# FINDING THE RELATIONSHIP BETWEEN THE T-NUMBER AND T-TOTAL USING DIFFERENT SIZED GRIDS.

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Introduction

AARUSHI MEHTA 10.3

T-TOTALS

FINDING THE RELATIONSHIP BETWEEN THE T-NUMBER AND T-TOTAL USING DIFFERENT SIZED 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 |

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 |

This is a T-Shape taken from the 9x9 grid shown above.

1 | 2 | 3 | |

11 | |||

20 |

The T-Total of a T-Shape is all the numbers in that T-Shape added together.

For example:

When we add the numbers inside this T-Shape which are 1+2+3+11+20 it equals 37. Therefore, the T-Total of this T-Shape is 37.

The number at the bottom of the T-Shape is the T-Number. Therefore, the T-Number of this T-Shape is 20.

The relationships between the T-Number and other numbers in the T-Shape will be:

1 | 2 | 3 | |

11 | |||

20 |

20-1=19 20-2=18

20-3=17 20-11=17

To see if these results apply on every T-Shape on a 9x9 grid, I will find the results for another T-Shape on my 9x9 grid.

61 | 62 | 63 | |

71 | |||

80 |

80-61=19 80-62=18

80-63=17 80-71=9

Since my results have been proven correctly, I will now work out a formula for any T-Shape on a 9x9 grid.

If I give my T-Number a letter (n) to represent it, the formula will be:

1 | 2 | 3 | |

11 | 11 | ||

20 | |||

n-19 | n-18 | n-17 | |

n-9 | |||

n |

=

*THIS FORMULA APPLIES ON ANY T-SHAPE ON A 9x9 GRID ONLY.*

After this, I decided to work out another formula for the T-Shape. This is the formula between the T-Total and T-Number.

Middle

75

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79

80

81

82

83

84

85

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87

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90

91

92

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100

T-Total= N + N-21

+ N-20

+ N-19

+ N-10

= 5N-70.

This is the formula for finding the T-Total on a 10x10 grid.

N-21 | N-20 | N-19 | |

N-10 | |||

N |

To find the relationship between the T-Shape T-Total and Grid Size:

N= T-Number

G = Grid Size

NO. INSIDE T-SHAPE | FORMULA | JUSTIFICATION | |

5 | N-2G-1 | 23-18-1 | |

6 | N-2G | 23-18 | |

7 | N-2G+1 | 23-18+1 | |

14 | N-G | 23-10 | |

23 | N | 23 | |

5 | 6 | 7 | |

14 | |||

23 |

This T-Shape is

taken from a

9x9 grid.

To prove these formulas, I’ll try them on T-Shapes from other sized grids.

NO. INSIDE T-SHAPE | FORMULA | JUSTIFICATION | |

34 | N-2G-1 | 51-16-1 | |

35 | N-2G | 51-16 | |

36 | N-2G+1 | 51-16+1 | |

43 | N-G | 51-8 | |

51 | N | 51 | |

34 | 35 | 36 | |

43 | |||

51 |

1. This T-Shape is

taken from an 8x8 grid.

31 | 32 | 33 | |

42 | |||

52 | |||

NO. INSIDE T-SHAPE | FORMULA | JUSTIFICATION | |

31 | N-2G-1 | 52-20-1 | |

32 | N-2G | 52-20 | |

33 | N-2G+1 | 52-20+1 | |

42 | N-G | 52-10 | |

52 | N | 52 |

2. This T-Shape is taken from a

10x10 grid.

I HAVE FOUND THAT MY FORMULAS WORK ON ALL GRID SIZES.

i.e.8x8,9x9,10x10 ETC.

Now I have to find the general formula for all these formulas.

N-2G-1 | N-2G | N-2G+1 | |

N-G | |||

N |

N-2G-1 + N-2G + N-2G+1 + N-G + N = 5N -7G.

To find the general formula, we have to add all the N’s i.e. 5N, and all the G’s i.e. 7G. Therefore, the formula is 5N-7G.

To prove this formula, I will use two 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 |

This is a 5x5 grid.

2 + 3 + 4 + 8 + 13 = 30

N=13 therefore 5N= 65

G=5 therefore 7G= 35

65-35=30. ←This is the T-Total for my T-Shape.

Now I will try this on a 6x6 grid.

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 |

8 + 9 + 10 + 15 + 21 = 63

N=21 therefore 5N = 105

G=6 therefore 7G = 42.

105-42=63.← This is the T-Total for my T-Shape.

I HAVE NOW PROVED THAT MY FORMULA WORKS.

VECTORS & TRANSLATIONS

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 |

Conclusion

= (110-15) + 120 – 56 = 159

When I do this manually, 26 + 27 + 28 + 35 + 43 also equals 159.

Now I am going to try this on another sized grid.

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 |

EXAMPLE: 7x7 GRID

RED TOBLUE

(-3) x

(-4) Y

5 x 20 + (5 x -3) – (5 x 7 x -4) – (7 x 7)

= (100-15) + 140 - 49 = 176

When I do this manually, 30 + 31 + 32 + 38 + 45 also equals 176.

I have now proved my formula and noticed that when the selected T moves horizontally, it increases or decreases by 1. When it moves vertically, it increases or decreases by the grid size.

Constraints

What are constraints?

→ Constraints are the maximum no. of spaces movable in a grid.

This is a table of constraints for different sized grids.

Grid Size | Constraint (max. no. of spaces movable) |

3 | 0 |

4 | 1 |

5 | 2 |

6 | 3 |

7 | 4 |

8 | 5 |

9 | 6 |

10 | 7 |

11 | 8 |

Looking at this table I found two things.

- Each time the grid size increases by 1, so does the constraint.
- In each case, the constraint is 3 less than the grid size (G).

To prove my results, I have made four grids.

1 | 2 | 3 |

5 | ||

8 |

1. 3x3 Grid

There can’t be any movement in this T-Shape. Therefore, the constraint is 0.

1 | 2 | 3 | |

6 | |||

10 | |||

2. 4x4 Grid

These arrows show that the T-Shape can move 1 place horizontally and 1 place vertically. Therefore, the constraint is 1.

1 | 2 | 3 | ||||

9 | ||||||

16 | ||||||

- 7x7 GRID

The arrows show that the T-Shape can move 4 places horizontally and 4 places vertically. Therefore, the constraint is 4.

- 11x11 GRID

1 | 2 | 3 | ||||||||

13 | ||||||||||

24 | ||||||||||

The arrows show that the T-Shape can move 8 places horizontally and 8 places vertically. Therefore, the constraint is 8.

PAGE

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

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