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• Level: GCSE
• Subject: Maths
• Word count: 3569

# T-totals, Relationships between the T-number and the T-total on a 9 x 9 grid.

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Introduction

## Mathematics GCSE      T-totals     Alex Pavlou

1).  Relationships between the T-number and the T-total on a 9 x 9 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 T-totals T-numbers 21 42 26 67 42 147 66 267 80 337 86 367

The difference in each T-shape is:  N-19  N-18  N-17

N-9

N

If we take the T-shape:  2  3  4

12

21

we can create the sum:   t=21+(21-9)+(21-19)+(21-18)+(21-17)

As there are 5 numbers in the T-shape we need 5 lots of 21, the number above 21 is 12, which is 9 less than 21, the other numbers are 2,3 and 4 which is 9 less than 21.  Therefore we arrive to the conclusion:  N-19  N-18  N-17

N-9

N

To prove this we use the T-shape:  61  62  63

71

80

T=80-19+80-18+80-17+80-9+80

T=337

We can do the same for the T-shape:  47  48  49

57

66

T=66-19+66-18+66-17+66-9+66

T=267

To find the formula of the relationship between the T-number and the T-total we use N for the T-number.

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

T=5N-9-54

T=5N-63

Examples of this formula are :

In the case of the T-shape:  52  53  54

64

71

T=5(71)-63

T=355-63

T=292

When we add the numbers on the calculator we get 292

In the case of the T-shape:  64  65  66

74

83

T=5(83)-63

T=425-63

T=352

When we add the numbers on the calculator we get 352

2).  Relationship between the T-number, T-total and the grid size using different size grids.

 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

If we use the same formula as in question 1 for the T-shape:  10  11  12

19

27

we get 79 as the T-total.  To achieve this we use the formula:

Middle

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
 Centre of rotation Rotation (degrees) T-total Difference compared to original (41) 11 0 41 0 11 90 57 +16 11 180 69 +28 11 270 53 +12

To find the formula we must break the results down.

180°

As the T is upside down the equation is easy to find.

The formula is 5c+2g

We get this by:                C-7

C

C+6  C+7  C+8

T=C+C-7+C+6+C+7+C+8

T=5N+14

T=5N+(14divided by 7) (the sum is divided by 7 because this is the grid size)

T=5N+2g

To prove this we use a grid size 4 x 4.

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

3+7+10+11+12=43

Through these numbers we find the formulas

T=(5x7)+(2x4)

To double check we use the grid size 5 x 5

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2+7+11+12+13=45

T=(5x7)+(2x5)

This proves that T=5c+2g can be used to find the T-shape at 180°.

90° and 270°

 1 2 3 4 5 6 7 8 9

3+4+5+6+9=27

The formula can be found from:           c-2

c-1  c   c+1

c+4

Therefore if we substitute c for numbers we get:

T=(5+1)+(5+4)+(5-2)+(5-1)+5

T=27

Therefore we find the formula T=(c-1)+c+(c+1)+(c-2)+(c+4)

T=5c+2

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

To prove the formula we use a 4 x4 grid:

3+7+5+6+11=32

T=5c+2

T=(5x6)+2

T=30+2

T=32

This proves that the formula T=5c+2 can be used to find the T-total of any 270° shape where c is the centre of rotation.

The relationship between the 90° and the 270°is very similar so for a 90° flip the formula must be T=5c-2.

To prove this we use the grid size 6 x 6.

3+9+15+10+11=48

When we use the formula:

T=5c-2

T=(5x10)-2

T=50-2

T=48

Therefore proving that the formula is T=5c-2 for a 90° flip and can find the T-total

Now we can form the table:

 Rotation (Degrees) Direction Equation 0 Clockwise T=5c-2g 90 Clockwise T=5c+2 180 Clockwise T=5c+2g 270 Clockwise T=5c-2

We can also find different formulas.

180°

To find this formula we use the 3 x 3 grid.

 1 2 3 4 5 6 7 8 9
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

2+5+7+8+9=31

We can find the T-total through:  N

N+3

N+5  N+6  N+7

T=2+(2+3)+(2+5)+(2+6)+(2+7)

T=31

Therefore we discover the formula.

T=N+(N+3)+(N+5)+(N+7)+(N+6)

T=5N+21

T=5N+7g

This formula is also a T-shape formula turned around.  To prove this we use the grid size 4 x 4.

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

2+6+9+10+11=38

T=5N+7g

T=5(2)+7(4)

T=10+28

T=38

Therefore proving that we can use the formula T=5N+7g to find the total of any grid size.

90° and 270°

To find the formula for this we use grid size 4 x 4.

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

3+5+6+7+11=32

N-2

N  N+1  N+2

N+6

T=N+(N+1)+(N-2)+(N+2)+(N+6)

T=5N+7

To prove this we use the grid size 7 x 7.

 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 21 22

5+10+11+12+19=57

By using the formula we find: T=5N+7

T=5(10)+7

T=50+7

T=57

We get the T-total.

By this we know that by using the formula T=5N+7 we can the T-total on any grid size.

To find the T-total of a 90° flip we simply turn the equation around giving us T=5N-7.

To prove this we us the grid size 5 x 5.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2+7+12+8+9=38

Using the formula: T=5N-7

T=5(9)-7

T=45-7

T=38

Thus proving the formula T=5N-7 can be used on any grid size to find the T-total.  Now we can create a table:

 Rotation (degrees) Direction Equation 0 Clockwise T=5N-7g 90 Clockwise T=5N-7 180 Clockwise T=5N+7g 270 Clockwise T=5N+7

45° and 225°

The rule for a diagonal shape is much the same as a straight shape.  First I will find the relationship on a 9 x 9 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

45°

19+29+39+21+13=121

The difference in each T-shape is      N

N+6    N+8

N+16

C+26

T=N+N+6+N+8+N+16+N+26

T=5N+56

On a 7 x 7 grid the formula is:

 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

8+16+10+4+24=62

The difference is:     N

N+4   N+6

N+12

N+20

T=N+N+2+N+6+N+12+N+20

T=5N+42

The overall formula of a 45° is T=5N+(7g-7)

To prove this we use grid six 10 x 10:

 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

11+22+33+13+4=83

T=5N+(7g-7)

T=20+(70-7)

T=20+63

T=83

This proves that the formula for a 45° angle is T=5N+(7g-7)

225°

To find the formula for this we will start with a 9 x 9 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

7+17+27+25+33=109

The difference is:  N-26

N-16

N-8          N-6

N

T=N+N-8+N-16+N-26+N-6

T=5N-56

On a 7 x 7 grid the formula is:

 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

Conclusion

T=25+(70+7)

T=25+77

T=102

Now we can create a table of results for rotation:

 Rotation (degrees) Direction Equation 0 Clockwise T=5N-7g 45 Clockwise T=5N+(7g-7) 90 Clockwise T=5N-7 135 Clockwise T=5N-(7g+7) 180 Clockwise T=5N+7g 225 Clockwise T=5N-(7g-7) 270 Clockwise T=5N+7 315 Clockwise T=5N+(7g+7)

### Larger T

 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

Now that we have worked out the formulas of normal T’s we enlarge the shape.  If we double the T-shape so the volume is 4 times bigger the grid shows the new shape.

1+2+3+4+5+6+9+10+11+12+13+14+19+20+27+28+35+36+43+44=342

The T-number is 158 as the bottom 4 numbers make up the T-number and the T-total is 342.

The difference in each number (the 4 groups of numbers) is: N-136  N-128  N-120

N-64

N

T=N+N-64+N-136+N-120+N-128

T=5N-448

To prove this we use:

T=5N-448

T=6x158-448

T=342

Now we must find the overall formula.  We now use a 12 x 12 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

1+2+3+4+5+6+13+14+15+16+17+18+27+28+39+40+51+52+63+64=478

The T number is 230 and the T-total 478.  The difference in the T-shape is:

N-200  N-192  N-184

N-96

N

T=N+N-96+N-192+N-184+N-200

T=5n-672

To prove this:

T=5N-672

T=1150-672

T=478

To find the overall formula we must make a table of results

 Grid size Equation Difference 6 x 6 T=5N-336 0 7 x 7 T=5N-392 +56 8 x 8 T=5N-448 +56 9 x 9 T=5N-524 +56 10 x 10 T=5N-560 +56 11 x 11 T=5N-616 +56

With this table we notice a pattern and that is that every the difference in each is 56.

The formula can be easily formed:

T=5N-(7gx8)

To prove this we use a 15 x 15 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

1+2+3+4+5+6+16+17+18+19+20+21+33+34+48+49+63+64+78+79=580

T=5N-(7gx8)

T=1420-(105x8)

T=1420-(840)

T=580

This proves that our formula is correct.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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