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T - Total. 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

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Introduction

Mathematics GCSET – TotalJames De Souza

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, the T-numbers and the grid size.

3) Use grids of different sizes again. Try other transformations and combinations of transformations. Investigate relationships between the T-total, the T-numbers 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

T22=3+4+5+13+22

=47

T69= 50+51+52+60+69

=282

In the diagram below it shows the difference between the T-number and the other numbers. First is the T-shape in question:

1 2 3

  11

  20

This is the T-shape and here is the Difference T-shape:

N-19 N-18 N-17

          N-9

           N

This shows the difference N= T-number

Simplified this can be written as 5N - 63

In the T

...read more.

Middle

Aim3- Transformations, (stretches) and there effects on the formula

I’ll do this with a 12 by 12 first, as this will give me enough accuracy to start with.

 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

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+(4*36)=216

216 / 12 = 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.

Below is a sheet with more simple things on it satisfying the same aim roughly explaining the above.

...read more.

Conclusion

79

           -

32

167

image01.png

135

56

32

209

image02.png

177

42

32

153

image03.png

121

- 56

36

124

image00.png

88

           -

36

187

image01.png

151

63

36

236

image02.png

200

49

36

173

image03.png

137

- 63

As you can see from the results the third difference of the difference is an inverse of the first difference of the difference.

So the formula for a T that has been rotated by 180 degrees should be an inverse of the formula used to work out a T in its upright position. Therefore

5N + 7W

It’s a plus sign as it’s the inverse of a minus sign.

Proof …

8 by 8 grid

1        2        3        4        5        6        7        8 (5 * 2) + (7 * 8) = 66

9        10        11        12        1314        15        16                2 + 10 +17 +18 +19 = 66

17        18        19        20        21        22        2324        It works !

25        26        27        28        29        30        31        32        

However the formula will only work for T’s rotated by 180 degrees.

In terms of the T – number the 270 degree rotation is equal to :

5N – 7

I found this out by accident I found out that it worked for one and then I tried it on other size grids and it worked.

5 * 21 – 7 = 98

21 + 22 + 23 + 13 + 29 = 98    

see it works. I then tested it on other places and it still worked. It wasn’t just a coincidence.

I took this further and used the same method as before in working out the inverse of this. To find out the T – total for a T that has been rotated by 90 degrees it will be the inverse of the 270 degree one.

5 * 14 + 7 = 77

14 + 15 + 16 + 24 + 8 =77   This also works.

image01.png        5N + 7

image03.png        5N - 7

image00.png        5N – 7W                                                                                                        

image02.png        5N + 7W

...read more.

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