I am going to investigate T-Shapes.

I have established a pattern in the results.  This will enable me to work out the first T-Number from the grid size.  The formula is as follows (S is grid size and N is T-Number): -

N=2S+2

To work out the first T-Number of any size grid you simply double the amount of squares along the top and add two.  Once the T-Number has been found the T-Total can be found using the formulas shown below: -

The first T-Shape

As you can see the above results prove my formula works.  I could now work out any T-Total from any T-Number for any size grid using these two formula.  

Now that I have worked out two separate formulas, I am going to try and combine the two.  If we go back to the original diagram which I used when working out the initial formulas (S=Grid Size).  

As you can see there is a definite relation ship between the highlighted number and the grid size.  We already know that the T-Number multiplied by five and subtracted by a number equal the T-Total.  The subtraction number is not always the same it is dependant on the size of the grid.  So if I can find out why this number changes I should be able to work out the overall formula.  

The above diagram shows the relationship between the size and the T-Number, the subtraction number varies according to the grid size.  

Below is a list of the different subtraction numbers for the different size

grids: -

There is a pattern between the differences they increase by seven each time.  If you times the grid number by seven you get the subtraction number (N=Number, T=Total, S=Size).

So if I add this to what we already know we get the formula: -

T=5N-7S

Now I must test my formula to make sure it works for all grid sizes.  

This proves my formula works for any size grid and any T-Number.

Movement

I am now going to find a formula which will enable me to work out the T-Total after the T-Shape has been moved around the grid.  As shown below: -

As you can see the shape has been moved two to the right and two down.  

 

When the shape is moved down the T-Total the shape increases on a number dependant on the Grid Size.  This is five times the grid size.  When moving up the grid you subtract the number.  

When the shape is moved to the right each number moves up one, as there are five numbers you simply add five for each time it’s moved.  When moving to the left you subtract five.  

The final formula is below (L=Left, R=Right, U=Up, D=Down, Note if the shape isn’t moved in a direction the letter should be changed to zero): -

I now have a formula which can work out any T-Total from any T-Number on any Grid Size.  I also have a formula which enables me to work out the T-Total after the shape, have been moved.

E.g. If I was told that a T-Shape with the T-Number 20 had moved two right and two down I could work out the new T-Total using algebra.  

Rotations and Reflections

I am now going to investigate how rotating and reflecting the T-Shape affects the T-Total and T-Number.  I have chosen to investigate the other three possible rotation or reflections.

Upside down T-Shapes

   

T=5N+7S

Now that I have a formula I must test it to make sure it works.  

My formula works.

T-Shapes on the right

T=5N+(S-2)

I have tested the formula, as shown above and it works.  

T-Shapes on the left

T=5N-(S-2)

I have tested the formula and it works.  

The above formulas enable me to work out any T-Total for any T-Number on any grid size in every possible rotation and reflection.  

Validation

I now have a several formulas, which can work out the T-Total from various arrangements, but the formulas will still give me an answer even if the shape is only partly on the grid.  As shown below: -

I will now try and find a formula which checks if the T-Number is a valid for that size grid.  As you can see only some squares can be used as T-Numbers, the below diagram shows them: -

The grid is 9*9 but only the middle seven squares can be used as T-Numbers and only the bottom seven when the shape is moved up and down. This changes when the grid size changes its always the grid size minus two.  So I am going to divide the T-Number by the grid size.

 When the T-Number is divided by the grid size we are left with a number and remainder.  The number plus one is the row of the T-Number and the remainder is the column number.  So using this information I can work out were the T-Number is valid.  I will now try and prove this algebraically.  

E.g. 20/9=2.2

There are three steps to validating a T-Number.  

1. N/S – The T-Number is divided by the Grid Size, the answer will be a number and a remainder (R=Row, C=Column).  

N / S = R.S

2. Row – The row number is put into the following inequality to check whether it is valid.  

2 > (R+1) < S+1

3. Column – The column number is put into this inequality to check its validity: -

1 > C < S-1

If the two numbers fit the inequalities then the T-Number is valid.  I will now test the formula.  

As you can see the formula can successfully validate T-Numbers.  

I will now investigate rotations in more detail e.g. what happens if you rotate the T-Number left or right.  As shown below.  

The table below shows five full rotations.  

The difference between the Left and Right or Upside Down and Normal is double the number you add on or take off.  For example you add 63 to and upside down shape and take 63 off a normal shape.  If you add these together you get the difference.  This also true for the Left and Right, 7 is added on and subtracted and 14 is the difference.  

The difference between the normal and right or upside and left are 70.  So starting from the normal is you turn right you add 70.  Then if you turn right again you add 56.  Then if you turn right again you add 70.  Then is you turn right again bringing you to the top you take 56.  

(LR=Rotated Left, RR=Rotated Right, N=Normal T-Shapes, UD=Upside down T-Shapes, LT= Left T-shapes, RT=Right T-Shapes)

The table above shows what the T-Total is after a shape has been rotated.  So if you know where the shape is to start with and which way its is being rotated, you can work out the new T-Total.

Conclusion

I have worked out five formulas and a table.  I have worked out a formula for each type of rotation for any grid size.  I also have a formula which allows the T-Total to found out after the shape ahs been translated.  The table enables me to work out the T-Total after the shape has been rotated.  

I think I have successfully investigated T-Shapes to the best of my ability.