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'T' Totals Investigation.

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Introduction

'T' Totals Investigation Introduction We have been given the task to find the relationship between the 'T' ? and the 'T' Total 9x9 grid. The 'T' ? is the number at the bottom of the T shape and the 'T' Total is the sum of all the numbers inside the T. Example 'T' Total = 1+2+3+11+20 = 37 To find the relationship between the 'T' ? and the 'T' Total, algebra will be needed. Therefore the 'T' ? will be represented by 'X' and the 'T' Total will be represented by 'T'. I will substitute the numbers in the T and replace them with those numbers but with the 'T' ? (X) as the subject. Example From this we can come up with a formula. I will check the formula with one of my other results for that grid. If the relationship between the T' ? and the 'T' Total match the formula will be correct and the problem will be solved. Table of Results for a 9x9 Grid X 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 'T' ? (X) 'T' Total (T) 20 37 42 147 67 272 80 337 T = X+(X-9)+(X-17)+(X-18)+(X-19) T = 5X-63 Check T = 5X-63 T = 5x42-63 T = 147 Formula to find 'T' Total (T) if you have the 'T'? (X) in a 9x9 grid: =5X-63 Now I have solved the first problem I am going to further my investigation and vary the grid sizes. ...read more.

Middle

8x8 9x9 10x10 Therefore these squares on the grid could be made into X+G and X+2G: This is the same as the formula for the normal 'T' shape except it is addition instead of subtraction. The last two links are in the 'T' Shapes are: a) The square to the right of the middle square on the bottom is always the grid size multiplied by 2 and subtracted by 1. b) The square to the left of the middle square on the bottom is always the grid size multiplied by 2 and then 1 is added. 8x8 9x9 10x10 Therefore these squares on the grid could be made into X-2G-1 and X-2G+1: Now that the 'T' shape has been put into algebra that incorporates the grid size, a formula that incorporates the grid size can be made. T= X+(X+G)+(X+2G)+(X+2G-1)+(X+2G+1) T= 5X+7G Check 8x8 Check 9x9 Check 10x10 T = 5X+7G T = 5X+7G T = 5X+7G T = 5x2+7x8 T = 5x2+7x9 T = 5x2+7x10 T = 10+56 T = 10+63 T = 10+70 T = 66 T = 73 T = 80 Formula to find 'T' Total if you have the 'T'? (X) and the grid size (G) and the 'T' shape is turned 180� clockwise about the 'T' ? square : =5X+7G This formula is the similar to the normal 'T' shape except it is an addition instead of a subtraction. I will now turn the 'T' shape 270� clockwise about the 'T' ? square so the 'T'? will be in a different place, therefore the formula will be different. I will alter the grid sizes as well and find a formula for all grid sizes. Table of Results for an 8x8 Grid X 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 'T' ? ...read more.

Conclusion

square X 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 'T' ? (X) 'T' Total (T) Vector New 'T' Number (X1) New 'T' Total (T1) 24 127 (-4,-2) 38 197 48 247 (3,-2) 69 352 T1 = T+5(A-GB) T1 = 127+5(-4-9x-2) T1 = 127+5x14 T1 = 127+70 T1 = 197 T1 = T+5(A-GB) T1 = 247+5(3-9x-2) T1 = 247+5x21 T1 = 247+105 T1 = 352 Table of Results for a 10x10 Grid if 'T' shape has been turned 90� clockwise about the 'T' ? square X 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 'T' ? (X) 'T' Total (T) Vector New 'T' Number (X1) New 'T' Total (T1) 11 62 (2,-4) 53 272 77 392 (0,5) 27 142 T1 = T+5(A-GB) T1 = 62+5(2-10x-4) T1 = 62+5x42 T1 = 62+210 T1 = 272 T1 = T+5(A-GB) T1 = 392+5(0-10x5) T1 = 392+5x-50 T1 = 392-250 T1 = 142 Through these tests I can see that the formula does work if the 'T' shape is turned 90� clockwise about the 'T' ? square. Tom Ballard 1 ...read more.

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