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• Level: GCSE
• Subject: Maths
• Word count: 2538

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Introduction

Tom white

Page  1- intro

Page  2- Basic t-numbers, t-shapes and t-totals

Page  3- the basic t-formula

Page  4- Algebra in a t-formula

Page  5- -

Page  6- 90˚ rotation

Page  7- -

Page  8- 180˚ rotation

Page  9- -

Page 10- 270˚ rotations

Page 11- -

Page 12- translations

Page 13- -

Page 14- -

Page 15- transformations

Page 16- -

Page 17- -

Page 18- conclusion

Page 19- -

Intro

Over these next pages I will be investigating the formulas and patterns between all t-numbers, t-shapes and t-totals in different sized grids 9x9, 8x8, 7x7 and 6x6. I will also investigate translations and transformations in t-shapes.

1

Basic t-numbers, t-shapes and t-totals

A t-shape is 5 cells anywhere on grids in a t shape

 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 t-number is the number at the bottom of the t-shape e.g. 21, 34 and 68

The t-total is where you add up all the numbers in the t-shape e.g. 2+3+4+12+21=42

49+50+51+59+68=276

15+16+17+25+34=97

2

Basic t-formulas

This is of a 9x9 grid

 T-number t-total 57 222 58 227 59 232 60 237 61 242

You can see the t-total in going up 5 every time as the t-number is going up 1 every time their must be a link?

57x5=285

285-63=222 the t-total

What is the link between 63 and 9?

63 is 7x9

So the formula would be

T-number x 5 – (9 x 7)

5xt-number-7x9

3

Algebra in a t-formula

I am trying to find the number in each cell compared to the t-number

 A B C D T

7x7grid

 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

24-17=7

36-29=7

41-34=7

Every time it =7

 A B C D T

T represents the t-number

D= T-grid

B= T-2grid

A= T-2grid-1

C= T-2grid+1

4

Middle

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81

Using a basic t-shape I am trying to find a formula to get the t-total after turning it 180˚ clockwise

 T-number T-total Before 180  turn 32 97 After 180  turn 32 223

I will now relate each cell to the t-number (t)

 a b c d t E g f h

A=t-2g-1                       E=t+g

B=t-2g                         F=t+2g

C=t-2g+1                        G=t+2g-1

D=t-g                         H=t+2g+1

8

When you put this all together into a formula you get

5N-7g for the t-shape before the rotation

5N+g+2g+2g+2g-1+1 simplified would be 5n+7g

This formula will work on any size grid and any t-number for e.g.

7x7grid

 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

The t-number is 17 so using the formula the t-total would be 5x17-7g=26   2+3+4+10+17=26

After the 180˚ turn clockwise the t-total would be

5x17+7g=134   17+24+30+31+32=134

So the formula works on different size grids and different t-numbers

9

270˚ rotation

6x6grid

 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

Using a basic t-shape I am trying to find a formula to get the t-total after turning it 270˚ clockwise

 T-number T-total Before 270˚ turn 23 73 After 270˚ turn 23 108

I will now relate each cell to the t-number (t)

 a b c g d f e t h

A=t-2g-1

B=t-2g

C=t-2g+1

D=t-g

E=t-1

F=t-2

G=t-2-g

H=t-2+g

_                         10

When you put this all together into a formula you get

5N-7g for the t-shape before the rotation

5N-1-2-2-2-g+g simplifies would be 5n-7

This formula will work on any size grid and any t-number for e.g.

8x8grid

 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

The t-number is 46 so using the formula the t-total would be 5x48-7g=184   29+30+31+38+46=184

After the 270˚ turn clockwise the t-total would be

5x46-7=233   46+45+44+36+52=233

So the formula works on different size grids and different t-numbers

11

Translations

Translations are were we can move the t-shape around the grid but before we can do this we have to create a formula for the movements.

8x8grid

 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

To move the t-shape across to the right 1 cell you add 1 onto each square so you add 5 to the t-total

So to move the t-shape 3 cells across to the right you need to add 15 to the t-total

But when you move to the left you tack 1 from each cell so you take 5 off the t-total

So to move the t-shape 3 cells across to the left you need to take 15 off the t-total

This means that all movement on the x is equal to +or- 5 so you times the grid movement eg. +1, -5, +8 by 5

12

X can be a – number and a + number

- If you won’t to go left

+ If u won’t to go right

So -5 means move across 5 cells to the left

And +3 means move 3 cells to the right.

And the same for the Y but we need the formulas before we can put the numbers in the coordinates.

6x6grid

 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

Conclusion

The formula I will use is 5n+7g+5x-5gy the 5n+7g is from the 180˚!!" rotation and the 5x is the horizontal movement and the 5gy is vertical movement.

5n+7g+5x-5gy

(5 x t-number) + (7 x grid) + (5 x -6) – ((5 x 9) x +6)

Using this formula the answer is 73

2+11+19+20+21=73

So the formula is right but now we need to change it for the 270˚ rotation

16

I will now test the formula for the 270˚ turn and translation

9x9

 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

I am going to move the t shape with 66 as its t-number +6 across and +6 up an turning it 180’ clockwise

The formula I will use is 5n-7+5x-5gy the 5n-7 is from the 270˚ rotation and the 5x is the horizontal movement and the 5gy is vertical movement.

5n-7+5x-5gy

(5 x t-number) -7 + (5 x +6) – ((5 x 9) x +6)

Using this formula the answer is 83

7+25+16+17+18= 83

So the all the formulas are right and produce a table of results for all I have found

17

Conclusion

This table is all the formals you need to rotate a t-shape

 rotated Basic formulas 0˚ 5n-7g 90˚ 5n+7 180˚ 5n+7g 270˚ 5n-7

This table is for the movement or translation for a t-shape

 movement Number added to total 1 cell right +5 1 cell left -5 1 cell down +5g 1 cell up -5g

5N-7g+5X+5gY

Use X+Y as coronets

9x9

 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

18

X movement is +4     Y movement is +1

5N-7g+54+5g1

 start 21 12 2 3 4 X +4 +4 +4 +4 +4 Y +9 +9 +9 +9 +9 finish 34 25 15 16 17

T-total =    5N-7g+54+5g1

5x21-7x9+5x4+5x9

105-63+20+45

42+65

107

34+25+15+16+17=107

19

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