# T-Total. I will be looking at T-shape drawn on different size grids. I will also be comparing and finding link and formulas between T-number and T-total.

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Introduction

T-Total

In this investigation, I will be looking at T-shape drawn on different size grids. I will also be comparing and finding link and formulas between T-number and T-total. The different size grid will be: 10 by 10, 9 by 9, 8 by 8, 7 by 7, 6 by 6 and 5 by 5.

I will use grids of different sizes to try out transformation, combinations of transformations and reflection. I will also be making tables to identify formulae.

1.In this experiment I will for the formula of a T, with the T-number facing southwards. My first table is the result from a 10 by 10 grid, this on page 1.

Number | T-total | T-number |

1 | 40 | 22 |

2 | 45 | 23 |

3 | 50 | 24 |

4 | 55 | 25 |

5 | 60 | 26 |

6 | 65 | 27 |

Here is the grid formula.

n + n-10 + n-20 + n-21 + n-19 = 5n-70

and therefore formula for 10 by 10 is 5n-70.

Here is 9 by 9 grids result from page 2.

Number | T-number | T-total |

1 | 20 | 37 |

2 | 21 | 42 |

3 | 22 | 47 |

4 | 23 | 52 |

5 | 24 | 57 |

Here is the grid formula

n-19 + n-18 + n-17 + n-9 + n =5n-63

and therefore the formula for 9 by 9 grid is 5n-63.

Here is 8 by 8 grids result from page 2.

Number | T-number | T-total |

1 | 18 | 34 |

2 | 19 | 39 |

3 | 20 | 44 |

4 | 21 | 49 |

5 | 22 | 54 |

Here is the grid formula

n-17 + n-16 + n-15 + n-8 + n = 5n-56

therefore the formula is 5n-56.

Middle

T-total

1

10

43

2

11

48

3

12

53

4

13

58

5

14

63

Here is the grid formula for 7 by 7 grid.

n-9 + n-2 + n+5 + n-1 + n = 5n-7

From the results above I can predict that the formula for a 6 by 6 and 5 by 5 will be 5n-7.

I was able to find the general formula by looking at grids above.

All t-total with t-number facing eastward will have the formula 5n-7.

I found out the general grid formula which will be:

3) This time we will be looking at t-shape with t-number facing north.My first table will 10 by 10 grid from page 7.

Number | T-number | T-total |

1 | 2 | 80 |

2 | 3 | 85 |

3 | 4 | 90 |

4 | 5 | 95 |

5 | 6 | 100 |

6 | 7 | 105 |

Here is the grid formula for 10 by 10.

n + n +10 + n + 19 + n + 20 + n + 21 = 5n + 70

Here is grid 9 by 9 from page 8:

Number | T-number | ## T-total |

1 | 2 | 73 |

2 | 3 | 78 |

3 | 4 | 83 |

4 | 5 | 88 |

5 | 6 | 93 |

Here formula grid for 9 by 9 grid:

n + n + 9 + n + 17 + n + 18 + n + 19 = 5n+63

Results of grid 8 by 8 form page 8:

Number | T-number | ## T-total |

1 | 2 | 66 |

2 | 3 | 71 |

3 | 4 | 76 |

4 | 5 | 81 |

5 | 6 | 86 |

Here’s the grid formula for 8 by 8:

n + n + 8 + n + 15 + n + 16 + n + 17 = 5n+56

Here is the 7 by 7 grid from page 9:

Number | T-number | ## T-total |

1 | 2 | 59 |

2 | 3 | 64 |

3 | 4 | 69 |

4 | 5 | 74 |

5 | 6 | 79 |

The grid formula for 7 by 7 grid is:

n + n + 7 + n + 13 + n + 14 + n + 15 = 5n+49

From using information above I am able to find the general formula for any T-shapes with T-number facing. The general formula is 5n+7g (g = grid size, e.g. for 6 by 6 grid the g will be 6).

The general grid formula for this is:

4) This fourth experiments is about T-shapes with T-number facing west.

Here’s the 10 by 10 grid from page 10:

Number | T-number | ## T-total |

1 | 11 | 62 |

2 | 12 | 67 |

3 | 13 | 72 |

4 | 14 | 77 |

5 | 15 | 82 |

6 | 16 | 87 |

Here is the grid formula:

n + n + 1 + n + 2 + n – 8 + n + 2 + n + 12 = 5n+7

So the formula for 10 by 10 grid is 5n+7

Here’s 9 by 9 grid from page 11:

Number | T-number | ## T-total |

1 | 10 | 57 |

2 | 11 | 62 |

3 | 12 | 67 |

4 | 13 | 72 |

5 | 14 | 77 |

Conclusion

6) conclusion

The general formula for all sides.

The formula for T-shapes with T-number facing southwards is 5n-7g. The general formula for T-shapes with T-number facing northwards is 5n+7g. The general formula for T-shapes with T-number facing eastwards is 5n-7. The general formula T-shapes with T-number facing westwards is 5n+7. You can see from this results that when formula’s of vertically opposite T-shapes are nearly the same except it will be plus or an minus, e.g. the T-shapes with formula’s 5n-7g and 5n+7 are vertically opposite and the only difference is the plus and minus.

Look at these T-shapes I have only named two boxes the T-number and the on the opposite end, from I research I can say that whenever x is bigger than n there will be a plus sign in the formula, if n is bigger than x then there will be a minus sign in the formula. E.g. the T-shape with T-number facing southwards

The T-number is 22 and x is 2, the T-number is bigger than x and the formula is 5n-7g.

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

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