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Introduction

Introduction

My plan is to investigate the relationship between the t-total and the t-number on an assortment of different grid sizes. The grid sizes that I will use in my investigation are; 9x9, 8x8, 7x7, 6x6, and 5x5. This should allow me to develop an accurate idea about the relationship between the two numbers and find formula for them.

After completing the first stage of the investigation and collecting results I will try other ideas regarding the t-number and t-total. I shall try things such as transformations and combinations of transformations and translations. This should allow me to see if there are any patterns developing. To do this I will once again use a variety of grid sizes. My results will be set out neatly and my final formulae and answers also set out in a clear conclusion.

Finding The Formula for each Different Grid Size.

I will now find all the formulas for five separate grids. The grids I will use are 9x9, 8x8, 7x7, 6x6 and finally 5x5. I will set them out in a linear equation so that the formula for each should be easy to find. I will use consecutive t-numbers so that I have a pattern to go from.

9x9 grid.        1                2                3                4                5

T= 20                21                22                23                24

N=37                42                47                52                57

5                5                5                5

The second difference is constant so will include a 5. the formula for a 9x9 grid is Tn5-63

8x8 grid        1                2                3                4                5

T=18                19                20                21                22

N=34                39                44                49                54

5                5                5                5

The formula for this is Tn5-56

7x7 grid        1                2                3                4                5

T=16                17                18                19                20

N=31                36                41                46                51

5                5                5                5

The formula for this is Tn5-49

6x6 grid        1                2                3                4                5

T=14                15                16                17                18

N=28                33                38                43                48

5                5                5                5

The formula for this is Tn5-42

5x5 grid        1                2                3

T=12                13                14

N=25                30                35

1. 5

The formula for this is Tn5-35

These are the formulas to work out the t-totals on the grid sizes that they are written next to.

The Different Formula for each Grid

Middle

T= 20                21                22                23                24

N=37                42                47                52                57

+5                +5                +5                +5

There being a difference of +5 in the sequence means that the formula will have a 5 in it somewhere. This ‘5’ ends up being the 5 and is in the general formula for the T-total.

The 7:

Using gxg to represent any grid size we can prove where the 7 comes from by doing the following.

 n-2g-1    n-2g     n-2g+1 n-g n

Converting this into an actual t-shape off a real grid will give us the following.

For this example I will use a 9x9 grid.

 60       61           62 70 79

Using this example we can work out what the total for this equation will be.  It will be as follows:

(n-2g-1)+(n-2g)+(n-2g+1)+(n-g)+(n). This is equal to 5n-7g. To get this I added all the negative ‘g’s’ and that gave me a total of -7. This is how I got to my seven in the formula, this proves that there must be a seven in the formula for it to work correctly. You can also see that there are five lots of ‘n’ in this formula again. This five is from the general formula for the t-total. This is why there must be a five and a 7 in the general formula.

Translating the T and the effect it has on the T-total.

Now that I have found the general formula for the t-total on any grid size I must take the investigation one step further. I must now investigate the effect of translating the shape to a different point on the grid.

To do this I will use three different grid sizes, 9x9, 8x8 and finally 7x7. This will enable me to find a range of different formula for me to compare. After finding the different formula I will find the general formula for any grid size.

Conclusion

T number (41) stays the same

Difference:

12                                                                      g+3

0       10          20                                            0         g+1         2g+2

28                                                                     3g+1

If we add all this up, 0+(g+1)+(g+3)+(2g+2)+(3g+1), then it equals 7g+7

Using the vectors from the tables in the previous pages we can find the final formula. Looking at what the formula is for a translation on a 9x9 grid I should be able to find the formula.

T=5n-7g+5x-5gy. If I add the 7g+7 to this then I should have the final formula for a +90 degree rotation. The final formula is 5n-7g+5x-5gy+7g+7, this will simplify to;

5n+5x-5gy+7. I must now repeat this for -90 degrees. Then I will prove that my formulas are correct.

-90 degrees general formula for Rotation and Translation

I will now find the last formula that I will need. This will be for -90 degrees. I shall use the same procedure as I have done when I found the previous two general formulas. I will use a 9x9 grid, I will show my working out below.

 22        23            24 32 41 48         39           30 40 41

22 becomes 48- difference of 26

23 becomes 39- difference of 16

24 becomes 30- difference of 6

32 becomes 40- difference of 8

The t-number (41) stays the same.

Difference:

26                                                           3g-1

0           8            16                               0             g           2g-2

6                                                             g-3

If we add up all the differences g+(3g-1)+(2g-2)+(g-3), then we get 7g-7.

Using the vectors from the tables in the previous pages we can find the final formula. Looking at what the formula is for a translation on a 9x9 grid I should be able to find the formula.

T=5n-7g+5x-5gy. If I add the 7g-7 to this then I should have the final formula for a     -90 degree rotation. The final formula is 5n-7g+5x-5gy+7g-7, this will simplify to;

5n+5x-5gy-7.This is how I have found all of my formulas for the rotations and the translations.

So the final formulas are:

For 180 degree rotation; 5n+7g+5x-5gy

For +90 degree rotation; 5n+5x-5gy+7

For -90 degree rotation; 5n+5x-5gy-7

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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