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Introduction

Daniel Smith                Maths Coursework Maths Coursework

T Total Introduction

In my maths casework I am investigating the relationship between the T-Number and the T-Total, throughout a range of different size grids. I am going to work out the rule for any size grid that is by 10, below is a 9 by 10 grid, the t is the coloured in bit and the red number is the T-Number.

So the T-Number is 50 in this 9X10 grid is 50. The T-Total for this T is all the numbers in the T added up 50+41+31+32+33=187 The T-Total would be 187 as this is all the coloured in squares added up.

So in my coursework I am going to use different grid sizes to translate the T-Shape to different positions .I will then investigate the relationship between the T-Number and the T-Total, and the grid size.

I am then going to use different size grids to try to work out Ts in all different ways. Like the grid below, I am going to work out rules for all the T-Numbers with the T standing different ways.

This grid shows the ways in which I am going to work out the T- Number in different ways. Looking for patterns and predicting the next T

9X10 Grids I am first going to work out the T-Total for 5 consecutive Ts, starting at 20 and going up to 24. T-Number = 20

T-Total = 37 T-Number = 21

T-Total = 42 T-Number = 22

T-Total = 47 T-Number = 23

T-Total = 52 T-Number = 24

T-Total = 57

 T-Number T-Total 20 37 21 42 22 47 23 52 24 57

Middle

23

45

24

50

25

55

26

60

From this table I can see that each time I move the T one place to the right, this is because each digit increases by 1, making it 5 each time. From this I predict that for T-Number 27 the T-Total will be 65, I will now prove my theory. T-Number - 27

T-Total – 65

This shows that my prediction was correct, each time I move the T one place to the right that the T-Total increases by 5.  This is because each time the T moves one place to the right each number increase by 1, there are 5 numbers in the T, so the T-Total will increase by 5.

The formula for a T in the upright position in a 10X10 grid.

I am going to use the T with the numbers From this I can see that there is a rule for the way that the numbers in the T are arranged and I can see that it is From this I can see that on a 10 by 10 grid the formula equals

N + (N-10) + (N-20) + (N-21) + (N-19)

I can now simplify this by gathering up all the n terms and the numbers to make the formula 5N – 70

I will now prove this on a 10 by 10 grid using the numbers I will now use my formula to work out the T-Total

T-Total = 5N – 70

= (57X5) – 70

= 285 - 70

= 215

This shows that my formula works as the numbers in my T added up = 36+37+38+47+57 = 215

Conclusion

T-Total = 5N + 7 = (5X59) + 7

= 295 + 7

= 302

By using the formula my T-Total is 302 I will now prove this by adding the numbers up manually

59 + 60 + 61 + 52 + 70 = 302.

This shows that my formula for a T on its side is 5N + 7. I am now going to work out the formula for a T on its other side. Using these numbers I am going to work out the difference between the each number in the T, I will then put it into a diagram below. This diagram shows me the difference between the numbers in any 9X10 grid that is on its side, now I have the differences I am going to gather them up to get my formula.

T-Total = N + (N - 1) + (N - 2) + (N + 7) + (N - 11)

Now I will gather it all up and it comes to 5N - 7.

To prove this works I am going to randomly select a T from a 9X10 grid then work out its T-Total by adding it up, and then using my formula. T-Total = 5N – 7

= (5X62) – 7

= 310 – 7

= 303

By using the formula my T-Total is 303 I will now prove this by adding the numbers up manually

62 + 61 + 60 + 51 + 69 = 303.

This shows that my formula for a T on its side is 5N - 7.

I have now found out the formulas for Ts in a upright, upside down and on bots side positions I am now going to put them into a table below and try to work out a pattern.

 Which way up Rule 5N - 63 5N + 7 5N - 7 5N + 63

2007                T-Total coursework

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