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T-Shapes

Look at the T-shape drawn on the 9 by 9 number grid.

The total of the numbers inside the T-shape is 1+2+3+11+20=37

This is called the T-total.

The number at the bottom of the T-shape is called theT-number.

The T-number for this T-shape is 20.

1)  The relationship between the T-Total and the T-Number.

If you take the other numbers in the T-Shape away from the T-Number you get a T-Shape like this.

You will notice that the centre column of the T-Shape is going up in 9’s because of the table size. With the table set out like this a formula can be worked out to find any T-Total on this size grid. This is done in the working below:-

T-total = T-19+T-18+T-17+T-9+T

            =  5T-63

Now to test this formula to see if it works

For T-total I will use the letter X

For the T-Number I will use the letter T

So X = 5T-63

     T = 20

     X = 5x20-63

         = 100-63

         = 37

Now I will do two more to check to see if it will work anywhere on the grid.

T=43

X=5x43-63                         24+25+26+34+43=152

X=215-63

X=152

T=49

X=5x49-63                         30+31+32+40+49=182

X=245-63

X=182

I have tested the formula for three different T-numbers and the formula works for a 9 by 9 grid.

The full formula for this size grid is: - X=5T-63

2)    The T-Shape on different size grids, translated to different positions to Investigate relationships between the T-Total, the T-Numbers and the Grid sizes.

Below is a grid with parts of the formula in. I will use this in a minute to figure out a formula for any T-Total on any width grid.

For a “ T ” in the grid

T-total = T-2G-1

              T-2G

              T-2G+1

              T-G

              T

X=5T-7G

If the Grid Width (= G), and the T-Number (= T), the T-Total (= X) is calculated by

            X=5T-7G

Now I’m going to try the grid that I drew earlier i.e. G=9

The formula is X=5T-7G

Which changes to X=5T-7x9

                          X=5T-63

I will now try the formula on a 8 by 8 grid.

X=5x27-56 (10+11+12+19+27=79)

X=135-56

X=79

I will now try it on a 13 by 7 grid.

X=5x47-7x13 (20+21+22+34+47=144)

X=235-91

X=144

I will now try it on a 6 by 7 grid.

X=5x22-7x6 (9+10+11+16+22=68)

X=110-42

X=68

I have tried this formula out on three different grid widths and the formula works, the formula is: X=5T-7

3)  Grids of different size using other transformations and combinations of transformations to Investigate relationships between the T-Total, the T-Numbers, the Grid sizes and the transformations.

Below is a grid with parts of the formula in. I will use this in a minute to figure out a formula.

For a “ T ” in the grid

2 squares right

} How does the T-number change?

1 square down

From 2G+2 to 3G+4

This looks like adding numbers of squares down to the number of G’s and the number of squares right to the other number. If we start with a T-number (T) then moving 4 right and 3 down

The new T-number is:

             T+3G+4

We can use “h” for the number of places right and “Y” for the number of places down.

We could write this as (h/y), a column vector

Now X+(4/3)=X+3G+4

And X+(h/y)=X+yG+h

For all of the “ T’s ” the T-number is the same as shown below

The other “ T’s ” that I can work out the formula for are shown bellow.

A 3 by 3 grid can be used for the other 3.

T

T+G

T+2G

T+2G-1

T+2G+1

7G+5T

X=7G+5T=7x3+5x2

                =21+10 (2+5+7+8+9=31)

                =31

I will now try this new formula on a 7 by 4 grid

7G+5T

=7x7+5x5 (5+12+18+19+20=74)

=49+25

=74

7G+5T

7x6+5x3 (3+9+14+15+16=57)

42+15

57

I have tried this new formula for an inverted T and it works, the formula is: 7G+5T

T

T+1

T+2

T+2+G

T+2-G

5t+7

The formula is 5T+7

I will now try on a 5 by 5 grid

5x12+7=67

12+13+14+9+19=67

I will now check on a 7 by 7 grid

5T+7 (24+25+26+19+33=127)

5x24+7 127

I will now try the formula on a 8 by 3 grid

5T+7

5x12+7 (6+22+12+13+14=67)

67

I have tried this new formula for a side ways T, The formula worked it is: 5T+7

T

T-1

T-2

T-2-G

T-2+G

5T-7

I will now try it on a 5 by 5 grid

5T-7

5x14-7 (7+12+13+14+17=63)

63

And now on 4 by 4 grid

5T-7

5x12-7 (6+10+14+11+12=53)

53

And on a 6 by 3 grid

5T-7

5x9-7 (1+13+7+8+9=38)

45-7

38

I have tested this new formula for a side ways T the formula works it is: 5T-7

The formula for a T this way is: X=5T-7G

The formula for a T this way is: X=7G+5T

The formula for a T this way is: X=5T+7

The formula for a T this way is: X=5T-7

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