## T-Total and T-Number

PART 1

We have a grid nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the t-shape. This is highlighted in the colour red. This is shown below: -

The number 20 at the bottom of the t-shape will be called the t-number. All the numbers highlighted will be called the t-total. In this section there is an investigation between the t-total and the t-number.

## For this t-shape the

T-number is 20

And the

T-total is37

## For this t-shape the

T-number is 21

and the

T-total is 42

As you can see from this information is that every time the t-number goes up one the t-total goes up five.

Therefore the ratio between the t-number and the t-total is 1:5

This helps us because when we want to translate a t-shape to another position. Say we move it to here

We all ready know the answer to the one in red. To work out the one in green all we have to do is work out the difference in the t-number and in this case it is 54. We then times the 54 by 5 because it rises 5 ever time the t- number goes up. Then we + the t-total from the original t-shape and we come out with the t-total for the green t-shape. This is another way to work out the t-total.

What we need now is a formula for the relationship between the t-total and the t-number. I have found a formula which is 5t-number-63 = t-total.

The question is how did we work out this formula and what can we do with it?

The formula starts with 5* the t-number this is because there is a rise in the t-total by 5 for every t-number. We then –63 which do by working out the difference between the t-number and another number in the t-shape. This has to be done to the other 4 numbers in the t-shape. Here is an example: -

The t- shape has a t-number of 32. Now to work out the difference between the t-number and the rest of the numbers in this t-shape

Working out: -

32-13=19

32-14=18

32-15=17

32-23= 9

TOTAL= 63

This will happen to all the shapes this way up. To prove this I will do another.

The t-number is 70. Now to work out the difference between the t-number and the rest of the numbers in this t-shape

Working Out: -

70-51=19

70-52=18

70-53=17

70-61=9

TOTAL=63

Again the number turns out to be 63. This is where the 63 came from in this equation. There is also another place this 63 comes from. This is 9*7=63. The nine in this comes from the size of the grid this one been nine. If the grid size were 10 by 10 then it would be 10*7. At the end of this piece of coursework when we but all the formulas together we realise that the number we minus or plus by is divisible b y seven. This is where we get the seven from. The seven works with all the same sizes. The other method will also work with a different size grid.