n-17 + n-16 + n-15 + n-8 +n = 5n – 56 = T-Total
Above is a "T" representing itself in a 8x8 grid, and next to it as you can see is the formula for working out the T-Total knowing the T-Number.
I am now going to test the formula I have found.
5n-56
2 3 4 (5 x 19) -56 = 95-56 = 39 = T-Total FORMULA WORKS
11
19
26 27 28 (5 x 43) -56 =215-56 = 159 = T-Total FORMULA WORKS
35
43
46 47 48 (5 x 63) -56= 315-56= 259 = T-Total FORMULA WORKS
55
63
At this stage of my investigation, I can conclude that I have succeeded again in finding a formula to work out any number including the T-Total in an 8x8 grid. I have now figured out an occurring term in the formulas being found by myself. 5n is always in the formula simply because there are always 5 terms in any one "T".
7x7:
This is a 7x7 grid with 6 "T" shapes inside it. Again they're overlapping which does not matter. Next to the grid is a "T" property table. I now have to figure out a formula for working out any number in the "T" knowing the T-Number in a 7x7 grid. The formula for working this out is shown below.
n-15 + n-14 + n-13 + n-7 +n = 5n – 49 = T-Total
This is a "T" representing itself in a 7x7 grid, and next to it as you can see is the formula for working out the T-Total knowing the T-Number.
I am now going to test the formula I have found.
2 3 4 (5 x 17) -49 = 85-49 = 36 = T-Total FORMULA WORKS
10
17
10 11 12 (5 x 25) -49 = 125-49 = 76 = T-Total FORMULA WORKS
18
25
19 20 21 (5 x 34) -49 = 170-49 = 121 = T-Total FORMULA WORKS
27
34
By now, I have found 4 formulas for working out any numbers in 7x7, 8x8 and 9x9 grids, knowing the T-Number. Another pattern I have now spotted in the formulas being discovered by myself is that as the grid size decreases by a size the formulas decreases by 7. Basically the formula is 5n takeaway the grid size multiplied by 7.
5n –(9x7) = 5n - 63
5n –(8x7) = 5n – 56
5n – (7x7) = 5n - 49
Now that I know that the formula goes up and down in 7s, I am going to attempt to predict the formula for a 10 x 10 and a 5 x 5 grid.
10x10:
Knowing the pattern for the formulas, I predict that the formula for a 10 x 10 grid will be 5n – 63 plus 7. Therefore it will be 5n – 70, because 63 + 7 = 70 and because 10 x 7 = 70. I will know check my prediction. Below is a part of a 10 x 10 grid.
Above is part of a 10 x 10 grid and next to it is a table showing the properties of the 4 "T"s present in the part of the grid. Below is the working out for the formula, so then I can check if my predictions were correct.
n-21 + n-20 + n-22 + n-10 +n = 5n – 70 = T-Total
The formula I have found matches the one I predicted it would be. But that's not it; I now have to check if the formula is correct and works positively.
Prediction check:
(5 x 22) -70 = 110-70 = 40 = T-Total PREDICTION CORRECT
(5 x 26) -70 = 130-70 = 60 = T-Total PREDICTION CORRECT
(5 x 32) -70 = 160-70 = 90 = T-Total PREDICTION CORRECT
(5 x 28) -70 = 140-70 = 70 = T-Total PREDICTION CORRECT
As you can see, my predicted formula totally matched the one I found by investigating the 10 x 10 grid, and positively worked. Just to make sure, I am now going to do the same for a 5 x 5 grid as mentioned earlier on.
5x5:
Above is a 5 x 5 grid and next to it is a table with the "T"s properties. Looking at these, and knowing the formula for the 10 x 10 and 7 x 7 grid, I can predict that the formula will be 5n – 70 divided by 2 because 10/5 = 2, or 5n – 49 takeaway 14 because 7 – 5 = 2 and 2 x 7 = 14, or 5n – (5x7), which would be 5n – 35. Basically, I predict that the formula for the 5 x 5 grid will be 5n – 35. I am now going to work out the formula to see if my prediction is correct and afterwards, I shall see if the formula works.
n-11 + n-10 + n-9 + n-5 +n = 5n – 35 = T-Total
The formula I have just figured out matches the one I predicted. But I know need to check if the formula works. I have done so below by trying out the formula on the "T"s present on the 5x5 grid.
1 2 3 (5 x 12) -35 = 60-35 = 25 = T-Total PREDICTION CORRECT
7
12
2 3 4 (5 x 13) -35 = 65-35 = 30 = T-Total PREDICTION CORRECT
8
13
6 7 8 (5 x 17) -35 = 85-35 = 50 = T-Total PREDICTION CORRECT
12
17
13 14 15 (5 x 24) -35 = 120-35 = 85 = T-Total PREDICTION CORRECT
19
24
Once again, I can say that not only my prediction matched but also worked. I can now conclude that I have found formulas for a number of grids of different sizes. I have not had any problems so far in my investigation. I am now going to further my investigation by translating the "T" into different directions. For this part of my investigation I am going to use a 9x9, 8x8 and 5x5 grid.
9x9:
This is a 9x9 grid with "T"s on it. Unlike earlier on, the "T"s are in different directions. Next to it is the "T" property table. I am going to work out 3 formulas for these "T"s. There will be a formula for each "T"s because they are each in different directions. I have formulas for "T"s upright, I am now going to work formulas for "T"s to the left, to the right and downwards. I have worked out each of these 3 formulas below.
n-1 + n-2 + n-11 + n+ 7 +n = 5n – 7 = T-Total
Above is the formula for a "T" rotated 90 degrees anti-clockwise. After finding the remaining 2 formulas, I shall test each of those formulas to see if they are correct.
n +n+ 1+n+ 2+n- 7+n+ 11 = 5n + 7 = T-Total
This is the formula for a "T" rotated 90 degrees clockwise. I have noticed that if you rotate it clockwise the formula will have a positive term, and if you rotate it anti-clockwise, it will have a negative term. As you can see my first formula is 5n – 7 and my second one is 5n + 7. Knowing the formula for an upright "T", I am going to have little prediction for the next formula. I predict the formula will be
5n + 63. I am now going to see if I am correct.
n+n+9+n+17+n+18+n+19=5n+ 63=T-Total
My prediction was correct. Above is the formula for a "T" rotated at 180 degrees and it's basically the same we found earlier on for an upright "T" but the term is positive instead of negative. I have now changed my mind and decided not to translate the "T" into different directions in a 5x5 grid. I have done 9x9 already and I am going to do it for an 8x8 grid and leave out the 5x5x grid. But first I am going to test these 3 formulas.
(5 x 12) -7 = 63 5n – 7 FORMULA WORKS
(5 X 15) +7 = 82 5n + 7 FORMULA WORKS
(5 X 4) +63 = 83 5n + 63 FORMULA WORKS
The 3 formulas turned out to be perfectly working. I can now say that I have succeded in finding formulas for working out any number in a "T" placed in any direction and position in a 9x9 grid.
8x8:
This again is an 8x8 grid but with "T"s in different directions. Next to it is the "T" property table. Its now time to work out 3 formulas for the 3 "T"s present on the 8x8 grid. I am going to do this in a similar manner I done for the 9x9 grid. I have worked out each of the 3 formulas on the next page.
n+n+1+n+2+n+10+n-6=5n+ 7=T-Total
n+n+8+n+15+n+16+n+17=5n+ 56=T-Total
n+n-1+n-2+n-9+n+7=5n- 7=T-Total
I have now done the formulas for all 3 of the "T"s on the 8x8 grid. I am now going to test these newly found formulas.
(5 x 12) -7 = 53 5n – 7 FORMULA WORKS
(5 X 54) +7 = 277 5n + 7 FORMULA WORKS
(5 X 12) +56 = 116 5n + 56 FORMULA WORKS
All these formulas are to work out the T-Total knowing the T-Number. The "n" stands for T-Number and the formulas in italic, bold and underlined are formulas that I have not worked out and/or tested during the course of my investigation. They are simply formulas that I predict are more or less 100% correct and working, its just that I simply have not had enough time to prove this. All of the other formulas are fully tested and working formulas.
A few patterns I noticed:
- A "T" rotated 90 degrees clockwise will have the same formula in any grid size, always 5n + 7
- A "T" rotated 90 degrees anti-clockwise will have the same formula in any grid size, always 5n - 7
- An upright "T" always has formula of 5n – (7 x grid size)
- A "T" rotated 180 degrees always has formula of 5n + (7 x grid size)
- If the T-Number is greater than most of the other 4 numbers in the "T", then the formula will have a negative sign
- If the T-Number is smaller than most of the other 4 numbers in the "T", then the formula will have a positive sign
- 7 (seven) has been the magic number throughout the course of my investigation
I have now proven that in whatever size grid, knowing the formula for an upright "T" and a "T" rotated 90 degrees in any direction; you can figure out the remaining 2 formulas. To figure out a "T" rotated 180 degrees formula, you just change the sign to its opposite and you do the same for a "T" rotated 90 degrees in any direction. For further comparison and understanding, please refer to my formula page on the previous page.
Conclusion:
I can conclude that in the time available to me, two weeks precisely, I think I have succeeded in investigating the relationship between the T-Total and the T-Number. I have used grids of different sizes and translated the "T"s to different directions and positions on the grids. By doing so I have found a number of formulas for working out any number including the T-Total knowing the T-Number in a number of grids.
Evaluation:
I would say that this investigation has been a success. I managed to find a number of rules and formulas to work out numbers in a "T". I do believe that I followed the guidelines given pretty well.
Unfortunately, my investigation was hindered by 1 thing – this was my lack of time to carry out the investigation as far as possible ( i.e. use more grids and test each of the formulas present in my formula page a few more times).
If I were to redo this investigation, I would make sure that I set aside enough time to do a proper job of it.
