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T-total Investigation

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Introduction

Habibur Rahman                        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

...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

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 t0number 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.

What you can also figure out with this method is the t-total of an upside-down t-shape. To do this, you do as above, but instead of subtracting the X value away from 5N, you add it on. Note that all of the above equations have been based on upright t-shapes and don’t work if the t-shape is on its side.

...read more.

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