To investigate the relationship between the T-total and the T-number. The problem of the question is based on a 9 by 9 number grid.
Kimberly Wong 10F
MATH COURSEWORK
T-Totals
Part I:
Aim:
To investigate the relationship between the T-total and the T-number. The problem of the question is based on a 9 by 9 number grid.
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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-shape drawn on a 9x9 grid:
A T-shape is drawn on the 9 by 9 grid and all the numbers the T-shape are added together, and the sum is the T-total. This 'T-total' is known as T. The problem is to figure out the relationship between the T-total and the position of the T-shape (the position of the T-shape is the number in the bottom part of the T-shape). This position of the T-shape is known as p.
Results:
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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-total:
T = 2 + 3 + 4 + 12 + 21
T = 14 + 15 + 16 + 24 + 33
T = 37 + 38 + 39 + 47 + 56
T = 52 + 53 + 54 + 62 + 71
Observations to help me find the Formula:
P
(position of T-shape)
T
(T-total)
20
+ 2 + 3 + 11 + 20
= 37
21
2 + 3 + 4 + 12 + 21
= 42
22
= 5 (22) - 63
= 47
23
= 5 (23) - 63
= 52
24
= 5 (24) - 63
= 57
Looking for Patterns to help find the equation:
Figure out a relationship ...
This is a preview of the whole essay
Observations to help me find the Formula:
P
(position of T-shape)
T
(T-total)
20
+ 2 + 3 + 11 + 20
= 37
21
2 + 3 + 4 + 12 + 21
= 42
22
= 5 (22) - 63
= 47
23
= 5 (23) - 63
= 52
24
= 5 (24) - 63
= 57
Looking for Patterns to help find the equation:
Figure out a relationship between the two numbers (the position of the T-shape, p and the T-total, T)
P
(position of T-shape)
T
(T-total = 5p - 63 )
20
37
21
42
22
47
23
52
24
57
Working out the formula:
Since the formula could not be found from finding patterns in the table above, using the T-shape diagram that I have created, I can figure out the formula using algebraic methods.
The formula for T if we know p is:
T = p + (p - 9) + (p - 19) + (p - 18) + (p - 17)
Simplified, this is:
T = 5 p - 63
T-Total
(figured out using the 9x9 grid)
P
(position of T-shape)
T
(T-total)
T= 2 + 3 + 4 + 12 + 21
T= 42
21
5 p - 63
= 5 (21) - 63
= 105 - 63
= 42
T= 14 + 15 + 16 + 24 + 33
T= 102
33
5 p - 63
= 5 (33) - 63
= 165 - 63
= 102
T= 37 + 38 + 39 + 47 + 56
T= 217
56
5 p - 63
= 5 (56) - 63
= 280 - 63
= 217
T= 52 + 53 + 54 + 62 + 71
T= 292
71
5 p - 63
= 5 (71) - 63
= 355 - 63
= 292
Checking the Formula:
I can check to see if the formula is correct using any number on the 9 x 9 grid. I decided to use the same numbers that I chose earlier on in the investigation.
Proving the Formula:
.
* Because it is a 9 x 9 grid, the difference between the three numbers is 9
* Each level increased means a decrease in 9
T = p (p - 9) + (p - 18) + (p - 17) + (p - 19)
decrease by 9 each time
This is shown in the equation. P represents the position of the T-shape, which can be shown as 0 because every other number has a relationship on some way with this number. This can be described as (9 x 0). 9 is because it is on a 9 x 9 number grid and zero is because it is the starting level of the T-shape.
The next box above is (p - 9) because it has increased one level on the 9 x 9 grid, and therefore the number is 9 less than before. So, since the number is 9 less than the p number, it is therefore (p - 9). Another way I got the 9, is from (9 x 1). 9 is because it is a 9 x 9 number grid, and 1 because it has increased one level.
The middle box in the top row is (p - 18) because it has increased two levels on the 9 x 9 grid. I got the 18 from (9 x 2). 9 is because it is a 9 x 9 number grid, and 2 is because it has increased 2 levels. Also, when there is an increase in level on the grid, it means that there is a decrease in number. So, the number is 2 times less than the p number, and is therefore (p - 18).
2.
* The top three numbers are always consecutive
T = p (p - 9) + (p - 19) + (p - 18) + (p - 17)
three numbers are consecutive
The boxes on the two sides are found using the box from the middle. Because I figured out the middle box earlier (see above explanation), we can use the equation (p - 18), to help me find the other two boxes. Since the three numbers are consecutive on the 9 x 9 number grid, the box on the right must be higher than the box in the middle and the box on the left must be lower than the box in the middle. To find the box in the middle, I used (9 x 2) because it was a 9 x 9 number grid and it increased 2 levels from p. I can use this same method to help me find the remaining side boxes. The box on the right would be (9 x 2 - 1). 9 is because it is on a 9 x 9 number grid. 2 is because it is 2 levels up from the p number. And (- 1) is because it is going to be a higher number than the box in the middle.
To find the box on the left, I can use this same method, and it would be (9 x 2 + 1). 9 is because it is on a 9 x 9 number grid. 2 is because it is 2 levels up from the p number. And (+1) is because it is going to be one less than the number in the middle.
From these equations, I put them all together to get:
T = p (p - 9) + (p - 19) + (p - 18) + (p - 17)
Then, I simplified it to get:
T= 5p - 63
The 5p is because there are 5 p's in the T-shape, one in each box because they are all related to the p number in the bottom box.
The (- 63) is the (-9), (-19), (-18) and (-17) added together.
Further Observations:
By using the method shown above, I can find out the formula for any number grid (for example, I can expect that on an 8 x 8 grid, I would do something like this):
* 8 x 0 is because it is on an 8 x 8 number grid, and 0 is because it is the p number, it is the starting level. Therefore, the equation is just p, because 8 x 0 is just 0.
* 8 x 1 is because it is on an 8 x 8 number grid, and 1 is because it has increased one level. 8 x 1 is 8, but because it has increased one level, the number has decreased by 8, so it is p - 8.
* 8 x 2 is because it is on an 8 x 8 number grid and 2 is because it has gone up 2 levels. 8 x 2 is 16, but because it has increased 2 levels, the number has decreased by 16, so it is p - 16.
* The top two side numbers are figured out using the middle top number. Since they are consecutive, the number on the left would be p - 17 (this is found from 8 x 2 +1, and the number on the right would be p - 15 (this if found from 8 x 2 - 1).
Also, since I know that T= 5p - 63, I now also know what p would be if I only knew T.
If T= 5p - 63
P= T + 63
5
