T  Total. Looking at the 99 grid below and the Tshape drawn on it, The total number of the numbers on the inside of the Tshape is called the Ttotal
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Introduction
Mathematics GCSET – TotalJames De Souza
Investigation into T shapes
Looking at the 99 grid below and the Tshape drawn on it,
The total number of the numbers on the inside of the Tshape is called the Ttotal
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 ttotal for this Tshape is:
1+2+3+11+20=37
So 37 = Ttotal
The number at the bottom is the Tnumber, so the Tnumber for this shape is 20
Aims:
1) Investigate the relationship between the Ttotal and the Tnumber
2) Use the grids of different sizes. Translate the Tshape to different positions. Investigate relationships between the Ttotal, the Tnumbers and the grid size.
3) Use grids of different sizes again. Try other transformations and combinations of transformations. Investigate relationships between the Ttotal, the Tnumbers 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 Tnumber and the other numbers. First is the Tshape in question:
1 2 3
11
20
This is the Tshape and here is the Difference Tshape:
N19 N18 N17
N9
N
This shows the difference N= Tnumber
Simplified this can be written as 5N  63
In the T
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:
8N18W=Ttotal
8N= number of integers in the Tshape
18W=difference number calculated
Conclusion:
The size of the Tshape 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 Tnumber 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.
Conclusion
79

32
167
135
56
32
209
177
42
32
153
121
 56
36
124
88

36
187
151
63
36
236
200
49
36
173
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.
5N + 7
5N  7
5N – 7W
5N + 7W
This student written piece of work is one of many that can be found in our GCSE TTotal section.
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