Number grid

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Samantha Whittaker

Number 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

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

00

Pick out 2x2 squares

Multiply the diagonals

Find the difference

34

35

44

45

INVESTIGATE

27

28

37

38

82

83

92

93

68

69

78

79

9

0

9

20

First number

Second number

Third number

Fourth number

stx4th

2ndx3rd

Difference

34

35

44

45

530

540

0

27

28

37

38

026

036

0

82

83

92

93

7626

7636

0

68

69

78

79

5372

5382

0

9

0

9

20

80

90

0

I have put my results into a table so that they are easier to analyse and compare.

What I have found is; when you take a 2x2 grid from a 10x10 grid, times the diagonals, the difference between the products of the diagonals is always 10.

36

37

46

47

If this rule is correct, then by using this grid:

I predict that the difference between the

Product of 36 and 47 compared with that of

37 and 46 will equal 10.

36x47=1692

37x46=1702

Difference=10

My prediction was correct as the difference between 36x47 (1692), and 37x46 (1702) was 10.

To prove that this theory will work for any 2x2 grid from a 10x10 number square, I am going to express it as an algebraic equation.

X

X+1

X+10

X+11

(X)(X+11)=(X+1)(X+10)

X²+11X=X²+11X+10

The equation is set out with the first half being the top left multiplied by the bottom right and the second half being the top right multiplied by the bottom left.

The 10 that is underlined in the above equation, proves that the latter part of the equation will always be 10 more than the first half of the equation.

So, by using a 2x2 grid and multiplying the diagonals, the difference will always be 10.

I am now going to investigate whether there is any relationship between the products of the diagonals of a 3x3 grid.

42

43

44

52

53

54

62

63

64

78

79

80

88

89

90

98

99

00

6

7

8

6

7

8

26

27

28

8

9

20

28

29

30

38

39

40

2

3

4

2

3

4

22

23

24

First number

Third number

Seventh number

Ninth number

stx9th

3rdx7th

Difference

42

44

62

64

2688

2728

40

78

80

98

00

7800

7840

40

6

8

26

28

68

208

40

8

20

38

40

720

760

40

2

4

22

24

48

88

40

I have put my results into a table so that they are easier to analyse and compare.

What I have found is; when you take a 3x3 grid from a 10x10 grid, times the diagonals, the difference between the products of the diagonals is always 40.

31

32

33

41

42

43

51

52

53

If this rule is correct, then by using this grid:

I predict that the difference between the

product of 31 and 53 compared with that of

33 and 51 will equal 40.

31x53=1643

33x51=1683

Difference=40

My prediction was correct as the difference between 31x53 (1643), and 33x51 (1683) was 40.

To prove that this theory will work for any 3x3 grid from a 10x10 number square, I am going to express it as an algebraic equation.

X

X+1

X+2

X+10

X+11

X+12

X+20

X+21

X+22

(X)(X+22)=(X+2)(X+20)

X²+22X=X²+22X+40

The equation is set out with the first half being the top left multiplied by the bottom right and the second half being the top right multiplied by the bottom left.

The 40 that is underlined in the above equation, proves that the latter part of the equation will always be 40 more than the first half of the equation.

I am now going to extend my investigations further and see whether there is any relationship between the products of the diagonals of a 4x4 grid.

24

25

26

27

34

35

36

37

44

45

46

47

54

55

56

57

62

63

64

65

72

73

74

75

82

83

84

85

92

93

94

95

64

65

66

67

74

75

76

77

84

85

86

87

94

95

96

97

3

4

5

6

3

4

5

6

23

24

25

26

33

34

35

36

27

28

29

30

37

38

39

40

47

48

49

50

57

58

59

60

First number

Fourth number

Thirteenth number

Sixteenth number

stx16th

4thx13th

Difference

24

27

54

57

368

458

90

62

65

92

95

5890

5980

90

64

67

94

97

6208

6298

90

3

6

33

36

08

98

90

27

30

57

60

620

710

90

I have put my results into a table so that they are easier to analyse and compare.

What I have found is; when you take a 4x4 grid from a 10x10 grid, times the diagonals, the difference between the products of the diagonals is always 90.

43

44

45

46

53

54

55

56

63

64

65

66

73

74

75

76

If this rule is correct, then by using this grid:

I predict that the difference between the

product of 43 and 76 compared with that of

46 and 73 will equal 90.

43x76=3568

46x73=3358

Difference=90

My prediction was correct as the difference between 43x76 (3568), and 46x73 (3358) was 90.

To prove that this theory will work for any 4x4 grid from a 10x10 number square, I am going to express it as an algebraic equation.

X

X+1

X+2

X+3

X+10

X+11

X+12

X+13

X+20

X+21

X+22

X+23

X+30

X+31

X+32

X+33

(X)(X+33)=(X+3)(X+30)

X²+33X=X²+33X+90

The equation is set out with the first half being the top left multiplied by the bottom right and the second half being the top right multiplied by the bottom left.

The 90 that is underlined in the above equation, proves that the latter part of the equation will always be 90 more than the first half of the equation.

I will now investigate further by taking a 2x2 grid from a 5x5 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

2

6

7

3

4

8

9

9

20

24

25

4

5

9

0

6

7

21

22

First number

Second number

Third number

Fourth number

stx4th

2ndx3rd

Difference

2

6

7

7

2

5

3

4

8

9

247

252

5

9

20

24

25

475

480

5

4

5

9

0

40

45

5

6

7

21

22

352

357

5

I have put my results into a table so that they are easier to analyse and compare.

What I have found is; when you take a 2x2 grid from a 5x5 grid, times the diagonals, the difference between the products of the diagonals is always 5.

3

4

8

9

If this rule is correct, then by using this grid:
Join now!


I predict that the difference between the

product of 3 and 9 compared with that of

4 and 8 will equal 5.

3x9= 27

4x8= 32

Difference= 32-27=5

My prediction was correct as the difference between 3x9 (27), and 4x8 (32) was 5.

To prove that this theory will work for any 2x2 grid from a 5x5 number square, I am going to express it as an algebraic equation.

X

X+1

X+5

X+6

(X)(X+6)= (X+1)(X+5)

X² +6X= X²+6X+5

The equation is set out with ...

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