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• Unlimited access from just £6.99 per month   # 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.

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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
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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

...read more.

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