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I will be using algebra to try and see if I can find a relationship between the T-number and the T-Total. 9x9

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Introduction

Introduction In this piece of coursework I have been asked to investigate the relationship between the T-number and the T-Total. As the diagram above shows, the number at the foot of the 'T' shape is called the T-number. If you add all of the numbers up inside the 'T' shape then you get the T-total. I will be using algebra to try and see if I can find a relationship between the T-number and the T-Total. 9x9 On the grid below, it shows the 'T' shape. I will move this shape across and down. The arrows show the direction the 'T' has moved. Across T-Number T-Total 20 37 21 42 22 47 23 52 24 57 The T-total goes up by 5 each time. This is because as the 'T' is moved each number increases by 1. Down T-Number T-Total 20 37 29 82 38 127 47 172 56 217 The T-total goes up by 45 each time. This is because as the 'T' is moved each number increases by 9. If I give each number inside the 'T' shape a value in terms of "n" then my 'T' would look like this: I can now gain a formula for the shape. ...read more.

Middle

If n = 104 The t-total should = 429 77 + 78 + 79 + 91 + 104 = 429 (5 x 104) - 91 = 429 Eg 2. If n = 92 The T-total should = 369 66 + 67 + 68 + 79 + 92 = 369 (5 x 92) - 91 = 369 These examples show my formula is correct. Below is a table that contains all the formulas I have collected: Grid Size Formula 9 x 9 5n - 63 10 x 10 5n - 70 11 x 11 5n - 77 12 x 12 5n - 84 13 x 13 5n - 84 With this I can find a master formula. When n is the T-number, and g is the grid size, the formula for ALL grids is as follows: 5n - 7g This 'T' shows the master formula: (n) + (n-2g-1) + (n-2g+1) + (n-2g+1) +(n-g) = 5n - 7g Rotations I am now going to rotate the 'T' shape and test my formula to see if it works. I will rotate it about point 'n'. The above, shows the rotated 'T' and all the values in terms of 'n'. ...read more.

Conclusion

+ 15 --> 190 + 108 + 15 = 313 Eg 2. If n = 56 The T-Total for the rotated "T" should = 403 68 + 77 + 85 + 86 + 87 = 403 (5 x 56) + (12 x 9) + 15 --> 280 + 108 + 15 = 403 These examples show that my formula is correct. Translations I will now translate my "T" shape, using vectors, and work out an algebraic formula. The formula for translation is as follows: 5n + 5a - 5bg - 7g n = T-number a = movement right b = movement up or down g = grid size. The above grid shows the "T" and all the values in terms of "n". The grid below shows a translated "T" in the vector of ( 2 ) (-3 ) Using the formula I can see that the T-Total should = 182 n = 20 5n = 100 5a = 10 g = 9 b = -3 -5bg = 135 -7g = -63 100 + 10 + 135 - 63 = 182 30 + 31 + 32 + 40 + 49 = 182. Double Transformation I will now rotate and translate my "T" shape in one single step and find a formula. ...read more.

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