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.