I will use the numbers inside this T-shape to find a formula.
I will refer these numbers to these letters below, with n being the T-number.
The value of a is n – 11
The value of b is n – 10
The value of c is n – 9
The value of d is n – 5
So this is how the table now looks
I know that that everything in this T-shape is equal to the original answer therefore I can add these up to find the T-total.
y = T-total and n = T-number
The T-total is = n – 11 + n – 10 + n – 9 + n – 5 + n
y = 5n – 35
Now I will test this formula with the T-total that I have already done.
n = 14 and the y = 35
If the total was not known then I would put n into the formula
y = 5 ( 14 ) – 35
y = 70 - 35
y = 35
The formula worked but I will do one more to make sure.
If n is 19 and my y = 60
y = 5 ( 19 ) – 35
y = 95 – 35
y = 60
Therefore the formula for any T-number with a 5 by 5 table is
n = T-number y = T-total
y = 5n - 35
Now I will investigate the T-total on a 6 by 6 grid.
The way I have worked out the formula on a 5 by 5 grid can also be used on a 6 by 6 grid.
n = 14
The value of a is n – 13
The value of b is n – 12
The value of c is n – 11
The value of d is n – 6
So this is how the T-shape looks
I will add up all the value inside the T-shape
y = n – 13 + n – 12 + n – 11 + n – 6 + n
y = 5n – 42
I will test this theory to see if it correct for any T-number
If n = 17 and y = T-total
y = 4 + 5 + 6 + 11 + 17
y = 43
So my formula is 5n - 42
y = 5n - 42
y = 5 ( 17 ) – 42
y = 85 -42
y = 43
The formula works however I will do a few more to make sure.
n = 35
y = 5n – 42
y = 5 ( 35 ) - 42
y = 175 – 42
y = 133
Check:
y = 22 + 23 + 24 + 29 + 35
y = 133
n = 15
y = 5n – 42
y = 5 ( 15 ) - 42
y = 75 – 42
y = 33
Check:
y = 2 + 3 + 4 + 9 + 15
y = 33
n = 20 and T-total = y
y = 5n – 42
y = 5 ( 20 ) - 42
y = 100 – 42
y = 58
Check:
y = 7 + 8 + 9 + 14 + 20
y = 58
Therefore the formula for any T-number with a 6 by 6 table is
n = T-number y = T-total
y = 5n - 42
I will now put all my results into a table
Looking at this table I notice that the T-total has a difference of 5 which the 5 by 5 table also has a difference of 5. My guess is that the 7 by 7 table also has a difference of 5.
Here is a table 7 by 7
I will find the formula for this table.
n = 16
The value of a is n – 15
The value of b is n – 14
The value of c is n – 13
The value of d is n – 7
So this is how the T-shape looks
I will add up all the value inside the T-shape
y = n – 15 + n – 14 + n – 13 + n – 7 + n
y = 5n – 49
So now I will test this to make sure that it is correct
If n = 16 and y = T-total
y = 1 + 2 + 3 + 16 + 9
y = 31
So my formula is 5n - 49
y = 5n - 49
y = 5 ( 16 ) – 49
y = 80 -49
y = 31
The formula works and now I will see if there is a difference of 5 between each T-total.
I now realise that because n is a multiple of 5 the difference is always going to be 5.
Therefore the formula for any T-number with a 7 by 7 table is
n = T-number y = T-total
T-total = 5n - 49
So from now I would not need to do detail working out, so I will just find out the formula for all the other grids because they all follow a same pattern in working out the formula.
Here is a table 8 by 8
n = 18
The value of a is n – 17
The value of b is n – 16
The value of c is n – 15
The value of d is n – 8
So this is how the T-shape looks
I will add up all the value inside the T-shape
y = n – 17 + n – 16 + n – 15 + n – 8 + n
y = 5n – 56
Instead of writing out the whole table to find a formula I will try to find it by using only a part of a table.
Therefore the formula for any T-number with a 8 by 8 table is
n = T-number and y = T-total
y = 5n - 56
I see that the way I have worked out the formula previously will be the same for the rest of the grids.
Looking at this formula table I found a relationship between each formula and that is the difference between each number after (5n - ) has a difference of 7, so if I divide each of these numbers by 7, The answer is the same number as the grid number.
So the formulas are:
If grid number = x then the formula for the T-total for any grid number is
Therefore the formula for any T-number with x by x grid is
n = T-number and y = T-total and x = grid number
y = 5n – 7x
I will now test this with a 15 by 15 grid.
So x = 15
y = 5n – 7x
y = 5n – 7 ( 15 )
y = 5n - 105
So now the T-total formula for a 15 by 15 grid is y = 5n - 105
If n = 32
Then y = 5 ( 32 ) – 105
y = 55
now I will check this to see if this formula is correct,
y = 1 + 2 + 3 17 + 32
y = 55
I have now proven that the formula works for any x by x table.
I will do one more to see if it is correct with a 4 by 4 grid.
y = 5n – 7x
y = 5n – 7 (4)
y = 5n – 28
If n = 10
y = 5 ( 10 ) – 28
y = 50 – 28
y = 22
To check to see if this is correct I will add up the numbers in the T-shape
y = 1 + 2 + 3 + 6 + 10
y = 22
I will now translate the T-shape by 180° with a 5 by 5 grid.
So the T-total for this shape is
y = 9 + 14 + 18 + 19 + 20
y = 80
Because I know there was a formula for the previous I am certain that there is a formula for this
y = n + n +5 + n + 9 + n + 10 + n + 11
y = 5n + 35
I will test this where n = 9
y = 5 (9) + 35
y = 45 + 35
y = 80
Therefore the formula for any T-number rotated by 180° with a 5 by 5 grid is
n = T-number and y = T-total
y = 5n + 35
I will now translate the T-shape by 180° with a 4 by 4 grid.
y = 2 + 6 + 9 + 10 + 11
y = 38
n = 2
Here I will show the formula.
y = n + n +4 + n + 7 + n + 8 + n + 9
y = 5n + 28
I will test this where n = 2
y = 5 (2) + 28
y = 10 + 28
y = 38
Looking at the T-shape with a grid 4 by 4, the formula was 5n - 28 however the formula for the T-shape rotated at 180° was 5n + 28. The formula is exactly the same except that there is only a sign change from + to -. So I will now make an assumption that all the formulas for the T-shape rotated at 180° are exactly the same as the normal T-shape except that there is a sign change from + to -.
The 8 by 8 formula was 5n – 56. I will assume that the T-shape rotated at 180° with an 8 by 8 grid has a formula 5n + 56.
I will now see if my theory is correct.
I will first add up everything in the T-shape,
y = 2 + 10 + 17 + 18 + 19
y = 66
Now I will see if I was correct, by putting my information into the formula.
If, n = 2
y = 5n – 56
y = 10 + 56
y = 66
The assumption has been proven correct, but I will do one more to make sure.
The formula for the 10 by 10 grid is 5n – 70. So for the T-shape rotated 180° is 5n + 70.
So y = 2 + 12 + 21 + 22 + 23
y = 80
Now I will test my formula to see if it works.
y = 5n + 70, where n = 2
y = 5 ( 2 ) + 70
y = 10 + 70
y = 80
The formula is correct. Here I have put all the formulas into a table.
So if all the formulas for each grid number with T-shape rotated at 180° are the same (except the sign) as the T-shape then the formula for any x by x grid must be exactly the same except that the sign is different.
So the previous formula for x by x grid is y = 5n - 7x, therefore the formula for the T-shape rotated at 180° is y = 5n + 7x.
Let’s check this with one that I have done to see if it works with a 10 by 10 grid, where n = 2 and the T-total is 80.
x = 10
n = 2
y = 5n + 7x
y = 5 ( 2 ) + 7 ( 10 )
y = 10 + 7
y = 70
So
T-shape rotated at 180°
Therefore the formula for any T-number with x by x grid is
n = T-number and y = T-total and x = grid number
y = 5n + 7x
Now I will rotate the T-shape 90° and find a formula for this because there were formulas for the previous ones.
I will start with a 5 by 5 grid.
n = 7
y = n + n +1 + n + 2 + n - 3 + n + 7
y = 5n + 7
I will see if this formula works,
If n = 16
y = 16 + 17 + 18 + 13 + 23
y = 87
y = 5n + 7
y = 5( 16 ) + 7
y = 87
The formula works.
Therefore the formula for any T-number rotated by 90° with a 5 by 5 grid is
n = T-number and y = T-total
y = 5n + 7
Now I will find a formula with a 10 by 10 gird
n = 13
y = n + n +1 + n + 2 + n - 8 + n + 12
y = 5n + 7
y = 13 + 14 + 15 + 5 + 25
y = 72
The formula is exactly as the same as before. I will see if this formula works,
If n = 13
y = 5n + 7
y = 5( 13 ) + 7
y = 72
The formula works.
Therefore the formula for any T-number rotated by 90° with a 10 by 10 grid is
n = T-number and y = T-total
y = 5n + 7
I will assume that for any x by x grid the formula is 5n + 7 because the previous two are exactly the same. But I will do one more to make sure.
15 by 15 grid
The T-total for this is shaded area is 127
The formula for this is y = 5n + 7
y = 5 ( 24 ) + 7
y = 120 + 7
y = 127
The formula works.
T-shape rotated at 90°
Therefore the formula for any T-number with any x by x grid is
n = T-number and y = T-total and x = grid number
y = 5n + 7
I will therefore assume that the T-shape rotated at 270° has a formula
y = 5n - 7 with any x by x grid, (only a sign change). This is because when the T-shape was flip the signs also changed but the formula was the same. So for the T-shape rotated at 90°, it is also being flipped therefore causing the sign to change as well from + to -.
I will do check one with a 15 by 15 grid to see if it is correct.
The T-total for this shaded area is 108
So now I will see if my formula is correct,
y = 5n – 7, where n = 23
y = 5 ( 23 ) – 7
y = 115 – 7
y = 108
The formula works.
T-shape rotated at 270°
Therefore the formula for any T-number with any x by x grid is
n = T-number and y = T-total and x = grid number
y = 5n + 7
Summery
Here are all the formulas for the T-Shape rotated at different degrees.
Therefore the formula for any T-number with x by x grid is
n = T-number and y = T-total and x = grid number
y = 5n – 7x
T-shape rotated at 180°
Therefore the formula for any T-number with x by x grid is
n = T-number and y = T-total and x = grid number
y = 5n + 7x
T-shape rotated at 90°
Therefore the formula for any T-number with any x by x grid is
n = T-number and y = T-total and x = grid number
y = 5n + 7
T-shape rotated at 270°
Therefore the formula for any T-number with any x by x grid is
n = T-number and y = T-total and x = grid number
y = 5n + 7
All the formulas are similar and the differences are only a sign change ( + or – ) or the formula either includes an x or not.
