# T-Totals. Firstly I am going to do a table of 5 x 5 and look at the T-totals and T-numbers.

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Introduction

Coursework Number One

T-Totals

I am going to investigate the relationship between the T-totals and the T-number. Firstly I am going to do a table of 5 x 5 and look at the T-totals and T-numbers.

Here is a 9 by 9 table

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 total number inside the T-shape is

1 + 2 + 3 + 11 + 20 = 37

The T-Number is 20

Now I am going to put the T-shape in another location

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-number is 21

The T-total for this T-shape is 2 + 3 + 4 + 12 + 21 = 42

I will continue to place the T-shape in different location.

The T-number is 22

3 + 4 + 5 + 13 + 22 = 47

T-total = 47

The T-number is 23

4 + 5 + 6 + 14 + 23 = 52

T-total = 52

The T-number is 24

5 + 6 + 7 + 15 + 24 = 57

T-total = 57

The T-number is 25

6 + 7 + 8 + 16 + 25 = 62

T-total = 50

The T-number is 26

7 + 8 + 9 + 17 + 26 = 67

T-total = 67

The rest of the answers will be put into a table.

T-number | T-total | T-number | T-total | T-number | T-total | ||

20 | 42 | 43 | 157 | 66 | 272 | ||

21 | 47 | 44 | 162 | 67 | 277 | ||

22 | 52 | 45 | 167 | 68 | 282 | ||

23 | 57 | 46 | 172 | 69 | 287 | ||

24 | 62 | 47 | 177 | 70 | 292 | ||

25 | 67 | 48 | 182 | 71 | 297 | ||

26 | 72 | 49 | 187 | 72 | 302 | ||

27 | 77 | 50 | 192 | 73 | 307 | ||

28 | 82 | 51 | 197 | 74 | 312 | ||

29 | 87 | 52 | 202 | 75 | 317 | ||

30 | 92 | 53 | 207 | 76 | 322 | ||

31 | 97 | 54 | 212 | 77 | 327 | ||

32 | 102 | 55 | 217 | 78 | 332 | ||

33 | 107 | 56 | 222 | 79 | 337 | ||

34 | 112 | 57 | 227 | 80 | 342 | ||

35 | 117 | 58 | 232 | 81 | 347 | ||

36 | 122 | 59 | 237 | 82 | 352 | ||

37 | 127 | 60 | 242 | 83 | 357 | ||

38 | 132 | 61 | 247 | 84 | 362 | ||

39 | 137 | 62 | 252 | 85 | 367 | ||

40 | 142 | 63 | 257 | 86 | 372 | ||

41 | 147 | 64 | 262 | 87 | 377 | ||

42 | 152 | 65 | 267 | 88 | 382 |

Here is a 5 by 5 table

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 |

The total number inside the T-shape is

1 + 2 + 3 + 7 + 12 = 25

The T-Number is 12

Now I am going to put the T-shape in another location

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 |

The T-number is 13

The T-total for this T-shape is 2 + 3 + 4 + 8 + 13 = 30

I will continue to place the T-shape in different location and once I have enough data I will put it into a table.

The T-number is 14

3 + 4 + 5 + 9 + 14 = 35

T-total = 35

The T-number is 15

4 + 5 + 6 + 10 + 15 = 40

T-total = 40

The T-number is 16

5 + 6 + 7 + 11 + 16 = 45

T-total = 45

The T-number is 17

6 + 7 + 8 + 12 + 17 = 50

T-total = 50

The T-number is 18

7 + 8 + 9 + 13 + 18 = 55

T-total = 55

I am now assuming that the T-total for the T-number 18 is 55 because looking at the T-total results, it is going up in 5.

T-number | T-total |

12 | 25 |

13 | 30 |

14 | 35 |

15 | 40 |

16 | 45 |

17 | 50 |

18 | 55 |

19 | 60 |

20 | 65 |

21 | 70 |

22 | 75 |

23 | 80 |

24 | 85 |

I will test two of my guesses to see whether or not my assumption is correct.

Middle

16

17

18

19

20

21

22

23

24

25

I will refer these numbers to these letters below, with n being the T-number.

a | b | c |

d | ||

n |

The value of a is n – 11

The value of b is n – 10

The value of c is n – 9

The value of d is n – 5

So this is how the table now looks

n -11 | n - 10 | n - 9 |

n - 5 | ||

n |

I know that that everything in this T-shape is equal to the original answer therefore I can add these up to find the T-total.

y = T-total and n = T-number

The T-total is = n – 11 + n – 10 + n – 9 + n – 5 + n

y = 5n – 35

Now I will test this formula with the T-total that I have already done.

n = 14 and the y = 35

If the total was not known then I would put n into the formula

y = 5 ( 14 ) – 35

y = 70 - 35

y = 35

The formula worked but I will do one more to make sure.

If n is 19 and my y = 60

y = 5 ( 19 ) – 35

y = 95 – 35

y = 60

Therefore the formula for any T-number with a 5 by 5 table is

n = T-number y = T-total

y = 5n - 35

Now I will investigate the T-total on a 6 by 6 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 |

The way I have worked out the formula on a 5 by 5 grid can also be used on a 6 by 6 grid.

a | b | c |

d | ||

n |

n = 14

The value of a is n – 13

The value of b is n – 12

The value of c is n – 11

The value of d is n – 6

So this is how the T-shape looks

n -13 | n - 12 | n – 11 |

n - 6 | ||

n |

I will add up all the value inside the T-shape

y = n – 13 + n – 12 + n – 11 + n – 6 + n

y = 5n – 42

I will test this theory to see if it correct for any T-number

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 |

If n = 17 and y = T-total

y = 4 + 5 + 6 + 11 + 17

y = 43

So my formula is 5n - 42

y = 5n - 42

y = 5 ( 17 ) – 42

y = 85 -42

y = 43

The formula works however I will do a few more to make sure.

n = 35

y = 5n – 42

y = 5 ( 35 ) - 42

y = 175 – 42

y = 133

Check:

y = 22 + 23 + 24 + 29 + 35

y = 133

n = 15

y = 5n – 42

y = 5 ( 15 ) - 42

y = 75 – 42

y = 33

Check:

y = 2 + 3 + 4 + 9 + 15

y = 33

n = 20 and T-total = y

y = 5n – 42

y = 5 ( 20 ) - 42

y = 100 – 42

y = 58

Check:

y = 7 + 8 + 9 + 14 + 20

y = 58

Therefore the formula for any T-number with a 6 by 6 table is

n = T-number y = T-total

y = 5n - 42

I will now put all my results into a table

n | 5n - 42 | y |

14 | 5 ( 14 ) - 42 | 28 |

15 | 5 ( 15 ) - 42 | 33 |

16 | 5 ( 16 ) - 42 | 38 |

17 | 5 ( 17 ) - 42 | 43 |

18 | 5 ( 18 ) - 42 | 48 |

19 | 5 ( 19 ) - 42 | 53 |

20 | 5 ( 20 ) - 42 | 58 |

21 | 5 ( 21 ) - 42 | 63 |

22 | 5 ( 22 ) - 42 | 68 |

23 | 5 ( 23 ) - 42 | 73 |

24 | 5 ( 24 ) - 42 | 78 |

25 | 5 ( 25 ) - 42 | 83 |

26 | 5 ( 26 ) - 42 | 88 |

27 | 5 ( 27 ) - 42 | 93 |

28 | 5 ( 28 ) - 42 | 98 |

29 | 5 ( 29 ) - 42 | 103 |

30 | 5 ( 30 ) - 42 | 108 |

31 | 5 ( 31 ) - 42 | 113 |

32 | 5 ( 32 ) - 42 | 118 |

33 | 5 ( 33 ) - 42 | 123 |

34 | 5 ( 34 ) - 42 | 128 |

35 | 5 ( 35 ) - 42 | 133 |

Looking at this table I notice that the T-total has a difference of 5 which the 5 by 5 table also has a difference of 5. My guess is that the 7 by 7 table also has a difference of 5.

Here is a table 7 by 7

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 |

I will find the formula for this table.

a | b | c |

d | ||

n |

n = 16

The value of a is n – 15

The value of b is n – 14

The value of c is n – 13

The value of d is n – 7

So this is how the T-shape looks

n -15 | n - 14 | n – 13 |

n - 7 | ||

n |

I will add up all the value inside the T-shape

y = n – 15 + n – 14 + n – 13 + n – 7 + n

y = 5n – 49

So now I will test this to make sure that it is correct

If n = 16 and y = T-total

y = 1 + 2 + 3 + 16 + 9

y = 31

So my formula is 5n - 49

y = 5n - 49

y = 5 ( 16 ) – 49

y = 80 -49

y = 31

The formula works and now I will see if there is a difference of 5 between each T-total.

n | y |

16 | 31 |

17 | 36 |

18 | 41 |

19 | 46 |

20 | 51 |

21 | 56 |

22 | 61 |

23 | 66 |

24 | 71 |

25 | 76 |

26 | 81 |

27 | 86 |

28 | 91 |

29 | 96 |

30 | 101 |

31 | 106 |

32 | 111 |

33 | 116 |

34 | 121 |

35 | 126 |

36 | 131 |

37 | 136 |

38 | 141 |

39 | 146 |

40 | 151 |

41 | 156 |

42 | 161 |

43 | 166 |

44 | 171 |

45 | 176 |

46 | 181 |

47 | 186 |

48 | 191 |

I now realise that because n is a multiple of 5 the difference is always going to be 5.

Therefore the formula for any T-number with a 7 by 7 table is

n = T-number y = T-total

T-total = 5n - 49

So from now I would not need to do detail working out, so I will just find out the formula for all the other grids because they all follow a same pattern in working out the formula.

Here is a table 8 by 8

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 |

a | b | c |

d | ||

n |

n = 18

The value of a is n – 17

The value of b is n – 16

The value of c is n – 15

The value of d is n – 8

So this is how the T-shape looks

n -17 | n - 16 | n – 15 |

n - 8 | ||

n |

I will add up all the value inside the T-shape

y = n – 17 + n – 16 + n – 15 + n – 8 + n

y = 5n – 56

Instead of writing out the whole table to find a formula I will try to find it by using only a part of a table.

Therefore the formula for any T-number with a 8 by 8 table is

n = T-number and y = T-total

y = 5n - 56

I see that the way I have worked out the formula previously will be the same for the rest of the grids.

Grid | Formula |

5 by 5 | 5n - 35 |

6 by 6 | 5n - 42 |

7 by 7 | 5n - 49 |

8 by 8 | 5n - 56 |

9 by 9 | 5n - 63 |

10 by 10 | 5n - 70 |

11 by 11 | 5n - 77 |

12 by 12 | 5n - 84 |

13 by 13 | 5n - 91 |

14 by 14 | 5n - 98 |

15 by 15 | 5n - 105 |

16 by 16 | 5n - 112 |

Conclusion

y = 5n - 7 with any x by x grid, (only a sign change). This is because when the T-shape was flip the signs also changed but the formula was the same. So for the T-shape rotated at 90°, it is also being flipped therefore causing the sign to change as well from + to -.

I will do check one with a 15 by 15 grid to see if it is correct.

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 |

The T-total for this shaded area is 108

So now I will see if my formula is correct,

y = 5n – 7, where n = 23

y = 5 ( 23 ) – 7

y = 115 – 7

y = 108

The formula works.

T-shape rotated at 270°

Therefore the formula for any T-number with any x by x grid is

n = T-number and y = T-total and x = grid number

y = 5n + 7

Summery

Here are all the formulas for the T-Shape rotated at different degrees.

Therefore the formula for any T-number with x by x grid is

n = T-number and y = T-total and x = grid number

y = 5n – 7x

T-shape rotated at 180°

Therefore the formula for any T-number with x by x grid is

n = T-number and y = T-total and x = grid number

y = 5n + 7x

T-shape rotated at 90°

Therefore the formula for any T-number with any x by x grid is

n = T-number and y = T-total and x = grid number

y = 5n + 7

T-shape rotated at 270°

Therefore the formula for any T-number with any x by x grid is

n = T-number and y = T-total and x = grid number

y = 5n + 7

All the formulas are similar and the differences are only a sign change ( + or – ) or the formula either includes an x or not.

Zarlasht Sardar

Mathematics GCSE

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

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