Number Grids Investigation

30

2*5

40

2*6

50

2*7

60

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

67

68

76

77

78

86

87

88

22

23

24

32

33

34

42

43

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

8

9

20

27

28

29

30

37

38

39

40

47

48

49

50

65

66

67

68

75

76

77

78

85

86

87

88

95

96

97

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

10*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.

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

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

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

I will begin by doing some 3*2 rectangles

21

22

23

30

31

32

57

58

59

66

67

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.

2

0

1

9

20

40

41

49

50

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

41

42

43

49

50

51

52

69

70

71

72

78

79

80

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

29

30

31

32

37

38

39

40

41

64

65

66

67

68

73

74

75

76

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

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

2

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

22

30

31

3*3

41

42

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)

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

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

9

20

21

22

23

24

25

26

28

29

30

31

32

33

34

35

37

38

39

40

41

42

43

44

46

47

48

49

50

51

52

53

55

56

57

58

59

60

61

62

64

65

66

67

68

69

70

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.

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

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

61

62

63

64

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

3

4

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

47

48

54

55

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

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

4

5

9

0

1

5

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

Name: Sophie Johnson Form: 10LK Class: 10A6 Teacher: Mrs Hindley Centre number: 34519 Candidate number: 8146 R0 number: R04240