There is a definite pattern. The z value is increasing by 5 each time. I will try and find a formula.
I have found a formula that works this sequence perfectly
5n – 63
I will check this by placing a T in the middle of the grid.
Z = 49+50+51+59+68= 277
My Formula is correct.
To further my investigation I wanted to see the relationship between the t-number and the t-total on different grid sizes.
I first put the T in the top left side of an 8 by 8 grid.
But from my earlier mistakes, I will move the t shape along a number of times to identify a pattern on an 8 x 8 grid.
Right away I can see the t-total is increasing by 5. I will look at the difference of the numbers after I do the first part of the equation (5n)
So we can see that the formula is 5n - 56
Now I will see if there is a way to find more patterns so we can find out more equations.
5n - 70
So far we have 10 x 10 = 5n - 70
9 x 9 = 5n - 63
8 x 8 = 5n - 56
Here, we can see a pattern, every time we go down a grid size the red numbers are decreasing by 7. The reason why it is always 5n is because each square increases by one every time you move it across one space.
So now we can make a new formula... one for the general t-shape, no matter the grid size. But, we need a new value
g = grid size Z = 5n – 7g
After more experiments of t-shapes in different grids, I noticed that you don’t always have to be a 9 x 9 or a 10 x 10. It can be 9 x 10 or 10 x 12.
So we have to make g a value that doesn’t change (the width).
To extend the experiment more, I flipped the t-shape so it is upside down. To find a formula, I will simple move the T to the right on a 9 x 9 grid. The t-number will still be at the bottom of the T.
The formula for this is 5n + 63
I will do what I did when the t-shape was the right way up (look for a general formula)
4 x 4 grid 7 x 7 grid
5n + 28 5n + 49
4 x 4 = 5n + 28
5 x 5 = 5n + 35
6 x 6 = 5n + 42
7 x 7 = 5n + 49
8 x 8 = 5n + 56
9 x 9 = 5n + 63 ……
The formula is 5n + 7g
It the same as the original formula but is + instead of – because the T is upside down.
To extend my investigation I will look at the relationship between the numbers IN the t-shape.
I will look at the squares one by one on a 9 x 9 grid.
The t-number is 20 and the number above is 11, which is 9 lower than 20, and so is the number above that! So is 2 is 18 less than 20, it is 2 x 9.
The number to the left of 2 is 1 which is one less. So it must be -1. The number to the right of 2 is one more than to so it’s going to be +1.
Now I will check this in the middle of a 9 x 9 grid.
50 – 9 = 41
50 – 2 x 9 = 32
50 – 2 x 9 – 1 = 31
50 – 2 x 9 + 1 = 33
This works perfectly.
Now I will try it in a different grid size, an 8 x 8 grid.
18 - 9 = 9
Straight away we can see that this doesn’t work. But, you can notice 41 is 9 lower that 50 and is also the grid width. So if we try this on the 8 x 8 grid.
18 – g = 10
18 – 8 = 10
This method works. So we have a new all round formula…
I will now try and find formulas when the T is upside down on a 9 x 9 grid.
Like when the T was the right way up, the number below the t-number is 9 more and so is the number below that. The number to the left is 1 less and the number to the right is 1 more. As the difference between the numbers (2, 11, 20) is 9 and is also the grid width the formula will be
Now I will check this in the middle of a 10 x 10 grid.
16 + g = 26 16 + 2g = 36 16 + 2g + 1 = 37 16 + 2g – 1 = 35
16 + 10 = 26 16 + 20 = 36 16 + 20 + 1 = 37 16 + 2g – 1 = 35
So this formula works for the upside down T on different grid sizes.
So all of the formulas that I found are:
T-Number / T-Total relationship = 5n - 7g
T-Number / T-Total relationship (upside down) = 5n + 7g
Inside the T-Shape =
Inside the T-Shape (upside down) =