T -Totals. From this 9*9 grid I will collect a number of T Shapes with T Numbers after collecting these I will put them into a table and will investigate to find any relationship between them.

Sohaib Mohammad 11H -- Maths Coursework -- T Totals

T –Totals

For this coursework I have been given a task which is to investigate the relationship between the T-total and the T-number. To start my investigation I will do this by working on a 9*9 grid. From this 9*9 grid I will collect a number of T –Shapes with T –Numbers after collecting these I will put them into a table and will investigate to find any relationship between them.

Definition of the T –Total and T –Numbers

Consider the numbers in the T –Shaped figure shown in the grid by thick lines, redrawn underneath separately for clarity. The integer at the bottom of the T –Shape is 20 this I known as the T –number for the T –Shape:

The T –Total is defined as the sum of all the integers shown in the above T –Shaped figure. For example the T –Total for the T –Shape above = 1+2+3+11+20 = 37

I firstly begin the investigation by starting of with a T shape, the following shows 4 T –Shapes collected from the first row:

T –Number = 21

T –Shape = 2+3+4+12+21 = T –Total = 42

T –Number = 22

T –Shape = 3+4+5+13+22 = T –Total = 47

T –Number = 23

T –Total = 4+5+6+14+23 = 52

The next part shows 4 more T –Shapes collected from a different position on the grid, these T –Shapes are from the second row:

T –Number = 29

T –Total = 82

T –Number = 30

T –Total = 87

T –Number = 31

T –Total = 92

T –Number = 32

T –Total = 97

After collecting these I have put them in a table to see if there is any relationship between the T –Number and the T –Total:

From both these tables I can see a pattern and that is that the T –Total increases by 5 every time the T –Number increases.

37, 42, 47, 52, 82, 87, 92, 97

5 5 5 5 5 5

For this investigation I am able to calculate and algebraic equation which would enable me to find any T –Total in this 9*9 grid to do this, I firstly I use “n” to represent the T –Number in the equation and to figure out the rest of the numbers in the T –Shape I will simply subtract to figure out the equation for that certain number eg: 11 = n-9 this is because n = 20 and 20-9 = 11 so therefore 11 would be represented as 11 = n-9:

Now to find out the equation I would have to add up the T –Shape as if I was finding the T –Total:

T –Number = n

T –Total = n+n-9+n-18+n-19+n-17

After simplifying the equation we end up with the following equation:

T –Total = 5n -63

This is the equation for the grid which enables us to find any T –Total on this 9*9 grid, to test if this equation is right I firstly choose the following T –Shape to experiment upon:

T –Number = 29

T –Total = 82

From this I substitute 29 into the equation as “n” to find the T-Total = 5(29) -63

= 145 -63

= 82

As you can see this equation is correct as I have managed to figure out the T –Total of this T –Shape which happens to be 82:

T –Number = 29

T –Total = 82

The next part is to do to the same thing on an 8*8 grid which is to once again find the relationship between the T –Total and T –Number and to also see if the equation works for an 8*8 grid.

Once again I have collected four T- Shapes from the first row:

T –Number = 18 T –Number = 19

T –Total = 1+2+3+10+18 = 34 T –Total = 2+3+4+11+19 = 39