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Introduction

T-total Coursework

1. To investigate the relationship between the T-total and the T-number.
2. Use grids of different sizes.  Translate the T-shape to different positions.  Investigate the relationship between the T-total and the T-number and the grid size.
3. Use grids of different sizes.  Try other transformations and combinations of translations.  Investigate relationships between the T-total, the T-number and the translations.

Relationships between T-number (x) and T-total (t) on a 9 x 9 grid.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

From the 9 by 9 grid we can see that the first T-shape highlighted in green has a T-number of 20 which is the number located at the bottom of the T-shape and the T-total (t) which is all the numbers in the T-shape added together equals 37 (20+11+1+2+3).  With the second T-shape with a T-number of 23, the T-total adds up to 52, you can see that the larger the T-number the larger the total.

If you plot all the other T-shapes and put the information into a table about the T-total and T-number you can really see a pattern and start to work out the 1st part of the formula.

 T-number (x) T-total (t) 20 37 21 42 22 47 23 52 24 57 25 62 26 67 29 82 30 87 31 92 32 97 33 102

The table proves that the bigger the T-number is bigger the T-total is larger; the T-numbers are arranged in order of size and the T-totals gradually get larger with the T-number.  From this we are able to work out some parts of the formula for a 9 by 9 grid. Taking the T-number of 20 as an example, we can say that the T-total is gained by:

t = (20 - 19) + (20 - 18) + (20 - 17) + (20 - 9) + (20 - 0) = 37

As there are 5 numbers in each T-shape, we have to use five lots of twenty.

Middle

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If we move the translation up one square you should get the result of:

t = 5x – 7g

t = (5 x 79) – (7 x 10)

t = 395 – 70.

t = 325 – 5g

t = 325 – 50

t = 275.

We can now check this by using the original formula filling it in with the data for a T-shape with a T-number of 69.

t = 5x – 7g

t = (5 x 69) – ( 7 x 10)

t = 345 – 70

t = 275

This proves that you have to use -5g for an upward translation of 1. If you wanted to do a translation of up3 you would then have a formula of p(–5g) where p= number of translations. So if you wanted to do a translation of up3 you would have to put the 3 in where the p is and then work out the rest of the formula.

The full formula looks like this.

t = (5x – 7g) – p(5g)                                p = number of translations

x = T-number

g = grid size

t = T-total

Translations (Horizontal)

We can then use the same method to work out the formula for a translation horizontally. If we start with a grid of 9 by 9.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

If we start with the T-number of 39 this gives us a T-total of

t = 5x – 7g

t = (5 x 39) – (7 x 9)

t = 195 – 63

t= 132

And then translate it horizontally by 1 to the right this gives us a T-number of 40 and a T-total of

t = 5x – 7g

t = (5 x 40) – (7 x 9)

t = 200 – 63

t= 137

As we can see there is a difference of +5 as the T-shape moves horizontally right 1 square. If we then do more translations of +1 right from the T-shape with a T-number of 40, and put all the results into a table you can get an idea of where to start a formula.

 Translation number T-number T-total Equation used t = 5x – 7g Difference by moving Right 0 39 132 (5 x 39) – (7 x 9) +5 Right1 40 137 (5 x 40) – (7 x 9) +5

Conclusion

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Again using the old method we get a T-total of the rotated shape is 92.  We have to start from scratch to make a formula so we can follow the steps used to find the original formula for the T shape.

t = (17 + 0) + (17 + 1) + (17 + 2) + (17 – 3) + (17 + 7)

So if we substitute 17 with r we get:

t = r + (r + 1) + (r + 2) + (r + 1) + (r + 4)

t = 5r + 8

t = 5r + 8

On any grid the formula 5r + 8 can be used to work out the T-total (t) where r is the T number. The formula t = 5r + 8 can be used to find the T-total (t) of any T shape that has been rotated 90 degrees clockwise where r is the T number.

Following the 180 degree flip rule we can predict that for T shapes which has been rotated by 270 degrees the formula is t = 5r – 8. As it is a flip of the T shape with a rotation of 90 degrees clockwise.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Working this out by hand, we get a T-total of 63 (7 + 12 + 13 + 14 + 17) for this T-shape.  If we use the formula, we get:

t = (5 x 14) + 8

t = 70 + 8

t = 78

The formula t = 5x + 8 can be used to find the T-total (t) of any shape rotated 90 degrees clock wise on any sized grid, where x is the T number. The formula t = 5x – 7 can be used to find the T-total (t) of any shape rotated 270 degrees clock wise on any sized grid, where x is the T-number. Therefore we can state from a standard T-shape position the following equations can be used to generate the T-total from x.

These are all of the formulas used to find any rotation.

 Rotation (degrees) Direction Ending (y) 0 Clockwise -7g 90 Clockwise +8 180 Clockwise +7g 270 Clockwise -8 0 Anti-clockwise -7g 90 Anti-clockwise -8 180 Anti-clockwise +7g 270 Anti-clockwise +8

I have also rotated and flipped the formulas so that we can have formulas for an anti clockwise rotation.

Oliver Lamb

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