T-Totals.Plan: Part 1 I will investigate the relationship between the T-total and the T-number on a 9x9 grid. Part 2 I will investigate the relationship between the T-total, T-number and grid size.
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Introduction
T - Totals Investigation
T-Totals
Introduction:
This is an investigation on T-totals. By starting with a 9x9 grid, and numbering it, so the top left corner numbered with one and work downwards. An example of the grid is shown below.
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 |
The investigation will be done with a standard T-shape which is 3 along and from the centre 2 more down. As shown below.
2 | 3 | 4 |
12 | ||
21 |
This is the t-number for a T-shape.
The T-total is all the numbers in the T-shape added together. The T-total for this specific example is: 2+3+4+12+21=42.
Plan:
Part 1 – I will investigate the relationship between the T-total and the T-number on a 9x9 grid.
Part 2 – I will investigate the relationship between the T-total, T-number and grid size.
Part 1:
Diagrams:
20 | 21 | 22 |
30 | ||
39 | ||
52 | 53 | 54 |
62 | ||
71 |
20+21+22+30+39=13252+53+54+62+71=292
T-total = 132 T-total = 292
4 | 5 | 6 |
13 | 14 | 15 |
22 | 23 | 24 |
4+5+6+14+23=52
T-total = 52
From now on:
Let T-total be T!
Let T-number be n!
Now I have shown some examples, a table is required so the formula can be solved between the relationship of the T and n for a 9x9 grid.
n T
1 X
X
19 X
20 37
21 42
22 47
23 52
24 57
25 62
26 67
27 X
28 X
29 82
38 127
80 337
Middle
5n 100 105 110 115 120 125
T 37 42 47 52 57 62
5n to T - 63 - 63 - 63 - 63 - 63 - 63
Therefore the table above shows the formula will have – 63 in, because – 63 is a constant difference.
The formula comes out to be 5n – 63 = T for a 9x9 grid only.
As a test if n = 38
With the formula: (5 x 38) – 63 = T
- – 63 = T
127 = T
This is correct; the table shows that the formula works.
I predict that if the T-number is 40, with the formula of 5n – 63 = T for a 9x9 grid, T will equal: 137
T = 5n – 63
= (5 x 40) – 63
= 200 – 63
T = 137
To prove that the T-total for the T-number of 40 is 137:
21 | 22 | 23 |
31 | ||
40 |
40 + 31 + 22 + 23 + 21= T
137 = T
So the formula is correct as the predicted T-total with the formula was correct, but only for a 9x9 grid.
I think that the formula will also work with incomplete T-shapes too, if 19 is used as n, I can use the formula obtained to prove it:
T = (5 x 19) – 63
= 95 – 63
T = 32
To show that this is correct:
1 | 2 |
10 | |
19 |
T=19+10+1+2
T= 32
It is correct the formula also got 32, showing the formula works for T-shapes even if they are not complete.
Part 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 | 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 |
Conclusion
I think that I have found the relationship between grid size, T-number and T-total. Because of the table at the top of the page, shows how found that there is always 5n in the formula and then the grid size is multiplied by -7 to give T. So I think the formula is:
5n – 7G = T
To test the new formula of T = 5n -7G, I will have the grid size of 10 and n being 24.
The formula gets: T = (5 x 24) – (7 x 10)
T = 120 – 70
T = 50
To check this:
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= 3+4+5+14+24= 50
Excellent, the formula obtained which links the T-total, T-number and grid size is:
T = 5n – 7G
Whilst doing this investigation a number of questions sprung to mind, what would happen if the T-shape was turned round, through 90o, 180o and 270o? I also wondered, what would happen to the T-total once the T-shape had been rotated around a number in the grid, how this would relate to the new T-total, the old T-number and the number the T-shape is rotated around? How would the T-total change when the T-shape is reflected? How would the T-total change when it is enlarged or elongated? I also thought that what would happen to the T-total if the T-shape is translated about the grid?
This student written piece of work is one of many that can be found in our GCSE T-Total section.
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