From this table the first major generalisation can be made,
The larger the T-Number the larger the T-Total
The table proves this, as the T-Numbers are arranged in order (smallest first) and the T-Totals gradually get larger with the T-Number.
From this we are able to make a formula to relate T-Number (x) and T-Total (t) on a 9x9 grid. Taking the T-number of 20 as an example we can say that the T-Total is gained by:
t = 20 + 20 – 9 + 20 – 19 + 20 – 18 + 20 – 17 = 37
The numbers we take from 20 are found, as they are in relation to it on the grid, as the T-Shape spreads upwards all numbers must be less by a certain amount, these are found by the following method;
As there are 5 numbers in the T-Shape we need 5 lots of 20, the number adjacent to 20 is 11 which is 9 less than 20, the other numbers in the T-Shape are 1,2 & 3 which are 19, 18 & 17 less than 20. Thus the above basic formula can be generated.
If we say that 20 is x and x can be any T-Number, we get:
t = x + x – 9 + x – 19 + x – 18 + x – 17
To prove this we can substitute x for the values we used in are table, we get the same answers, for example taking x to be 80:
t = 80 + 80 – 9 + 80 – 19 + 80 – 18 + 80 – 17 = 337
And x as 52;
t = 52 + 52 – 9 + 52 – 19 + 52 – 18 + 52 – 17 = 197
Thus proving this equation can be used to find the T-Total (t) by substituting x for the given T-Number. The equation can be simplified more:
t = x + x – 9 + x – 19 + x – 18 + x – 17
t = 2x – 9 + 3x – 54
t = 5x – 63
Therefore, we can conclude that:
On a 9x9 grid any T-Total can be found using t = 5x – 63 were x is the T-Number.
We can also say that on a 9x9 grid that;
- A translation of 1 square to the right for the T-Number leads to a T-total of +5 of the original position.
- A translation of 1 square to the left for the T-Number leads to a T-total of -5 of the original position.
- A translation of 1 square upwards of the T-Number leads to a T-total of -45 of the original position.
- A translation of 1 square downward of the T-Number leads to a T-total of +45 of the original position.
Finding relationships on grids with sizes other than 9x9
If we take this 8x8 grid with a T-Number of 36 we get the T-Total of 124 (36 + 28 + 20 + 19 + 21), if we generalise this straight away using the same method’s used in before for a 9x9 grid we achieve the formula:
t = x – 8 + x – 17 + x – 16 + x – 15
t = 2x – 8 + 3x – 48
t = 5x – 56
This can also be shown in this form,
Testing this out using 36 as x we get:
t = (5 × 36) – 56
t = 180 – 56
t = 124
Which is the same answer as before proving this formula works.
On a 4x4 grid we can try the same method of generalisation to find the T-Total (47 with a T-Number of 15 o n this 4x4 grid), if we do this we get:
t = x + x – 4 + x – 9 + x – 8 + x – 7
t = 2x – 4 + 3x – 24
t = 5x – 28
If we test this:
t = (5 × 15) – 28
t = 75 – 28
t = 47
We get the same answer proving it works, from these results I have obtained I can state that:
All formulas for grid sizes have 5x (T-Number) – a multiple of 7 with a value dependent on the grid size.
We should now try and find the rule that governs the “magic number” that has to be taken from 5x to gain t. If we say g is the grid size (e.g. 4 for 4x4 or 5 for 5x5).
If we work the T-Total on this 3x3 grid the “old” way, we get 19 (8 + 5 + 1 + 2 +3), that proves a formula of:
t = x + x – 3 + x – 7 + x – 6 + x – 5
t = 5x – 21
Again the “magic number” is a multiple of 7, thus I can predict the “magic number” for any grid as 27 is the magic number found by 3 x 7 or g x 7.
From this a grid of a T-Number based on the relationships between x (T-Number) and g (grid size);
In addition, from this we can simplify the above and generate the formula of:.
t = 5x – (g × 7 )
t = 5x – 7g
As we have taken 5x (the number of numbers), from that we take the grid size times 7 (as they are all multiples of 7), as that gives the “magic number”, and therefore we can state:
Any T-Total of a T-Shape can be found if you have the 2 variables of a T-Number and a grid size for T shapes that extends upwards, using the formula t = 5x – 7g.
Another area that we can investigate is that of differing grid such as 4x7 and 6x5 will make a difference to this formula.
So if we work this grid out using the “old” method we get a T-Total of 67, and a formula of:
t = x + x – 4 + x – 9 + x – 8 + x – 7
t = 5x – 28
This formula is the same as the 4x4 grid as it has a “magic number” of 28, identical to a 4x4 grid, we can predict that grid width is the only important variable, but we will need to prove this.
On a 6x5 grid with a T-Number of 15 the T-Total is 33, again working out the formula leads to
t = 5x – 42
Again this is the same “magic number” as found by the predictions for a 6x6 grid found earlier, therefore we can state that:
5x – 7g can be used to find the T-Total (t) of any Grid save, regular (e.g. 5x5) or irregular
(e.g 5x101),
with the two variables of grid width (g) and the T-Number (x).
