'T' Totals Investigation

Introduction

We have been given the task to find the relationship between the 'T' ? and the 'T' Total 9x9 grid. The 'T' ? is the number at the bottom of the T shape and the 'T' Total is the sum of all the numbers inside the T.

Example

'T' Total = 1+2+3+11+20

= 37

To find the relationship between the 'T' ? and the 'T' Total, algebra will be needed. Therefore the 'T' ? will be represented by 'X' and the 'T' Total will be represented by 'T'. I will substitute the numbers in the T and replace them with those numbers but with the 'T' ? (X) as the subject.

Example

From this we can come up with a formula. I will check the formula with one of my other results for that grid. If the relationship between the T' ? and the 'T' Total match the formula will be correct and the problem will be solved.

Table of Results for a 9x9 Grid

X

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

'T' ? (X)

'T' Total (T)

20

37

42

47

67

272

80

337

T = X+(X-9)+(X-17)+(X-18)+(X-19)

T = 5X-63

Check

T = 5X-63

T = 5x42-63

T = 147

Formula to find 'T' Total (T) if you have the 'T'? (X) in a 9x9 grid:

=5X-63

Now I have solved the first problem I am going to further my investigation and vary the grid sizes. I am going to use the same method to find the formulas. The formulas should been linked and if I can find that link I cam make a formula that incorporates grid size. This would mean I could find the 'T' Total if I had the T'? in any grid size.

Table of Results for an 8x8 Grid

X

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

'T' ? (X)

'T' Total (T)

8

34

29

89

50

94

61

244

T = X+(X-8)+(X-15)+(X-16)+(X-17)

T = 5X-56

Check

T = 5X-56

T = 5x50-56

T = 194

Formula to find 'T' Total if you have the 'T'? (X) in a 8x8 grid:

=5X-56

Table of Results for a 10x10 Grid

See next page

X

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

77

88

89

90

91

92

93

94

95

96

87

89

99

00

'T' ? (X)

'T' Total (T)

22

40

37

15

69

275

75

305

T = X+(X-10)+(X-19)+(X-20)+(X-21)

T = 5X-70

Check

T = 5X-70

T = 5x69-70

T = 275

Formula to find 'T' Total if you have the 'T'? (X) in a 10x10 grid:

=5X-70

I have now found the formulae for three grid sizes:

8x8 = 5X-56

9x9 = 5X-63

0x10 = 5X-70

The link between these formulae is that the number being subtracted is always the grid size multiplied by 7:

8x8 = 5X- (8x7) = 5X-56

9x9 = 5X- (9x7) = 5X-63

0x10 = 5X- (10x7) = 5X-70

Two other links are:

) The square above the 'T' ? is always the 'T' ? subtracted by the grid size.

2) The square above that is always the 'T' ? subtracted by the grid size which has been multiplied by 2.

8x8

9x9

0x10

Therefore these squares on the grid could be made into X-G and X-2G ('G' meaning grid size):

The last two links are in the 'T' Shapes are:

a) The square to the right of the middle square on the top is always the grid size multiplied by 2 and subtracted by 1.

b) The square to the left of the middle square on the top is always the grid size multiplied by 2 and then 1 is added.

8x8

9x9

0x10

Therefore these squares on the grid could be made into X-2G-1 and X-2G+1:

Now that the 'T' shape has been put into algebra that incorporates the grid size, a formula that incorporates the grid size can be made.

T= X+(X-G)+(X-2G)+(X-2G-1)+(X-2G+1)

Introduction

We have been given the task to find the relationship between the 'T' ? and the 'T' Total 9x9 grid. The 'T' ? is the number at the bottom of the T shape and the 'T' Total is the sum of all the numbers inside the T.

Example

'T' Total = 1+2+3+11+20

= 37

To find the relationship between the 'T' ? and the 'T' Total, algebra will be needed. Therefore the 'T' ? will be represented by 'X' and the 'T' Total will be represented by 'T'. I will substitute the numbers in the T and replace them with those numbers but with the 'T' ? (X) as the subject.

Example

From this we can come up with a formula. I will check the formula with one of my other results for that grid. If the relationship between the T' ? and the 'T' Total match the formula will be correct and the problem will be solved.

Table of Results for a 9x9 Grid

X

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

'T' ? (X)

'T' Total (T)

20

37

42

47

67

272

80

337

T = X+(X-9)+(X-17)+(X-18)+(X-19)

T = 5X-63

Check

T = 5X-63

T = 5x42-63

T = 147

Formula to find 'T' Total (T) if you have the 'T'? (X) in a 9x9 grid:

=5X-63

Now I have solved the first problem I am going to further my investigation and vary the grid sizes. I am going to use the same method to find the formulas. The formulas should been linked and if I can find that link I cam make a formula that incorporates grid size. This would mean I could find the 'T' Total if I had the T'? in any grid size.

Table of Results for an 8x8 Grid

X

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

'T' ? (X)

'T' Total (T)

8

34

29

89

50

94

61

244

T = X+(X-8)+(X-15)+(X-16)+(X-17)

T = 5X-56

Check

T = 5X-56

T = 5x50-56

T = 194

Formula to find 'T' Total if you have the 'T'? (X) in a 8x8 grid:

=5X-56

Table of Results for a 10x10 Grid

See next page

X

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

77

88

89

90

91

92

93

94

95

96

87

89

99

00

'T' ? (X)

'T' Total (T)

22

40

37

15

69

275

75

305

T = X+(X-10)+(X-19)+(X-20)+(X-21)

T = 5X-70

Check

T = 5X-70

T = 5x69-70

T = 275

Formula to find 'T' Total if you have the 'T'? (X) in a 10x10 grid:

=5X-70

I have now found the formulae for three grid sizes:

8x8 = 5X-56

9x9 = 5X-63

0x10 = 5X-70

The link between these formulae is that the number being subtracted is always the grid size multiplied by 7:

8x8 = 5X- (8x7) = 5X-56

9x9 = 5X- (9x7) = 5X-63

0x10 = 5X- (10x7) = 5X-70

Two other links are:

) The square above the 'T' ? is always the 'T' ? subtracted by the grid size.

2) The square above that is always the 'T' ? subtracted by the grid size which has been multiplied by 2.

8x8

9x9

0x10

Therefore these squares on the grid could be made into X-G and X-2G ('G' meaning grid size):

The last two links are in the 'T' Shapes are:

a) The square to the right of the middle square on the top is always the grid size multiplied by 2 and subtracted by 1.

b) The square to the left of the middle square on the top is always the grid size multiplied by 2 and then 1 is added.

8x8

9x9

0x10

Therefore these squares on the grid could be made into X-2G-1 and X-2G+1:

Now that the 'T' shape has been put into algebra that incorporates the grid size, a formula that incorporates the grid size can be made.

T= X+(X-G)+(X-2G)+(X-2G-1)+(X-2G+1)