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• Level: GCSE
• Subject: Maths
• Word count: 1673

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

Extracts from this document...

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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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.

1. Each time the grid size increases by 1, so does the constraint.
2.  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
1. 7x7 GRID

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

1. 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

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