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investigation into T-shapes

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Introduction

investigation into T-shapes

Looking at the 9-9 grid below and the T-shape drawn on it,

The total number of the numbers on the inside of the T-shape is called the T-total

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 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

The t-total for this T-shape is:

1+2+3+11+20=37

So 37 = T-total

The number at the bottom is the T-number, So the T-number for this shape is 20

Aims:

1) Investigate the relationship between the T-total and the T-number

2) Use the grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total and the T-number and the grid size.

3) Use grids of different sizes again, try other transformations and combinations of transformations. Investigate relationships between the T-total and the T-number and the grid size and the transformations.

Aim 1- the solution

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 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

T69= 50+51+52+60+69

=282

T22=3+4+5+13+22

=47

...read more.

Middle

I´ll put that idea into another T. Note W= width number(9)

N-(2W-1) N-2W N-(2W+1)

N-W

N

This is the same thing as before but shown algebraically.

The formula for the Value of the T-total now is shown as:

5N-7W=T-total

Aim 2- different sizes and relationship

I know this works for the grid 9 by 9 but I´m not sure if it´ll work for any other grids.

Here is a test for a 10 by 10 grid

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 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 99 100

T22=1+2+3+12+22

=42 I notice this is 5 more than 9 by 9

T69=48+49+50+59+69

=275 Obviously no pattern there.

Method test?

(695)-70=275 YES it worked

My method seems to have worked out as it is logical and fairly straight forward to explain.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 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 99 100

...read more.

Conclusion

1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17 18 19 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 99 100 101 102 103 104 105 106 107 108

109 110 111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130 131 132

133 134 135 136 137 138 139 140 141 142 143 144

Stretch A will be called ST64 as it starts at 64, it´s a stretch of 2 in both directions.

St64=26+27+28+29+30+40+52+64

=296

I think I can work out the formula using my previous method so:

12+24+36+(436)=216

21612=18

This means the formula is:

8N-18W=T-total

8N= number of integers in the T-shape

18W=difference number calculated

Conclusion:

The size of the T-shape calculates the number before N in the formula and the grid size calculates the value of W. the number before W is calculated by looking at the rows and finding how many rows away from the T-number they are. If the T is regular then the W number is negative but if the T is flipped upside down the W number is positive.

...read more.

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