T-Total. I will take steps to find formulae for changing the position of the T in many ways using methods such as translation and rotation.

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T-Totals

This grid has a T-shape in the top left corner of it.  The numbers inside of the T are 1, 2, 3, 11 and 20.  To find the T total of the T I must add these 5 numbers together.  By adding these numbers the T total is 37.  The T number in each T drawn is the bottom of the T, whichever way the T faces.  This T number is 20.  From this information I will take steps to find formulae for changing the position of the T in many ways using methods such as translation and rotation.

                                         

The stages I will take for my investigation are the following: -

  • Firstly I will find a formula for finding the T total using the T number and including the grid size
  • Once I have found this formula, I will extend it and use it to find formulae to explain and solve translating the T
  • I will translate the T
  • Using what I have found from my formulae, my next step will be to move the T to the right and down, to the right and up, to the left and down and finally to the left and up
  • These are all the possible translations with this shape
  • Once I have found those formulae I will move on to rotation
  • I will be rotating the T shape 90 degrees, 180 degrees and 270 degrees
  • I will start with 90 degrees, then 270 degrees and then I will work on 180 degrees
  • Rotation will be my final part of my investigation
  • All the way through my work I will be including explanations and diagrams
  • As well as using explanations of what I am doing, I will explain why I am doing it and why I get the answers I do
  • I will be stating all the variables and when I add a new variable I will clearly state what it is.

Stage One – Formula For The T Total Of An Upright T Shape

I started the investigation by drawing up a 9 x 9 grid and drawing the T shape in the top left hand corner of the grid.  In this T were the numbers 1, 2, 3, 11 and 20.  I moved the T along one square so that the T number became 21.  I tried this out five times and here are my results: -

When the T is moved across one square, 5 is added to the T total because there are 5 squares in the T shape meaning 1 has been added to each number in the T adding up to a total of 5.  To find the rest of the formula I went back to my original T shape in the top left-hand corner of the grid.  I replaced the T with x, which meant that the rest of the numbers in the T had to be substituted with something else to relate it to x.  This is what I came up with for any T total in the grid size of 9 x 9: -

With these numbers, I added them altogether and I got the number 63.  I divided 63 by the grid size and got the answer of 7.  

I used the same method for the 8 x 8 grid above as I did for the 9 x 9 grid.  The T is in the same place but the numbers have changed.  I substituted the T number of 18 with x and changed the rest of the numbers appropriately.  Here are my results :-

I added up the numbers as I did with the other T shape, and got the answer of 56.  When dividing 56 by the grid size (8) again I got 7.  This explains that somewhere in the formula there must be 7 x the grid size.

A general way to look at replacing the T number with x, which would work in all grids, is the following: -

(g is the variable for grid size)

This table is proving that the T shape above with the equations in it works.  This example is from the T shape in a 9 x 9 grid found on the previous pages.

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The number 7 comes up when using each grid size because of the amount of squares each number is away from the T.  The T shape below will make this clearer:-

From the investigations I have done so far I have found the information for my formula and need to assemble it and make sure it works.   The formula I put together, using all the relevant information, is 5x – y.

(The x variable stands for the T number and the y variable stands for 7 x grid size).

I tried this ...

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