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T-total.During this coursework, I will be investigating the relationships between the T-shapes and how they relate to grid size.

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Introduction

Jacob Lie                        Maths Coursework

T-total

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On the grid on the right, you can see a 9 by 9 grid. On the grid, we see a “T” shape highlighted. The sum of the numbers within the T-shape is 1+2+3+11+20 = 37. This is known as the T-total.

The T-number is the number that is at the bottom of the T-shape. In this example, 20 is the T-number.

During this coursework, I will be investigating the relationships between the T-shapes and how they relate to grid size. I will also be looking closely into the significance of the T-number and how it could be used to figure out the T-total.

9 by 9 Grid

We have already figured out the t-total for one t-shape in the 9 by 9 grid. Here are some more results.

34+35+36+44+53 = 202

46+47+48+56+65 = 262

5+6+7+15+24 = 57

58+59+60+68+77 = 322

In this investigation, I’ll be implementing the use of equations. Here is how I started off.

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If I bring all these figures together, I should get a correct equation.

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

   = 5N – 63

Now if I replace “N” with the T-number, I should get a positive result.

N = 20

T = (5x20) – 63

   = 100 – 63

   = 37

...read more.

Middle

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As I did with the 9 by 9 grid, I worked out the t-total for various t-shapes in the grid the long way (adding each individual digit in the t-shape, one by one). I will the see

if the equation for the 9 by 9 grid would work on this

grid.

1+2+3+10+18 = 34

5+6+7+14+22 = 54

33+34+35+42+50 = 194

37+38+39+46+54 = 214

Then I attempted to use the equation in the previous grid to work out the t-total of a t-shape.

T = 5N - 63

N = 22

T = (5x22) – 63

   = 110 –63

   = 47

As we can see above, the result was wrong. So I decided to do what I did in the previous grid and work out a new equation.

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Now I bring all the terms together to get an equation.

T = N – 17+N – 16+N – 15+N – 7+N

   = 5N – 56

I then check if the equation is correct.

N = 18

T = (5x18) – 56

   = 34

We find that this new equation work in this case, but try it with a different t-number to confirm its validity.

N = 50

T = (5x50) –56

   = 194

Again, the equation has produced another correct result. I try it again one more time to make sure.

N = 22

T = (5x22) – 56

   = 54

So I have produced another equation that works with in this grid, but would it work in the next grid size?

7 by 7 Grid

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...read more.

Conclusion

63 = 7 x 9 or 42 = 7 x 6.  What we now notice from this is that if you divide the last number in each equation by 7, you will see that the result would be the grid size. An example of this is shown below.

For the 9 by 9 grid the formula is: -

T = 5N – 63

63 / 7 = 9

So now what I can say is to figure out the t-total of a t-shape in any size grid, you need the formula  “T = 5N – X”. The way you find out the t-total is as follows.

You firstly choose the t-shape you wish to find the t-total for. You then look at the grid size e.g. 7 by 7, and take the “7” from the grid size and multiply it by 7. This is your “X” value. You then multiply the t-number of the t-shape you are working out the t-total for and subtract the X value away from it. You should see, if done correctly, that the result would correspond with that of the real t-total.

...read more.

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