t = 30 + 30 – 9 + 30 – 19 + 30 – 18 + 30 – 17 = 87
The numbers we take from 30 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 30, the number adjacent to 30 is 13 which is 9 less than 30, the other numbers in the T-Shape are 3, 4 & 5 which are 19, 18 & 17 less than 30. Thus the above basic formula can be generated.
If we say that 30 is n and n can be any T-Number, we get:
t = n + n – 9 + n – 19 + n – 18 + n – 17
To prove this we can substitute n for the values we used in the table, we get the same answers, for example taking n to be 31:
t = 31 + 31 – 9 + 31 – 19 + 31 – 18 + 31 – 17 = 92
n as 32;
t = 32 + 32 – 9 + 32 – 19 + 32 – 18 + 32 – 17 = 97
n as 33;
t = 33 + 33 – 9 + 33 – 19 + 33 – 18 + 33 – 17 = 102
n as 34;
t = 34 + 34 – 9 + 34 – 19 + 34 – 18 + 34 – 17 = 107
Thus proving this equation can be used to find the T-Total (t) by substituting n for the given T-Number. The equation can be simplified more:
t = n + n – 9 + n – 19 + n – 18 + n – 17
t = 2n – 9 + 3 n – 54
t = 5n – 63
On this grid I have plotted the T Shape going vertically to see if I will gain any different results.
By looking at the grid I plotted a table of results for the T Shape going vertically.
When going vertically on the 9x9 grid the table of result shows a ratio of 1:5 between the T-Number and the T-Total.
So no matter how the T-Shape is moved, either horizontally or vertically the T-number and the T-Total are in the same ratio of 1:5.
Therefore, we can conclude that: On a 9x9 grid any T-Total can be found using t = 5 n – 63 were n is the T-Number.
Finding relationships on grids using 8x8
If we take this 8x8 grid with a T-Number of 29 we get the T-Total of 89 (12+13+14+21+29), if we generalize this straight away using the same method’s used in before for a 9x9 grid we achieve the formula:
t = n – 8 + n – 17 + n – 16 + n – 15
t = 2n – 8 + 3n – 48
t = 5n – 56
Testing this out using 29 as n we get:
t = (5 × 29) – 56
t = 145 – 56
t = 89
This proves the formula because I obtain the same answer as before.
This can also be shown in this form,
From the table below we can see that the T-number increases by 1 and the T-Total increases by 5 giving it a ratio of 1:5 this is the same as in the 9x9 grid.
I plotted the same grid but going vertically to see if it will any different to what I obtained in the 9x9 grid.
I also plotted a table of results to shows the trend between the T-Number and the T-Total and this is what I obtained.
By looking at this table we can see that the T-Number increases by 8 and the T-Total increases by 40 giving it a ratio of 1:5.
Finding relationships on grids using 7x7
On a 7x7 grid we can try the same method of generalization to find the T-Total (41 with a T-Number of 18 on this 7x7 grid), if we do this we get:
t = n + n – 7 + n – 15 + n – 14 + n – 13
t = 2n – 7 + 3n – 42
t = 5n – 49
If we test this:
t = (5 × 18) – 49
t = 90 – 49
t = 41
Finding relationships on grids using 6x6
On a 6x6 grid we can try the same method of generalization to find the T-Total (118 with a T-Number of 32 on this 7x7 grid), if we do this we get:
t = n + n – 6 + n – 11 + n – 12 + n – 13
t = 2n – 6 + 3n – 36
t = 5n – 42
If we test this:
t = (5 × 32) – 42
t = 160 – 42
t = 118
By using this method we still obtain the same answer proving that it works therefore I can state that all the formulas for the grid sizes have 5n – a multiple of 7 with a value dependent on the grid size.
Now that I obtained different formulas for each grid I can come up with the general formula that can be used in any grid and will give me the same answer calculated before:
each time the number after 5n is going up by 7 so 7 times 9 in the (9x9) grid its equal to 63, therefore t=5n -7g, n is the t-number t is the t-total and g can be any grid size.
The general formula for the t in terms of n in any grid size is t=5n -7g
Therefore the number for each grid is a multiple of 7; hence I can predict that the number for any grid is found by 7 x g.
From this, a grid of a T-Number is based on the relationships between n (T-Number) and g (grid size);
In addition, from this we can add and simplify the above:
t = n – 2g +1 + n – 2g + n – 2g – 1 + n – g + n
t = 5n – (7 x g)
t = 5n – 7g
As we have taken 5n (the number of numbers), from that we take the grid size times 7 (as they all are multiples of 7), 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 = 5n – 7g.
Another area that we can investigate is that of differing grid sizes such as 3x8 and 4x6 to see if they will make a difference to the formula.
So if we work this grid out using the same method as before we get a T-Total of 47, and a formula of:
t = n + n – 4 + n – 9 + n – 8 + n – 7
t = 5n – 24
This formula is the same as the 4x4 grid as it has a number of 24, 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 4x6 grid with a T-Number of 17 the T-Total is 64, again working out the formula leads to
t = n + n – 3 + n – 7 + n – 6 + n – 5
t = 5n – 21
Again this is the same number as found by the predictions for a 6x6 grid found earlier, therefore we can state that:
5n – 7g can be used to find the T-Total (t) of any Grid size, regular (e.g. 7x7) or irregular (e.g. 7x11), with the two variables of grid width (g) and the T-Number (n).
