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T-Shape investigation.

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Introduction

T-Shape

Firstly, I am going to look at the relationship between the t-number and the t-total. I am going to refer to these terms using the letters N and Z:

        n = t-number

        z = -total

I will take the first t-shape at the top left of a 9 x 9 size grid.

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10

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37

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40


                                                             
image00.png

I predict that if I move the t-shape to a different location the t-total will be the

t-number + 17   (n+17)

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25image01.png

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If my prediction is correct, t-total should equal 40 (23 + 17 = 40)

                                                                                             Z = 4+5+6+14+23=52

My prediction is not correct. I will move the t-shape to the right and tabulate my results to see if there is a pattern.

n

20

23

26

z

37

52

67

The t-total is in creasing by 15. Using first differences I will try to find a formula.

n

20

23

26

15n

300

345

390

-263

-293

-323

z

37

52

67

I cannot find a formula for this. However, I have noticed that my n value is increasing by 3 every time. I will re-tabulate my results so it increases by 1 each time. This will make the numbers smaller and closer together and easier to spot a pattern.

n

20

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z

37

42

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52

57

62

There is a definite pattern. The z value is increasing by 5 each time.

...read more.

Middle

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But from my earlier mistakes, I will move the t shape along a number of times to identify a pattern on an 8 x 8 grid.

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n

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z

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49

Right away I can see the t-total is increasing by 5. I will look at the difference of the numbers after I do the first part of the equation (5n)

n

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5n

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-56

-56

-56

-56

z

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49

So we can see that the formula is 5n - 56

Now I will see if there is a way to find more patterns so we can find out more equations.

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n

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5n

110

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-70

-70

-70

-70

z

40

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60


                                        
5n - 70

So far we have 10 x 10  = 5n - 70

        9 x  9    = 5n - 63

        8 x  8    = 5n - 56

Here, we can see a pattern, every time we go down a grid size the red numbers are decreasing by 7. The reason why it is always 5n is because each square increases by one every time you move it across one space.

 So now we can make a new formula... one for the general t-shape, no matter the grid size. But, we need a new value

        g = grid size                                Z = 5n – 7g

After more experiments of t-shapes in different grids, I noticed that you don’t always have to be a 9 x 9 or a 10 x 10. It can be 9 x 10 or 10 x 12.

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49image03.png

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...read more.

Conclusion

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18 - 9 = 9

Straight away we can see that this doesn’t work. But, you can notice 41 is 9 lower that 50 and is also the grid width. So if we try this on the 8 x 8 grid.

18 – g = 10

18 – 8 = 10

 This method works. So we have a new all round formula…

n-2g-1

n-2g

n-2g+1

n-g

n

I will now try and find formulas when the T is upside down on a 9 x 9 grid.

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Like when the T was the right way up, the number below the t-number is 9 more and so is the number below that. The number to the left is 1 less and the number to the right is 1 more. As the difference between the numbers (2, 11, 20) is 9 and is also the grid width the formula will be

n

n+g

n+2g-1

n+2g

n+2g+1

Now I will check this in the middle of a 10 x 10 grid.

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37

16 + g = 26        16 + 2g = 36         16 + 2g + 1 = 37        16 + 2g – 1 = 35

16 + 10 = 26        16 + 20 = 36        16 + 20 + 1 = 37        16 + 2g – 1 = 35

So this formula works for the upside down T on different grid sizes.

So all of the formulas that I found are:

T-Number / T-Total relationship = 5n - 7g

T-Number / T-Total relationship (upside down) = 5n + 7g

Inside the T-Shape =

n-2g-1

n-2g

n-2g+1

n-g

n

Inside the T-Shape (upside down) =

n

n+g

n+2g-1

n+2g

n+2n+1

...read more.

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