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  • Level: GCSE
  • Subject: Maths
  • Word count: 8309

Investigating the links between the T-number and the T-total on a size 9 grid

Extracts from this document...

Introduction

Investigating the links between the T-number and the

T-total on a size 9 grid

Richard Smith 5α

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I will be using the number at the bottom of the T as the T-number, so in these two examples the T-numbers are 20 and 51. This will be the layout of each T: -

T2

T3

T4

T1

T-number

Take the differences between the T-number and T1, 2, 3 and 4.

n-19

n-18

n-17

n-9

n

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11

20

These differences are the same throughout the grid (size 9).

Examples

n-19

n-18

n-17

n-9

n

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51

n-19

n-18

n-17

n-9

n

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If you take all the differences, which add up to be -63 and take that from 5 (the amount of numbers in one T) multiplied by the T-number, it gives you the T-total. Here is the formula: -

5n-63=T-total

I will now test this formula using some of the T shapes above.

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5n-63 = T-total

5(20)-63 = T-total

37 = T-total

Also:

1 + 2 + 3 + 11 +20 = 37

5n-63 = T-total

5(51)-63 = T-total

192 = T-total

Also:

32 + 33 + 34 + 42 + 51 = 192

5n-63 = T-total

5(35)-63 = T-total

112 = T-total

Also:

16 + 17 + 18 + 26 + 35 = 112

As I have proved, the formula is correct for the T shape in a grid size of 9.

Rotating T through 900 clockwise

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I will use the same principal in naming each number as I did it the up-right T shape: -

T2

T-number

T1

T3

T4

I predict that I will be able to use the same theory of the differences between the T-number and T1, 2, 3 and 4. For example, I will add up the differences and add it from the T-number x 5 to get the formula.

n-7

n

n+1

n+2

n+11

Testing my prediction

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n+1

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The differences add up to +7, so I believe the formula will be: -

5n+7 = T-total

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5n+7 = T-total

5(10)+7 = T-total

57 = T-total

Also:

10 + 11 + 12 + 21 + 3 = 57

5n+7 = T-total

5(28)+7 = T-total

147 = T-total

Also:

21 + 28 + 29 + 30 + 39 = 147

5n+7 = T-total

5(41)+7 = T-total

212 = T-total

Also:

34 + 41 + 42 + 43 + 52 = 212

This proves that the formula was correct.

Rotating T through 1800 clockwise

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...read more.

Middle

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Differences

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n-5

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n-5

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n+1

n+2

n+9

The differences add up to +7, so I think the formula will be: -

5n+7 + T-total

Testing the Formula

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5n+7 = T-total

5(8)+7 = T-total

47 = T-total

Also:

3 + 8 + 9 + 10 + 17 = 47

5n+7 = T-total

5(19)+7 = T-total

102 = T-total

Also:

14 + 19 + 20 + 21 + 28 = 102

5n+7 = T-total

5(39)+7 = T-total

202 = T-total

Also:

34 + 39 + 40 + 41 + 48 = 202

This proves that the formula was correct.

Rotating T through 2700 clockwise

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Differences

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n-9

n-2

n-1

n

n+5

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n-9

n-2

n-1

n

n+5

The differences add up to –7, so I think the formula will be: -

5n-7 = T-total

Testing Formula

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5n-7 = T-total

5(10)-7 = T-total

43 = T-total

             Also:

1+ 8 + 9 + 10 + 15 = 43

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5n-7 = T-total

5(21)-7 = T-total

98 = T-total

Also:

12 + 19 + 20 + 21 + 26 = 98

5n-7 = T-total

5(41)-7 = T-total

198 = T-total

Also:

32 + 39 + 40 + 41 + 46 = 198

This proves that the formula was correct.

Size 7 Grid Formulas

Upright T :

900 T:

1800 T:

2700 T:

5n-49 = T-total

5n+7 = T-total

5n+49 = T-total

5n-7 = T-total

My Prediction that the numbers in the formula are multiples of 7 seems to be correct. This means that a general formula for any grid size can be created, by replacing the number (i.e. 63, 56, 49) by 7g (seven multiplied by the grid size). Here are the general formulas for each rotation: -

General Formulas

Upright T :

900 T:

1800 T:

2700 T:

5n-7g = T-total

5n+7 = T-total

5n+7g = T-total

5n-7 = T-total

Testing the General Formulas - Size 10 Grid

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5n-7g = T-total

5(22) -7(10) = T-total

40 = T-total

Also:

1 + 2 + 3 + 12 +22 = 40

5n-7g = T-total

5(45) -(10) = T-total

155 = T-total

Also:

24 + 25 + 26 + 35 + 45 = 155

5n-7g = T-total

5(59) -(10) = T-total

225= T-total

Also:

38 + 39 + 40 + 49 + 59 = 225

This proves that the general formula is correct.

Rotating T through 1800 clockwise

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5n+7g = T-total

5(2)+7(10) = T-total

80 = T-total

Also:

2 + 12 + 21 + 22 + 23 = 80

5n+7g = T-total

5(25)+7(10) = T-total 195 = T-total

Also:

25 + 35 + 44 + 45 + 46 = 195

5n+7g = T-total

5(39)+7(10) = T-total 265 = T-total

Also:

39 + 49 + 58 + 59 + 60 = 265

This proves that the general formula was correct.

Rotating T through 900 clockwise

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5n+7 = T-total

5(11)+7 = T-total

62 = T-total

Also:

3 + 11 +12 + 13 + 23 = 62

5n+7 = T-total

5(34)+7 = T-total

177 = T-total

Also:

14 + 34 + 35 + 36 + 46 = 177

5n+7 = T-total

5(48)+7 = T-total

247 = T-total

Also:

40 + 48 + 49 + 50 + 60 = 247

This proves that the general formula was correct.

Rotating T through 2700 clockwise

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5n-7 = T-total

5(12)-7 = T-total

53 = T-total

Also:

1 + 10 + 11 + 12 + 19 = 53

5n-7 = T-total

5(18)-7 = T-total

83 = T-total

Also:

7 + 16 + 17 + 18 + 25 = 83

5n-7 = T-total

5(43)-7 = T-total

208 = T-total

Also:

32 + 41 + 42 + 43 + 50 = 208

This proves that the formula was correct.

All this proves that the general formulas I predicted were correct. Here are the correct general formulas for any grid size: -

Upright T:

900 T:

1800 T:

2700 T:

5n-7g = T-total

5n+7 = T-total

5n+7g = T-total

5n-7 = T-total

Further Investigation

        The formulas I have found can be made simpler to use by making both signs the same. By using sine and cosine we can do this.

For the angle 00, sine in equal to 0 and cosine in equal to 1

For the angle 900, sine in equal to 1 and cosine in equal to 0

For the angle 1800, sine in equal to 0 and cosine in equal to -1

For the angle 2700, sine in equal to -1 and cosine in equal to 0

Because these values of sine and cosine are negative and positive they can be used to make the signs in the formulas the same. This is because of the rule of signs: like signs give a positive value and unlike signs give a negative value. Here are the revised formulas: -

Upright T:

900 T:

1800 T:

2700 T:

5n-7g(cos 00) = T-total

5n+7(sin 900) = T-total

5n-7g(cos 1800) = T-total

5n+7(sin 2700) = T-total

As you can see the formulas are now easier of follow as the two sets of like formulas have the same signs. Taking the negative values of sine and cosine, then multiplying them by the original formula has achieved this.

These equations can now be used in incorporate vectors. A vector is a translation of the T which if put into the formula in the correct way can allow you to find the T-total which first finding that T’s T-number. I will now investigate vectors on a size 9 grid. Here is the vector I will be using in the formulas:      . So in the vector       1 is represented by x and 0 is represented by y.image00.pngimage01.png

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52image00.png

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...read more.

Conclusion

+7. I will now test the formula, which I think will look like this: -

5n-7g(sin 900)+5x-(5g)y -70= T-total

Rotating T through 2700 clockwise

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image18.png

Here the original T is in red and translation through the vector       is in blue.

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Using the general formula the T-total of the blue T should be: -

5n-7g(sin 2700)+5x-(5g)y –70 = T-total

(5x29)-(7x9) x (-1) + (5x6) – ([5x9]x1) –70 = T-total

(5x29)-(7x9) x (-1) + (5x6) – (45x1) –70 = T-total

145-(63x-1)+30-45 –70 = T-total

(145+63)+30-45 –70 = T-total

82+30-45 –70 = T-total

123 = T-total

Using the old formula the T-total of the blue T is: -

5n-7 = T-total

(5x26)-7 = T-total

130-7 = T-total

123 = T-total

This proves the formula is correct.image19.png

Here the original T is in red and translation through the vector       is in blue.

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Using the general formula the T-total of the blue T should be: -

5n-7g(sin 2700)+5x-(5g)y –70 = T-total

(5x43)-(7x9) x (-1) + (5x-4) – ([5x9]x3) –70 = T-total

(5x43)-(7x9) x (-1) + (5x-4) – (45x3) –70 = T-total

215-(63x-1)-20-135 –70 = T-total

(215+63)-20-135 –70 = T-total

278-20-135 –70 = T-total

53 = T-total

Using the old formula the T-total of the blue T is: -

5n-7 = T-total

(5x12)-7 = T-total

60-7 = T-total

53 = T-total

This proves my general vector formula is correct. There is, again, no need to test the 900 T and the 1800 T general vector formulas, as they both follow the patterns of the earlier 900  and 1800 formulas, which work for any grid size Here are all the general vector formulas: -

Upright T: 5n-7g(cos 00)+5x-(5g)y = T-total

900 T: 5n-7ga(sin 900)+5x-(5g)y +70= T-total

1800 T: 5n-7g(cos 1800)+5x-(5g)y = T-total

2700 T: 5n-7g(sin 900)+5x-(5g)y +70= T-total

These formulas do not work if the T goes outside the grid. This is because the shape is no longer a T shape. Using these formulas it is possible to work out any T-total of any rotation of the T shape, on any grid size, from the T-number of an upright T and a vector to the desired T-number.

...read more.

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