T - Totals Investigation

T-Totals

Introduction:

        This is an investigation on T-totals. By starting with a 9x9 grid, and numbering it, so the top left corner numbered with one and work downwards. An example of the grid is shown below.

The investigation will be done with a standard T-shape which is 3 along and from the centre 2 more down. As shown below.

This is the t-number for a T-shape.

The T-total is all the numbers in the T-shape added together. The T-total for this specific example is: 2+3+4+12+21=42.

Plan:

        Part 1 – I will investigate the relationship between the T-total and the  T-number on a 9x9 grid.

        Part 2 – I will investigate the relationship between the T-total, T-number and grid size.

Part 1:

        Diagrams:

20+21+22+30+39=13252+53+54+62+71=292

         T-total = 132                                    T-total = 292

4+5+6+14+23=52

                                        T-total = 52

From now on:

        Let T-total be T!

Let T-number be n!

Now I have shown some examples, a table is required so the formula can be solved between the relationship of the T and n for a 9x9 grid.

                        n                                T

1                                X

                                                        X

                        19                                X

                        20                                37

                        21                                42

                        22                                47

                        23                                52

24                                57

                        25                                62

        26                                67

27                                X

28                                X

        29                                82

        38                                127

80                                337

5n      100    105     110     115     120    125

T         37        42        47        52        57        62

  5n to T    - 63    - 63     - 63     - 63    - 63     - 63

Therefore the table above shows the formula will have – 63 in, because – 63 is a constant difference.

The formula comes out to be 5n – 63 = T for a 9x9 grid only.

As a test if n = 38

                With the formula: (5 x 38) – 63 = T

190. – 63 = T

127 = T

This is correct; the table shows that the formula works.

I predict that if the T-number is 40, with the formula of 5n – 63 = T for a 9x9 grid, T will equal: 137        

T = 5n – 63

= (5 x 40) – 63

   = 200 – 63

T = 137

To prove that the T-total for the T-number of 40 is 137:

40 + 31 + 22 + 23 + 21= T

                       137 = T

        So the formula is correct as the predicted T-total with the formula was correct, but only for a 9x9 grid.

        I think that the formula will also work with incomplete T-shapes too, if 19 is used as n, I can use the formula obtained to prove it:

        T = (5 x 19) – 63

           = 95 – 63

        T = 32

To show that this is correct:

                        T=19+10+1+2

T= 32

It is correct the formula also got 32, showing the formula works for T-shapes even if they are not complete.

Part 2:

        I think that I have found the relationship between grid size, T-number and T-total. Because of the table at the top of the page, shows how found that there is always 5n in the formula and then the grid size is multiplied by -7 to give T. So I think the formula is:

5n – 7G = T

        To test the new formula of T = 5n -7G, I will have the grid size of 10 and n being 24.

        The formula gets: T = (5 x 24) – (7 x 10)

                               T =    120    –     70

                                  T = 50  

        To check this:

T= 3+4+5+14+24= 50

Excellent, the formula obtained which links the T-total, T-number and grid size is:

T = 5n – 7G

Whilst doing this investigation a number of questions sprung to mind, what would happen if the T-shape was turned round, through 90o, 180o and 270o? I also wondered, what would happen to the T-total once the T-shape had been rotated around a number in the grid, how this would relate to the new T-total, the old T-number and the number the T-shape is rotated around? How would the T-total change when the T-shape is reflected? How would the T-total change when it is enlarged or elongated? I also thought that what would happen to the T-total if the T-shape is translated about the grid?