# T-Total Investigation

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Introduction

Maths GCSE Coursework 2000 - T-Total Introduction In this investigation I aim to find out relationships between grid sizes and T shapes within the relative grids, and state and explain all generalizations I can find, using the T-Number (x) (the number at the bottom of the T-Shape), the grid size (g) to find the T-Total (t) (Total of all number added together in the T-Shape), with different grid sizes, translations, rotations, enlargements and combinations of all of the stated. Relations ships between T-number (x) and T-Total (t) on a 9x9 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 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 From this we can see that the first T shape has a T number of 50 (highlighted), and the T-total (t) adds up to 187 (50 + 41 + 31 + 32 + 33). With the second T shape with a T number of 80, the T-total adds up to 337, straight away a trend can be seen of the larger the T number the larger the total. 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 From these Extra T Shapes we can plot a table of results. ...read more.

Middle

We know with will not make a difference to the final answer as proved in question 2. 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 As we can see, we have a horizontal translation of the first T-Shape (where v =17) by +4. Where v = 17, t = 63, and where v = 21, t = 83 (both found by using t = 5v - 2g), if be draw up a table in the same format as the one we used for the 9x9 grid, we should be able to find some relationships (from 21 to 17). Middle number (v) T-Total (t) Equation used Difference 21 83 t = (5 x 21) + ( 2 x 11 ) 5 (83 - 78) 20 78 t = (5 x 20) + ( 2 x 11 ) 5 (78 - 73) 19 73 t = (5 x 19) + ( 2 x 11 ) 5 (73 - 68) 18 68 t = (5 x 18) + ( 2 x 11 ) 5 (68 - 63) 17 63 t = (5 x 17) + ( 2 x 11 ) N/a From this we can see that 5 is the "magic" number again as for a grid width of 9 for horizontal translations. From this an obvious relation ship can bee seen that for all grid sizes, a horizontal translation of a T-Shape by +1, makes the T-Total +5 larger, but this is only a prediction. To verify this we can see what the "magic" number is on a grid width of 10. 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 ...read more.

Conclusion

Using the formula we get; T=5(36+2x10+0)+2 T=5(56)+2 T=282 Thus proving this formula works and it is obvious that it will work in the same fashion as my static (middle number as the centre) rotations, as they both find the position as the new V number then generate the t-total based on that number, therefore I can state; The T-Total of any rotation of a T-Shape with any centre of rotation on any grid size can be found by using the formula of t=5(c+d(g)+b)+y were t is the T-Total, c is the centre of rotation (grid value) d is the horizontal difference of v from the relative centre of rotation , is the grid width, and b is the vertical difference of v from the relative central number. y is to be substituted by the ending required by the type of rotation, these are : Rotation (degrees) Direction Ending (y) 0 Clockwise - 2g 90 Clockwise + 2 180 Clockwise + 2g 270 Clockwise - 2 0 Anti-Clockwise - 2g 90 Anti-Clockwise - 2 180 Anti-Clockwise + 2g 270 Anti-Clockwise + 2 In terms of x (T-Number); The T-Total of any rotation of a T-Shape with any centre of rotation on any grid size can be found by using the formula of t=5(c+d(g)+b)+y were t is the T-Total, c is the centre of rotation (grid value) d is the horizontal difference of x from the relative centre of rotation, g is the grid width, and b is the vertical difference of x from the relative centre of rotation. y is to be substituted by the ending required by the type of rotation, these are : Rotation (degrees) Direction Ending (y) 0 Clockwise - 7g 90 Clockwise + 7 180 Clockwise + 7g 270 Clockwise - 7 0 Anti-Clockwise - 7g 90 Anti-Clockwise - 7 180 Anti-Clockwise + 7g 270 Anti-Clockwise + 7 [PP1]i ?? ?? ?? ?? 1 Philip Price 07/10/01 ...read more.

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