# Maths T-Shapes Investigation

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Introduction

Anand Patel Mathematics Coursework Page of

## QUESTION 1

A T-shape is drawn onto grid like the one below (fig.1.1). The grid can be of any size, e.g. 9x9, 4x4, 100x100 etc. In question 1 the only grid size that will be used will be the 9x9 size.

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 |

(fig.1.1)

The T-shape has a T-number, this the number that is at the bottom of the T-shape. The T-number of the T-shape in fig.1 is 18.

The T-total of the T-shape is all of the numbers the total of all of the numbers in the T-shape. The T-total of T-shape in fig.1.1 is:

7+8+9+13+18= 55

I will now investigate the relationship between the T-total and the T-number.

The coloured section in Fig.1.2 shows all of the possible positions that the T-number could be located in.

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 |

(Fig.1.2)

I noticed that if the number at the bottom of the T-shape is T

Middle

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(Fig.1.5)

Fig.1.6 shows the T-totals worked out by adding all of the numbers in the T-shape together.

1+2+3+11+20= 37

13+14+15+23+32= 97

7+8+9+17+26= 67

48+49+50+58+67= 272

52+53+54+62+71= 292

(Fig.1.6)

Fig.1.7 the T-totals of the T-shapes in Fig.1.5 worked out using the general formula.

5T- 63 | = T-TOTAL |

(5 X 20) – 63 | = 37 |

(5 X 32) - 63 | = 97 |

(5 X 26) - 63 | = 67 |

(5 X 67) - 63 | = 272 |

(5 X 71) - 63 | = 292 |

(Fig.1.7)

Comparing figures 1.6 and 1.7, I noticed that the T-totals are the same. This proves that my general formula is correct.

QUESTION 2

In Fig.2.1 I have positioned the T-number in different places on the grid. The table in Fig.2.2 shows the T-numbers from Fig.2.1 and their T-totals. I aim to find a connection between them.

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 |

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 |

(Fig.2.4)

Fig. 2.4 shows the grid of 5x5 and I have randomly picked two different T-shapes from the grid and highlighted them. The T-numbers are 12 and 18 and in Fig.2.5 I will test my formula with these two T-shapes.

1+2+3+7+12 = 25

5x12 – 5x7 = 25

7+8+9+13+18 = 55

5x18 – 5x7 = 55

(Fig.2.5)

Fig.2.5 shows that my formula works for the 5x5 grid. I can now change the number that represents the grid size and change it to just one letter, G, which can be used for all grid sizes. Therefore one can find the T-total of any ant T-shape that is in any grid size using the following formula:

T= 5t-7n

I will now prove that my formula works for a few more grid sizes before confirming that it is definately correct. Fig.2.6 shows these tests:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

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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 |

1 | 2 | 3 |

4 | 5 | 6 |

7 | 8 | 9 |

1 | 2 | 3 | 4 | 5 | 6 | 7 |

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43 | 44 | 45 | 46 | 47 | 48 | 49 |

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