20 37
21 42
22 47
23 52
24 57
25 62
26 67
27 X
28 X
29 82
38 127
80 337
As I was writing out the table I noticed that as n increased by one T also increased by 5. This has happened because each time the T-number is increased by one, all 5 boxes in the T-shape move along one too.
The formula:
n = T-number 20 21 22 23 24 25
T = T-total 37 42 47 52 57 62
From this I can see that the formula will have 5n in because there is a difference of 5 each time for T as n is increased by 1. Also all 5 boxes in the T-shape are related to n because that is the only number we know, so it is 5n.
n 20 21 22 23 24 25
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:
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:
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:
As we already have the formula for the grid size 9, I will start with an 8x8 grid.
Colour n T
28 84
58 234
63 259
The 8x8 grid has the algebraic form of:
If all these are added together the formula comes to: 5n – 56 = T for an 8x8 grid.
To check this, if n=52 then T should be 204.
To prove this, so, 35+36+37+44+52=204
Showing the formula is correct.
Now we have the formula for a 9x9 grid and a 8x8 grid we need to find out the formula for a 7x7 grid before the grid size, t-number and t-totals relationship can be found.
For a 7x7 grid:
Colour n T
20 51
37 136
40 151
The 7x7 grid has the algebraic form of:
If all these are added together the formula comes to: 5n – 49 = T for a 7x7 grid.
To check this, if n=34 then T should be 121.
To prove this, so, 34+27+20+19+21=121
Showing the formula is correct.
Grid Size Formula
9 5n – 63 = T
8 5n – 56 = T
7 5n – 49 = T
There is a visible pattern, as the grid size decrease by one the formula keeps 5n but the subtracted figure from 5n decreases by 7, eg. Before it was –63 when decreased by one grid size it is –56.
I also saw that the end sector to the formula after 5n was the grid size multiplied by -7.
From this I predict that the formula when the grid size is 6x6 the formula will be 5n – 42 = T, because it is decreasing by 7 from the last formula and also if 6 is multiplied by -7 the answer is -42, so the formula comes to: 5n -42=T.
To prove this,
Colour n T
15 33
29 103
33 123
Using the formula, of 5n – 42=T
For when n= 15, T is equal to 33, which is correct.
For when n= 29, T is equal to 103, which is correct.
For when n= 33, T is equal to 123, which is correct.
Let G be grid size!
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:
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?