# T-shapes. In this project we have found out many ways in which to solve the problem we have with the t-shape being in various different positions with different sizes of grids.

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Introduction

## T-Total and T-Number

PART 1

We have a grid nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the t-shape. This is highlighted in the colour red. This is shown below: -

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The number 20 at the bottom of the t-shape will be called the t-number. All the numbers highlighted will be called the t-total. In this section there is an investigation between the t-total and the t-number.

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## For this t-shape the

T-number is 20

And the

T-total is37

## For this t-shape the

T-number is 21

and the

T-total is 42

As you can see from this information is that every time the t-number goes up one the t-total goes up five.

Therefore the ratio between the t-number and the t-total is 1:5

This helps us because when we want to translate a t-shape to another position. Say we move it to here

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We all ready know the answer to the one in red. To work out the one in green all we have to do is work out the difference in the t-number and in this case it is 54. We then times the 54 by 5 because it rises 5 ever time the t- number goes up.

Middle

9

10

11

12

13

14

15

16

T-number = 10

T-total = 1+2+3+6+10= 22

7 * 4 (grid size) = 28

5tn- 28= t-total

5*10-28=22

This has proven to work on a smaller scale. We can see that by changing the grid size we have had to change the formula but still managing to keep to the rule of how you get the number to minus in the formula.

PART 3

In this next section there is change in the size of grid. Also there is transformations and combinations of transformations. The investigation of the relationship between the t-total, the t-numbers, the grid size and the transformations.

If we turned the t- shape around 180 degrees it would look like this. When we have done this we should realise if we reverse the t-shape we should have to reverse something in the formula.

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It is obvious that we will have to change the minus sign to a different sign. We should try the opposite of minus which is plus

5tn + 63=t-total

5 * 2 + 63 = 73

Check to see if the formula has worked

T-number = 2

T-total = 2+11+19+20+21 =73

The reverse in the minus sign has worked.

The next step is to move the shape on its side. Again we nearly keep the same formula as we had at the beginning. Again we change the minus number. We can work out the number to minus by working out the difference in the t-number to each number in the t-shape.

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Difference

12-1 =11

12-10= 2

12-19= -7

12-11 = 1

TOTAL = 7

Formula

5tn - 7 =t-total

5*12 - 7= 53

Check to see if the formula is right

T-number = 12

T-total = 1 +10 +19 +11 +12 = 53

This formula has worked. If we rotated the t-shape 180 degrees, The same will happen, as what happened when the t-shape was turned 180 degrees from it is first original position. This is proven below.

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5tn + 7 = t-total

5* 70 + 7 = 357

Check

T-number = 70

T-total = 70+71+72+63+81 = 357

If we were to put the t-shape diagonally on the grid we find that the same rule applies again apart from you can not use the 2nd rule were you times the grid size by seven.

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The red t-shape has t-number of 33 and the t-total = 7+17+27+25+33 = 109

The difference between the t-number and the rest of the numbers in the t-shape.

33-25= 8

33-7= 26

33-17= 16

33- 27 = 6

TOTAL= 56

5tn+56= t-total

5 * 33 - 56 =109

The reverse triangle the sign should be reversed to a plus. The t-shape used here is the one in blue.

T-number is 13

T-total = 19+29+39+21+13 = 121

5tn+56= t-total

5*13+ 56= 121

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The t-shapes above holds more formulas as the rest they all work the same.

The red t-shape has a t-number of 32 and a t-total of 32+42+52+60+44= 230

This t-shape has a formula the formula is 5tn + 70 = t-total

To see if this formula works

First we work out the difference in between the t-number and the rest of the numbers in the t-shape.

Difference

42-32= 10

52-32= 20

60-32= 28

44-32= 12

TOTAL= 70

5*32 + 70 = 230

The blue shape is the opposite of the red t-shape so therefore the formula for the blue t-shape is 5tn – 70 = t-total. The sign has become the opposite of what it use to be. This has happened in many cases before.

Now that we have worked out all the formulas for the position in the normal sized t-shape. We can try enlarging the t-shape. If we double the t-shape (volume is four times bigger). The grid below shows the new shape. I have added all the numbers together in the squares of the t-shape. This leaves us with our original t-shape but with larger numbers in the grid.

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The t-number turns out to be 176. This is the bottom four numbers added together. The t-total is 356. I have worked out the differences between the t-number and the rest of the t-shape.

Difference

176-24 = 153

176-32 = 144

176-40 = 136

176-84 = 92

TOTAL= 524

Now we have the rest of the formula. The formula is very much the same apart from the number we minus or plus by is vaster.

Formula

5tn – 524 = t-total

5*176-524 = 356

Formula has proven to work.

We have seen that there is a relationship with all the transformations made to the t-shape. Everything that we have done the t-shape has seemed to link to the part that was discovered before. These still stays the same apart from we add an extra part on to the end of the formula. This is because we are not looking for a link between all the positions of the t-shape when it is a certain way up. Here we want to find out whether there is a link between only two t-shapes. Here first of all we are looking for a link when we rotate this t-shape 90 degrees.

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Conclusion

This formula should be tested. The t-total of the blue t-shape is 37 and the t-total of the red t-shape is 208.

Formula

5tn+(12*gridsize)= t-total

5*20+ 12* 9 = 208

The formula has worked.

#### CONCLUSION

In this project we have found out many ways in which to solve the problem we have with the t-shape being in various different positions with different sizes of grids. The way we have made the calculations less difficult is by creating a main formula that changes for all the different circumstances.

Here I have put all the formulas I have come up with. These formulas only apply to the nine by nine grids

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5tn-63= t-total D

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5tn+63 = t-total U

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5tn-7= t-total R

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5tn+7= t-total L

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5tn-70= t-total DR

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5tn+70 = t-total UL

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5tn-56= t-total DL

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5tn+56 = t-total UR

The different size of grid changes means the formula has to change slightly.

This is what happened.

T-shapes | number to x by 7 |

D & U | Grid size |

L & R | nothing |

DL & UR | Grid size -1 |

DR & UL | Grid size +1 |

We also have formula for rotation, which are

angle | formula |

45 degrees | 5tn-(7xG)+7= t-total |

90 degrees | 5tn-(7xG)+70= t-total |

135 degrees | 5tn-(7xG)+133= t-total |

180 degrees | 5tn-(7xG)+126 = t-total |

225 degrees | 5tn-(7xG)+119= t-total |

270 degrees | 5tn-(7xG)+56 = t-total |

315 degrees | 5tn-(7xG)-7= t-total |

We have a formula for reflection which is 5tn+(12*gridsize)= t-total.

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

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