# Investigate the relationship between the T-Total and the T-number for grid widths of different sizes.

Extracts from this document...

Introduction

PART 1

In this coursework I am going to investigate the relationship between the T-Total and the T-number for grid widths of different sizes. I am then going to rotate the T shape and do the investigation again for the rotated T shape.

We have a grid nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the t-shape. This is highlighted in the colour red. This is shown 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 |

55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 |

64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

The number 20 at the bottom of the t-shape will be called the t-number. All the numbers highlighted added together will be called the t-total. In this section there is an investigation between the t-total and the t-number.

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 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

## For this t-shape the

T-number is 20

And the

T-total is 1+2+3+11+20=37

## For this t-shape the

T-number is 21

And the

T-total is 2+3+4+12+21= 42

For this t-shape the

T-number is 22

And the

T-Total is 3+4+5+13+22=47

For this t-shape the

T-number is 23

And the

T-Total is 4+5+6+14+23=52

As you can see from this information, every time the t-number goes up one the t-total goes up five. This gives me another way to find the T-Total instead of adding up the numbers in the T all the time.

This helps us because when we want to translate a t-shape to another position. Say we move it to here

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 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

Now we know the T-Total and the T-number for the red T. to work out the green T-number I find the difference between both T-numbers, in this case it is 54.

Middle

T-number =52

T-Total =52+41+30+31+29=183

T-number =102

T-Total =102+91+80+79+81=433

Now I will find a formula for the 11*11 grid. First I will find the Difference between the T-Total and the other numbers in the T. then I know that it is 5n because there are still 5 squares in the T.

24-1= 23

24-2 = 22

24-3 =21

24-13 =11

TOTAL =77

So that means my formula is 5n-77

Try out the new formula

5n – 77= t-total

5*24-77=43

The same formula works for each grid with only changing the last number in the formula. This will be tried on a smaller grid size to prove that it works.

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

T-number = 10

T-total = 1+2+3+6+10= 22

Difference

10-1=9

10-2=8

10-3=7

10-6=4

10-10=0

9+8+7+4 =28

5n- 28= t-total

5*10-28=22

. We can see that by changing the grid size we have had to change the last part of the formula but still managing to keep to the rule of how you get the number to minus in the formula.

So I have found formulas for all grid sizes, which are

Grid width | Formula |

4*4 | 5n-28 |

5*5 | 5n-35 |

6*6 | 5n-42 |

7*7 | 5n-49 |

8*8 | 5n-56 |

9*9 | 5n-63 |

10*10 | 5n-70 |

11*11 | 5n-77 |

From looking at my results clearly I can see that the numbers at the end of the formulas are 7* the grid number. From this I should be able to find an overall formula that works on all grids. This formula is (g is grid width)

5n-7g

PART 3

In this next section there is change in the size of grid. Also there are transformations and combinations of T shapes. I will now be investigating the relationship between the t-total, the t-numbers, the grid size and the transformations.

Conclusion

This formula should be tested. The t-total of the blue t-shape is 37 and the t-total of the red t-shape is 208.

Formula

5n+(12*gridsize)= t-total

5*20+ 12* 9 = 208

The formula has worked.

#### CONCLUSION

In this project i have found out many ways in which to solve the problem i have with the t-shape being in various different positions with different sizes of grids. The way i have made the calculations less difficult is by creating a main formula that changes for all the different circumstances.

Here I have put all the formulas I have come up with. These formulas only apply to the nine by nine grids

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 |

5tn-63= t-total

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 |

5tn+63 = t-total

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 |

5tn-7= t-total

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 |

5tn+7= t-total

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 |

5tn-70= t-total

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 |

5tn+70 = t-total

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 |

5tn-56= t-total

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 |

5tn+56 = t-total

The different size of grid changes means the formula has to change slightly.

This is what happened.

T-shapes | number to x by 7 |

D & U | Grid size |

L & R | nothing |

DL & UR | Grid size -1 |

DR & UL | Grid size +1 |

We also have formula for rotation, which are

angle | formula |

45 degrees | 5tn-(7xG)+7= t-total |

90 degrees | 5tn-(7xG)+70= t-total |

135 degrees | 5tn-(7xG)+133= t-total |

180 degrees | 5tn-(7xG)+126 = t-total |

225 degrees | 5tn-(7xG)+119= t-total |

270 degrees | 5tn-(7xG)+56 = t-total |

315 degrees | 5tn-(7xG)-7= t-total |

We have a formula for reflection which is 5tn+(12*gridsize)= t-total.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month