My investigation is to see the relationship between the T-Total and the T-Number of T-Shapes and to find a rule for it so that the T-Total can be easily found.

 

TN= N+(N-9)+(N-19)+(N-18)+(N-17)   

6 + 7 + 8 + 16 + 25 = 62

Therefore the T-Total for 25 is 62.

My estimation was spot on.

To find the rule, first I multiplied 5 by the T-Number.

For example:-

Step 1

5 x 20 = 100

Then I took away the T-Total from the figure I got once I had multiplied 5 by the T-Number.

For Instance:-

Step 2

100 – 37 = 63

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                

As a result the rule in Algebraic form is:

TN= N+(N-9)+(N-19)+(N-18)+(N-17)

(By adding all the n’s together you get 5n)

TN= N+(N-9)+(N-19)+(N-18)+(N-17)

               

(by doing the equation you get –63)

5n – 63

In witch n represents the T-Number

I experimented this rule for another T-Shape selected at random here are the results :-

5 x 69 = 345

                              345 – 63 = 282                                                                                        

Hence I think the T-Total for T-Number 69 will be 282:-

TN= N+(N-9)+(N-19)+(N-18)+(N-17)

50 + 51 + 52 + 60 + 69 = 282

My rule for gird size 9 by 9 is precise.

Now I am going to investigate the relationship between the T-Total and the T-Number on a variety of grids of different sizes.

        

8 by 8 grid

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

1 + 2 +N- 3 +10 +18 =

T18=34

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

2 + 3 + 4 + 11 + 19=

T19=39

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

3 + 4 + 5 +12 + 20=

T20=44

Here is the results table for the 8 by 8 grid :-

The T-Total increase’s by 5 so the rule for the 8 by 8 grid would be found in the similar way to the 9 by 9 grid.

To work out the rule for this grid I multiplied 5 by the T-Number and took away the T-Total.

e.g.

 

5 x 18 = 90

90 – 34 = 56

As a result the rule for the 8 by 8 grid is :-

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

                        

(By adding together all the n’s you get 5n)

TN= N+(N-8)+(N-16)+(N-15)+(N-17)

                        

(By doing the equation you get left with –56)

5n – 56

Now I am going to do a 10 by 10 grid.

The working out for the rule

TN= N+(N-10)+(N-20)+(N-21)+(N-19)

1 + 2 + 3 + 12 + 22 =

T22=40

22 x 5 = 110

110 – 40 = 70

Therefore the rule for the 10 by 10 grid is:

TN= N+(N-10)+(N-20)+(N-21)+(N-19)

                

(By adding together all the n’s you get 5n)

TN= N+(N-10)+(N-20)+(N-21)+(N-19)

(by doing the equation you get –70)

5n – 70

Now I am going to do a 11 by 11 grid

TN= N+(N-11)+(N-23)+(N-22)+(N-21)

1 + 2 + 3 + 13 + 24 =

T24=43

5 X 24 =120

120 – 43 =77

The rule for the 11 by 11 grid Is :-

TN= N+(N-11)+(N-23)+(N-22)+(N-21)

                

                

(By adding together all the n’ s you will get 5n)

TN= N+(N-11)+(N-23)+(N-22)+(N-21)

                

(By doing the equation you get –77)

5n – 77

Now I am going to draw a results table for the rules of different grid sizes

Results table

As you can see from the results the rule increases by 7 in each grid. Also the last figure is obtained by multiplying the grid size by 7.

For example:-

The rule for the 8 by 8 grid is 5n – 56

7 x 8 = 56

Therefore to show this in algebraic form it would be:

7g

Were as the algebraic equation for all of it would be:        

5n – 7g

Now I am going to test this rule to see if it works on any grid size. I will do this on a 12 by 12 grid.

12 by 12 grid

5t – 7g is my rule here is the working out

5 x 26 =130

7 x 12 =84

Therefore I think the rule for the 12 by 12 grid will be :

130 – 84

Now I am going to see if I am right

TN=N+(N-12)+(N-24)+(N-23)+(N-25)

TN= 1+ 2+ 3 + 14 + 26=46

5 x 26 = 130

130 – 46 = 84

Therefore this proves that the rule works on any size grid.