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Introduction

GCSE T-totals Coursework

Introduction

In this project I will be investigating the formula, patterns and relationships between the t-numbers, t-shape and t-totals in different sized grids 10x10, 9x9, 8x8, 7x7.

I have a grid nine by nine starting with the numbers 1-54. There is a shape in the grid called the t-shape which is highlighted in red shown in the table below.

 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

The t-number is the number at the bottom of the t-shape which is 20

The t-total is all the numbers in the t added up together which is 1+2+3+11+20=37

 T-number T-total 20 37 21 42 22 47 23 52 24 57 25 62 26 67

As you can see:

The t-number increases by 1 each time.

The t-total increases by 5 each time is there a link?

20x5=100

100-63=37 the t-total

The link between 63 and 9 is 7 because 7x9=63

So the formula is T-number x 5 (7x9)

5n –number-7x9

How did I work out this and what can we do with this formula?

The formula starts with 5 as there is a rise between the t-total of 5 each time. We then -63. I got this number by working out the difference between the t-number and the other numbers in the t-shape. E.g.

 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

Middle

51

52

53

54

5x48-63=t-total

5x48-63=177

Check to prove formula

T-total=29+30+31+39+48=177

This proves my formula works.

I will now try using grids of different sizes and translating the t-shape into different positions. I will then investigate the relationship between the t-number, t-total and grid size.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 181 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

T-number=22

T-total=40

The t-number and t-total have both risen even though the t-shape is in the same place as it was before. The t-number has risen by 2 and the t-total by 3. This is because I am now using a ten by ten grid. I will now try and find some more t-numbers and t-totals for some more t-shapes on this grid.

T-number=55

T-total=34+35+36+45+55=205

Now I will try and find a formula for the ten by ten grid. I will first find the difference between the t-total and other numbers in the t-shape.

22-1=21

22-2=20

22-3=19

22-12=10

Total=70

Here is another way to figure out the formula

 n-21 n-20 n-19 n-10 n

n+n-10+n-20+n-21+n-19

So this would mean that my formula will be 5n-70

I will try and test my new formula

5n-70=t-total

5x22-70=40

The same formula can be used for each grid with only changing the last number in the formula. I will now try it on a different sized grid to prove this.

 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

T-number=18

T-total=1+2+3+10+18=34

Difference

18-1=17

18-2=16

18-3=15

18-10=8

Total=56

 n-17 n-16 n-15 n-8 n

n+n-8+n-17+n-16+n-15= 5n-56

5n-56=t-total

5x18-56=34

As you can see by changing the grid size we have to change the end of the formula but we still keep the rule of how you get the number to minus in the formula.

So I can find formulas for all grid sizes, which are shown below in the table

 Grid width Formula 4x4 5n-28 5x5 5n-35 6x6 5n-42 7x7 5n-49 8x8 5n-56 9x9 5n-63 10x10 5n-70

Conclusion

The difference between each t-total is 5 so again the formula will start with 5, I will then multiply 5 with 12 the t-number. 5x12=60 we then minus 7 to get 53 the t-total.

So therefore the formula will be:

5n-7

To check and make sure the formula works we do:

 n-11 n-2 n-1 n n+7

n+n-1+n-2+n-11+n+7=5n-7

I will now rotate the t-shape 180 degrees from its last position. I predict that the same will happen as before and the equation will stay the same but the minus sign will change to a plus sign. I will prove this below.

 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

T-number=10

T-total= 10+11+12+3+21=57

To find the formula 5n+7 that I predicted we do:

5x10(t-number) =50+7=57

To check the formula we do:

 n-7 n n+1 n+2 n+11

n+n+1+n+2+n+11+n-7=5n+7

The formula is correct.

Conclusion

In this coursework I found many ways to solve the problem with the t-shape being in various different positions and different grid sizes. I have made my calculations less difficult by creating a formula which changes according to the different circumstances.

Below I have shown the different formulas I have come up with these only apply to the 9x9 grid.

5n-63

After 180 degrees rotation

5n+63

270 degrees rotation

5n-7

90 degrees rotation

5n+7

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