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T-Total.I aim to find out relationships between grid sizes and T shapes within the relative grids, and state and explain all generalisations I can find, using the T-Number

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Introduction

Maths GCSE Coursework T-Total Lee Taylor

Introduction

In this investigation I aim to find out relationships between grid sizes and T shapes within the relative grids, and state and explain all generalisations I can find, using the T-Number (x) (the number at the bottom of the T-Shape), the grid size (g) to find the T-Total (t) (Total of all number added together in the T-Shape), with different grid sizes, translations, rotations, enlargements and combinations of all of the stated.

Relations ships between T-number (x) and T-Total (t) on a 9x9 grid

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From this we can see that the first T shape has a T number of 50 (highlighted), and the T-total (t) adds up to 187 (50 + 41 + 31 + 32 + 33).

...read more.

Middle

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From these Extra T Shapes we can plot a table of results.

T-Number (x)

T-Total (t)

20

37

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67

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182

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197

80

337

From this table the first major generalisation can be made,

The larger the T-Number the larger the T-Total

The table proves this, as the T-Numbers are arranged in order (smallest first) and the T-Totals gradually get larger with the T-Number.

From this we are able to make a formula to relate T-Number (x) and T-Total (t) on a 9x9 grid. Taking the T-number of 20 as an example we can say that the T-Total is gained by:

t = 20 + 20 – 9 + 20 – 19 + 20 – 18 + 20 – 17 = 37

The numbers we take from 20 are found, as they are in relation to it on the grid, as the T-Shape spreads upwards all numbers must be less by a certain amount, these are found by the following method;

As there are 5 numbers in the T-Shape we need 5 lots of 20, the number adjacent to 20 is 11 which is 9 less than 20, the other numbers in the T-Shape are 1,2 & 3 which are 19, 18 & 17 less than 20. Thus the above basic formula can be generated.

If we say that 20 is x and x can be any T-Number, we get:

t = x + x – 9 + x – 19 + x – 18 + x – 17

To prove this we can substitute x for the values we used in are table, we get the same answers, for example taking x to be 80:

t = 80 + 80 – 9 + 80 – 19 + 80 – 18 + 80 – 17 = 337

And x as 52;

t = 52 + 52 – 9 + 52 – 19 + 52 – 18 + 52 – 17 = 197

Thus proving this equation can be used to find the T-Total (t) by substituting x for the given T-Number. The equation can be simplified more:

t = x + x – 9 + x – 19 + x – 18 + x – 17

t = 2x – 9 + 3x – 54

t = 5x – 63

Therefore, we can conclude that:

On a 9x9 grid any T-Total can be found using t = 5x – 63 were x is the T-Number.

We can also say that on a 9x9 grid that;

  • A translation of 1 square to the right for the T-Number leads to a T-total of +5 of the original position.
  • A translation of 1 square to the left for the T-Number leads to a T-total of -5 of the original position.
  • A translation of 1 square upwards of the T-Number leads to a T-total of -45 of the original position.
  • A translation of 1 square downward of the T-Number leads to a T-total of +45 of the original position.

Finding relationships on grids with sizes other than 9x9

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If we take this 8x8 grid with a T-Number of 36 we get the T-Total of 124  (36 + 28 + 20 + 19 + 21), if we generalise this straight away using the same method’s used in before for a 9x9 grid we achieve the formula:

t = x – 8 + x – 17 + x – 16 + x – 15

t = 2x – 8 + 3x – 48

t = 5x – 56

This can also be shown in this form,

x-17

x-16

x-15

x-8

x

...read more.

Conclusion

Another area that we can investigate is that of differing grid such as 4x7 and 6x5 will make a difference to this formula.

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So if we work this grid out using the “old” method we get a T-Total of 67, and a formula of:

t = x + x – 4 + x – 9 + x – 8 + x – 7

t = 5x – 28

This formula is the same as the 4x4 grid as it has a “magic number” of 28, identical to a 4x4 grid, we can predict that grid width is the only important variable, but we will need to prove this.

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On a 6x5 grid with a T-Number of 15 the T-Total is 33, again working out the formula leads to

t = 5x – 42

Again this is the same “magic number” as found by the predictions for a 6x6 grid found earlier, therefore we can state that:

5x – 7g can be used to find the T-Total (t) of any Grid save, regular (e.g. 5x5) or irregular

 (e.g 5x101),

with the two variables of grid width (g) and the T-Number (x).

...read more.

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