6 by 6 grid. 5 numbers in the “T”
N= 14 15 16 17
T= 28 33 38 43
5 5 5
These results are similar hence the link is : 5n – 42
So far:
4 by 4 = 5n – 28
5 by 5 = 5n – 35
6 by 6 = 5n - 42
7 by 7 grid. 5 numbers in the “T”
N= 16 17 18 19 20
T= 31 36 41 46 51
5 5 5 5
Similarly the link is : 5n – 49
I predict that in an 8 by 8 grid the link will be 5n – 56 as there is a difference of seven between the second part of the equation:
5n – 42
← difference of 7
5n – 49
So far the link appears to be that the amount of squares in a T is the first number:
5 n
amount of numbers in the T
and the second number is the grid length times by 7.
8 by 8 grid. 5 numbers in the “T”
3
4
5
6
7
8
N= 18 19 20 21 22 23
T= 34 39 44 49 54 59
5 5 5 5 5
Link: 5n –56
9 by 9 grid. 5 numbers in “T”
N= 20 21 22 23 24 25 26
T= 37 42 47 52 57 62 67
5 5 5 5 5 5
Link: 5n – 63
Equations:
7 by 7 = 5n - 49
8 by 8 = 5n – 56
9 by 9 = 5n – 63
10 by 10 = 5n – 70
Each of these begins with 5n and ends with a multiple of 7. The exact multiple is the same as the length times 7. This makes the link between each total 5n – 7L, where L is length of one grid side This enables us to work out the “T” total and “T” number for that specific sized “T” on any grid size. The number 5 is the amount of numbers inside the “T”.
Second “T” size (6 numbers)
The first part of the equation for this set of results will be 6n.
4 by 4 grid. 6 numbers in the “T”
1
2
3
4
5
6
7
8
N= 14 15
T= 36 42
6
Link: 6n – 48
5 by 5 grid. 6 numbers in the “T”
N= 17 18 19
T= 42 48 54
- 6
Link: 6n – 60
6 by 6 =
N= 20 21 22 23
T= 48 54 60 66
6 6 6
Equations:
4 by 4 = 6n – 48
5 by 5 = 6n - 60
6 by 6 = 6n - 72
7 by 7 = 6n - 84
8 by 8 = 6n - 96
9 by 9 = 6n - 108
10 by 10 = 6n – 120
The link between each of these is the equation 6n – 12L
As before L equals the length of one of the grid sides, and the 6 is the amount of numbers in the T.
There is a pattern: an – bL
“a” is always the amount of numbers in the T.
Third “T” size (4 numbers)
4 by 4 grid. 4 numbers in the “T”
N= 6 7
T= 12 16
4
Link: 4n – 12
5 by 5: 4n – 15
6 by 6: 4n - 18
7 by 7: 4n - 21
8 by 8: 4n - 24
9 by 9: 4n - 27
10by 10: 4n – 30
The link between these is the equation: T = 4n – 3L
“T” number links
Equations for the sizes of “T”s so far
4 numbers: T = 4n – 3L
4
5 numbers: T = 5n – 7L 1
5
6 numbers: T = 6n – 12L 1
6
7 numbers: T = 7n – 18L 1
7
8 numbers: T = 8n – 25L 1
8
9 numbers: T = 9n – 33L 1
9
10 numbers: T = 10n – 42L
0 1 2 3 4
3 3 7 12 18
4 5 6
0.5 1 1
3 = 0.5 x 12 + b x 1 + 0
3 = 0.5 + b
3 – 0.5 = 2.5
a = 0.5
b = 2.5
c = 0
Changing the grid size:
0.5n2 + 2.5n + 0
This links with this number in the equations
T = 4n –3L
Formula for working out the “T” total on any grid size and any sized “T” (end solution)
T = 4n – (0.5n2 +2.5n) x L
1 1 2 3 4
3 4 5 6 7
1 1 1 1
n + 3
T = (X + 3) x n – (0.5n2 + 2.5n) x L
Y = x2 5x
+
2 2
T = 4n – 3L
T = ? n – ( x2 5x )
+
2 2
T = (y + X) x n – ((0.5n2 + 2.5n) x L)
Third Variable.
Width of the T.
1st T width
width of 5
grid size 6 by 6
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
N 15 16
T 39 46
7
The link is therefore: 7n – 66
Second grid size:
7 by 7
N: 17 18 19
T: 42 49 56
7 7
The link here is therefore 7n – 77
I predict that on an 8 by 8 grid the outcome will be 7n – 88 and also that the link with this sized T will be 7n – 11g
N: 19 20 21 22
T: 45 52 59 66
7 7 7
Prediction #1 correct: 7n – 88
6 by 6: 7n - 66
7 by 7: 7n - 77
8 by 8: 7n - 88
9 by 9: 7n - 99
10 by 10: 7n - 110
Second Prediction correct: 7n – 11g
2nd T size
9 numbers in the T
N: 20 21
T: 60 69
9
Therefore link: 9n – 120
Prediction: The link for a 9 by 9 grid will be 9n –135
N: 22 23 24
T: 63 72 81
- 9
Prediction correct: 9n – 135
The link for all this sized T is 9n-15g
5n – 7g
4
7n – 11g
4
9n – 15g
An - ?g A being amount of numbers in the t. N being the t number therefore the
Unknown. G being the length of the grid.
