Diagonal difference = 2537 - 2501 =36

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43

50

51

52

59

60

61

From this I have drawn up a table for squares on a 9*9 grid

Rectangle

Rows * columns

Diagonal difference

2*2

9 (1*9)

3*3

36 (4*9)

Multiples of 9

4*4

81 (9*9)

5*5

44 (19*9)

6*6

25 (25*9)

7*7

324 (36*9)

From this table I have found this formula for squares on a 9*9 grid

(X-1)*(X-1)*9

So I will test this: -

I predict an 8*8 square will be

(8-1)*(8-1)*9

= 7*7*9

=441

*71=71

8*64=512

Diagonal difference =512-71=441

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71

So my prediction and formula are right.

A smaller grid

I will now calculate the diagonal differences of rectangles and squares on an 8*8 grid.

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The past two formulas went like this: -

For a rectangle on a 10*10 grid

(R-1)*(C-1)*10

Square on a 10*10grid

(X-1)*(X-1)*10

Rectangle on a 9*9 grid

(R-1)*(C-1)*9

Square on a 9*9grid

(X-1)*(X-1)*9

So I predict the formula for an 8*8 grid will be

Rectangle: -(R-1)*(C-1)*8

Square: - (X-1)*(X-1)*8

I will test this.

3*2 rectangle on an 8*8 grid

I use my formula to predict that: -

(3-1)*(2-1)*8

=2*1*8

=16

2*12=24

4*10=40

Diagonal difference = 40 - 24=16

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0

1

2

My prediction is right.

So using my formula I predict that a 3*3 square on an 8*8 grid will be

(X-1)*(X-1)*8

(3-1)*(3-1)*8

=2*2*8

=32

46*64=2944

48*62=2976

Diagonal difference = 2976 - 2944 =32

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54

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56

62

63

64

So my prediction and formula are also right.

A pattern

A pattern has shown up in the formulas so I made this table: -

Grid rectangle

Formula

0*10

(R-1)*(C-1)*10

9*9

(R-1)*(C-1)*9

8*8

(R-1)*(C-1)*8

7*7

(R-1)*(C-1)*7

6*6

(R-1)*(C-1)*6

5*5

(R-1)*(C-1)*5

Grid Square

Formula

0*10

(X-1)*(X-1)*10

9*9

(X-1)*(X-1)*9

8*8

(X-1)*(X-1)*8

7*7

(X-1)*(X-1)*7

6*6

(X-1)*(X-1)*6

So on a 6*6 grid I would expect a 3*2 rectangle to be: -

(3-1)*(2-1)*6

=2*1*6

=12

So

8*16=128

0*14=140

Diagonal difference = 140 - 128 =12

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9

0

4

5

6

Also a 3*3 square on a 6*6 grid would be: -

(X-1)*(X-1)*6 => (3-1)*(3-1)*6

=2*2*6

=24

3*17=51

5*15=75

Diagonal difference = 75 - 51 =24

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9

0

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6

7

Both predictions and formulas are correct so I conclude that the overall formula is: -

R= rows C= columns X= square N° G= grid

Rectangle: - (R-1)*(C-1)*G ...AND... Square: - (X-1)*(X-1)*G