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.
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9
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9
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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
02
34
07
35
12
38
27
39
32
40
37
Prediction: I have found that there is a difference of five between each T-total. I found that there is a difference of 5 by subtracting one T-total from another, for example:
42 - 37 = 5
I subtracted the T-total 42 away from the T-total 37, which was the previous one above it. I think that if I multiply the T-number by five, the difference, and then subtract a number, that will give me the T-total. The number that I have to subtract is 63. I found this out by testing it out on the first T-number 20. I multiplied it by 5 and 37 I subtracted that answer by 37, which gave me 63. I am going to test this prediction on two more T-numbers to see whether it works or not. The equation I have come up with is:
Ttot = 5Tn - 63
Ttot = T-total
x Tn = T-number
Test: I am going to see if my prediction works by testing it with a T-number of 47. If I use the rule that I have found the sum would look like this:
5*47-63 = T-total
235-63 = 172
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9
20
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81
As you can see with a T-number of 47 the T-total does equal 172. I got this by adding the numbers inside the T together. The addition I did was:
47 + 38 + 28 + 29 + 30 = 72
I am now going to do another T with a T-number of 74, using the equation the sum will look like this:
5*74-63 = T-total
370-63 = 307
As you can see this does work with a T-number of 74.
74 + 65 + 56 + 55 + 57 = 307
Conclusion: In conclusion the rule that I have found, Ttotal = 5Tn - 63, does work. I am now going to see whether this rule works for any grid size or if there are different equations for each grid size, which have a different connection.
Some more T-totals: I am now going to try some more T-totals and find rules for them. I will start with a 4 by 4 grid. I will then do 5 by 5, 6 by 6 and finally 7 by 7. I am hoping to find a connection that links all these grids together including the
9 by 9.
4 by 4
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9
0
1
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T-number
T-total
0
22
1
27
4
42
5
47
Prediction: I have found that between the first two T-totals there is also a difference of 5 like I found in the 9 by 9. The difference occurs in each row in the grid, for example there is a difference of 5 between the T-totals 22 and 27, which have a T-number of 10 and 11 and they are on one row. Then 42 and 47 have a difference of 5 also and have T-numbers of 14 and 15, which are on the last row. Like in the 9 by 9 I am going to multiply the T-number by 5. I will then find the number I will have to subtract to get the T-totals for each T-number. I am going to multiply the first T-number in the table by 5, which gives me 50. I am now going to subtract 50 by 22, this gives me 28. The sum I have come up with by doing these calculations is:
Ttot = 5Tn - 28
Test: I am now going to test the rule to see if it works. Since there are know other T-totals to find I will use the rule to calculate the T-total for a T-number of 15 if the answer matches the answer I have in the table the rule has worked.
5*15 - 28 = T-total
75 - 28 = 47
I have found that this rule, Ttot = 5Tn - 28, also works.
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.
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
20
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61
62
63
64
65
66
67
68
69
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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
02
34
07
35
12
38
27
39
32
40
37
Prediction: I have found that there is a difference of five between each T-total. I found that there is a difference of 5 by subtracting one T-total from another, for example:
42 - 37 = 5
I subtracted the T-total 42 away from the T-total 37, which was the previous one above it. I think that if I multiply the T-number by five, the difference, and then subtract a number, that will give me the T-total. The number that I have to subtract is 63. I found this out by testing it out on the first T-number 20. I multiplied it by 5 and 37 I subtracted that answer by 37, which gave me 63. I am going to test this prediction on two more T-numbers to see whether it works or not. The equation I have come up with is:
Ttot = 5Tn - 63
Ttot = T-total
x Tn = T-number
Test: I am going to see if my prediction works by testing it with a T-number of 47. If I use the rule that I have found the sum would look like this:
5*47-63 = T-total
235-63 = 172
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3
4
5
6
7
8
9
10
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2
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9
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61
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63
64
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67
68
69
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71
72
73
74
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76
77
78
79
80
81
As you can see with a T-number of 47 the T-total does equal 172. I got this by adding the numbers inside the T together. The addition I did was:
47 + 38 + 28 + 29 + 30 = 72
I am now going to do another T with a T-number of 74, using the equation the sum will look like this:
5*74-63 = T-total
370-63 = 307
As you can see this does work with a T-number of 74.
74 + 65 + 56 + 55 + 57 = 307
Conclusion: In conclusion the rule that I have found, Ttotal = 5Tn - 63, does work. I am now going to see whether this rule works for any grid size or if there are different equations for each grid size, which have a different connection.
Some more T-totals: I am now going to try some more T-totals and find rules for them. I will start with a 4 by 4 grid. I will then do 5 by 5, 6 by 6 and finally 7 by 7. I am hoping to find a connection that links all these grids together including the
9 by 9.
4 by 4
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
T-number
T-total
0
22
1
27
4
42
5
47
Prediction: I have found that between the first two T-totals there is also a difference of 5 like I found in the 9 by 9. The difference occurs in each row in the grid, for example there is a difference of 5 between the T-totals 22 and 27, which have a T-number of 10 and 11 and they are on one row. Then 42 and 47 have a difference of 5 also and have T-numbers of 14 and 15, which are on the last row. Like in the 9 by 9 I am going to multiply the T-number by 5. I will then find the number I will have to subtract to get the T-totals for each T-number. I am going to multiply the first T-number in the table by 5, which gives me 50. I am now going to subtract 50 by 22, this gives me 28. The sum I have come up with by doing these calculations is:
Ttot = 5Tn - 28
Test: I am now going to test the rule to see if it works. Since there are know other T-totals to find I will use the rule to calculate the T-total for a T-number of 15 if the answer matches the answer I have in the table the rule has worked.
5*15 - 28 = T-total
75 - 28 = 47
I have found that this rule, Ttot = 5Tn - 28, also works.
