# The aim of this investigation is to find a relationship between the T-total and T-numbers. First by using a 9 by 9 number grid and then by changing the grid size.

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Introduction

Aim: The aim of this investigation is to find a relationship between the T-total and T-numbers. First by using a 9 by 9 number grid and then by changing the grid size. Method: I will first draw out a 9 by 9 grid and put Ts within it. I will place my results into a table and attempt to find a relationship between the two. I will incorporate this relationship into a rule using letters and numbers only. I will then do the similar thing for a 4 by 4, 5 by 5, 6 by 6 and 7 by 7 grid also. I will then try to find an overall rule to work out any grid size T-totals: I am going to do some T-totals and put the answers into tables. The first grid I am going to do is a 9 by 9 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 Table: This table shows all the T-numbers and T-totals possible in a 9 by 9 grid up to a T-number of 40. I have worked the T-totals out by drawing Ts in the grid and adding up the numbers within each T. T-number T-total 20 37 21 42 22 47 23 52 24 57 25 62 26 67 29 82 30 87 31 92 32 97 33 102 34 107 35 112 38 127 39 132 40 137 Prediction: I have found that there is a difference of five between each T-total. ...read more.

Middle

I have also done some extra T-totals because I am going to really see if my equation works by seeing if the answers I get with it match the ones that are already in the table. I will first use the T-number 19. 5*19 - 7*8 = T-total 95 - 56 = 39 This as you can see on the table is what I got when I added the numbers within the T to produce that table. I will now do the T-number 20. 5*20 - 7*8 = T-total 100 - 56 = 44 44 is also the answer which is on the table for the T-number 20. Conclusion: I have found out by doing these grids that as well as having one rule for each grid to work out the T-totals possible with in it, there can also be overall rules to work out any T-total of any grid size be it a 4 by 4 grid or a 65 by 65 grid. I also worked out that you always multiply the T-number 5 and to get the number which you have to subtract all you do is multiply the grid size by 7. Aim: The aim of this investigation is to find a relationship between the T-total and T-numbers, but in this case the T can be positioned in different ways, for example upside down or side ways. As the T can also be changed a relationship between the transformations also has to be found. Method: I will first draw out a 4 by 4 grid and put Ts within it first placing them upside down, then side ways on the right and then also on the left. I will place my results into a table and attempt to find a relationship between the T-total and T-number as I did before. I will incorporate this relationship into a rule using letters and numbers only. ...read more.

Conclusion

5*6 = 30 30 - 37 = 7 This is the equation. Ttot = 5Tn +7 Test: I will test it on a T-number of 16. 5*16 + 7 = T-total 80 - 7 = 73 I will now add the numbers in that T to see whether the answer is 73. 13 +16 + 17 + 18 + 23 = 73 This rule has also worked. There is a similarity with the equations to work out the T-totals on both grids. I think that it will be the same for any grid with the T on its right. To see whether this theory is true I will test it on a 6 by 6 grid. I will do this by using the rule to work the T totals. I will then draw Ts on the grid and add them up. Here is my working out: 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 T-number of 7 5*7 + 7 = T-total 35 + 7 = 42 T-number of 19 5*19 + 7 = T-total 95 + 7 = 102 T-number of 25 5*25 + 7 = T-total 125 + 7 = 138 Now I will add the numbers within the Ts to see whether the answers all match. 3 + 7 + 8 + 9 + 15 = 42 15 + 19 + 20 + 21 + 27 = 102 21 + 25 + 26 + 27 + 33 = 138 Conclusion: These answers all match. This means that the overall rule for a T 90 degrees to the right is Ttot = 5Tn + 7. There is another connection that I have found, which is for moving the T to the left or right the equation stays the same the only thing that has been changed is the sign. For the right it is addition and for the left it is subtraction. ...read more.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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