T-Shapes.I am going to investigate the T-Shapes grid. I will be first looking at the 9 x 9 grid. I will then show how many T-shapes I would be able to make, by only moving the shapes across only once each time.

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T-Shapes

I am going to investigate the T-Shapes grid. I will be first looking at the 9 x 9 grid.

I will then show how many T-shapes I would be able to make, by only moving the shapes across only once each time.

Once I have done this I will add up all the T-Shapes to find out the T-Total.

The last number at the bottom of the T-Shape is called the T-number.

The T-total for this T-shape is:

1 + 2 + 3 + 11 + 20 = 37

37 = T-Total

 

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

 

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

= 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

X = 5T - 63

T = 20

X = 5 x 20 - 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 = 5 x 43 - 63                        

 24 + 25 + 26 + 34 + 43 = 152

X = 215 - 63

X = 152

T = 49

X = 5 x 49 - 63                      

30 + 31 + 32 + 40 + 49 = 182

X = 245 - 63

X = 182

Conclusion

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

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

I then tried to find out the T-Total for a 6 by 6 grid,

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

 

This time the centre column of the T-Shape is going up in 6’s because of the table size. The formula for this size grid is:

T-Total      = T - 13  + T - 12  + T – 11 + T – 6 + T

                   = 5T - 42

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