In this investigation I will be looking at the relationship between the T-total and the T-number with simple drawings and try to identify a general rule.

My prediction is already included on the 9x9 grid it is the Green T-shape.

Testing my prediction

My prediction was correct as you can see on the grid the green T-shape =337.

T-n= 81 + 72 + 62 + 63 + 64 = 337 (T-t)

General Rule

The general rule for finding the Nth term for the T-total is,

T-total = n + n – 9 + n – 19 + n – 18 + n – 17 = 5 n - 63

Using the nth term 5n – 63 I should be able to choose a T- number randomly (has to be n) and use this new rule to work

out the T-total.

Testing the general rule

I will now choose a random T-number on a 9x9 grid to test my rule.

T-n = 63 + 54 + 44 + 45 + 46 = 252 (T- t)

 

I will now test this with my rule: n = 63

Nth term = 5n – 63=   (5 x 48) 315 – 63 =252

T-total = 252    

Part 2  

With the results from part 1 where I looked at the relationship between the t-total and the T-number. I will now take this investigation further by using different grid sizes. I am going to use grid sizes 7 x 7 and 8 x 8 to see if there is any difference in the relationship between the T-total and T-number.    

Drawings

I am going to draw the following grids 7x7 and 8x8. I produce 3 T-shapes on each grid the 4th   

T - shape being my prediction.

T - n = 16 + 9 + 1 + 2 + 3 = 31(T-t)

T - n = 24 + 17 + 9 + 10 + 11 = 71(T-t)

T – n = 32 + 25 + 17 + 18 + 19 = 111 (T-t) 

           

 T – n = 18+ 10 + 1 + 2 + 3 = 34    (T- n)

 T – n = 27 + 19 +  + 10 + 11 + 12 = 79   (T- n)  

 T – n = 36 + 28 + 19 + 20 + 21 = 124    (T- n)

Table of Results

My results for the 7x7 and 8x8 grid will be put into a table, so that I can see if there are any patterns emerging.

7 x 7 Grid                                                                             8 x 8 Grid

                             

General Rule

These are the nth terms for Grids 7x 7 and 8 x 8.

7 x 7                                                                                      8 x 8

T-total = n + n – 7 + n – 15 + n – 14 + n – 13                T-total = n + n – 8 + n + 17+ n – 16 +n -15

                          =5n – 49                                                          = 5n – 56

Identifying the pattern

There is a clear pattern on the 7 x 7 grid as the T- n goes up by 8 the T- t increases by 40 every time. On the 8 x 8 grid the T-n goes up by 9 every time and the T-t increases by 45.

I have also noticed that every time the grid size gets smaller the nth term decreases by 7 but the 5n does not change.  

 

 

Prediction

I predict that the nth term for the 6 x 6 grid will be 5n – 42, I am basing this on the results from my other grids.

Testing prediction

I will now test my prediction:

                                         t-n = 22 + 16 + 9 + 10 + 11 = 68 (T-t)

                                                               

T- total = n + n – 6+ n – 13 + n – 12 + n – 11 = 5n  - 42

Part 3

Introduction

I will now try to find an overall rule that will allow me to work out the T – total on any grid size. I am going to use the evidence that I have found from the three grid sizes 7 x 7, 8 x 8 and 9 x 9 to help investigate the relationship between the grid size and the T-total.

Overall Rule

I am going to use all the nth terms from the 3 grids and put them in a table and try to find a rule that will work on any grid size.

           7 x 7                                          8 x 8                                                    9 x 9

n = T - number

G = Grid size

         

General Rule

I will now solve the rule.

T-total=n + n–g + n–2g +1 + n–2g + n-2g – 1 = 5n – 7g

The rule to find any t – total in any grid size is 5n – 7g.

Testing overall rule

To prove that my rule does work I will test it on 6 x 6 and  10 x 10 grids

 

T- n = 32

5n – 7g =(5 x 32) 160 – (7 x 6) 42 = 118 (T-t)

T-t = 32 + 26 + 19 + 20 + 21  = 118

T-n = 19

5n – 7g = (5 x 19) 95 – (7 x 5) 35 =  60 ( T- t)

T – t = 19 + 14 + 8 + 9 + 10 = 60