# Investigate the relationship between the T-total and the T-number

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Introduction

Maths Coursework

Aim: Investigate the relationship between the T-total and the T-number. Use grids of different sizes and translate the T-shape to different positions. Investigate relationships between the T-total and the T-number and the grid size.

Method:In order to find this rule we will have to go through a trial and error process. To secure an algebraic rule expressed in the nth term we will have to test it in many variables. These variables consist of random placement of the T-shapes; another is the shape being placed at different degrees. This will provide us with numerous results, which will help us, gain a conclusive overall algebraic term.

9x9 Number grid (Across)

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 |

T-Number | T-Total | Difference |

20 | 1+2+3+11+20=37 | - |

21 | 2+3+4+12+21=42 | 5 |

22 | 3+4+5+13+22=47 | 5 |

23 | 4+5+6+14+23=52 | 5 |

24 | 5+6+7+15+24=57 | 5 |

25 | 6+7+8+16+25=62 | 5 |

As the T-Shape translates across by 1, the T-Number increases by 1. The difference in T-Total is 5 each time so far. Therefore the T-Total for the T-Number 24 should be 57 if the pattern follows. The T-Total for the T-Number 25 should be 62.

The results show the predictions made earlier are correct proving that the pattern follows.

9x9 Square Grid Patterns down

T-Number | T-Total | Difference |

20 | 1+2+3+11+20=37 | - |

29 | 10+11+12+20+29= | 45 |

38 | 19+20+21+29+38= | 45 |

47 | 28+29+30+38+47= | 45 |

56 | 37+38+39+47+56=217 | 45 |

65 | 40+41+42+50+59=262 | 45 |

I have noticed patterns within my table of results.

Middle

T-Total=41+42+43+51+60=237

T-Number=60

As the other example shows, my formula is correct as the same T-Total was found for each T-Number when using both methods.

8x8 Number grid (Across)

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 |

T-Number | T-Total | Difference |

18 | 1+2+3+10+18= | - |

19 | 2+3+4+11+19= | 5 |

20 | 3+4+5+12+20= | 5 |

21 | 4+5+6+13+21= | 5 |

22 | 5+6+7+14+22=54 | 5 |

23 | 6+7+8+15+23=59 | 5 |

As the T-Shape translates across by 1, the T-Number increases by 1. The difference in T-Total is 5 each time so far. Therefore the T-Total for the T-Number 22 should be 54 if the pattern follows. The T-Total for the T-Number 23 should be 59.

The results show the predictions made earlier are correct proving that the pattern follows

8x8 Square Grid patterns down

T-Number | T-Total | Difference |

18 | 1+2+3+10+18=34 | - |

26 | 9+10+11+18+26=74 | 40 |

34 | 17+18+19+26+34=114 | 40 |

42 | 25+26+27+34+42=154 | 40 |

50 | 33+34+35+42+50=194 | 40 |

58 | 41+42+43+50+59=234 | 40 |

I have noticed patterns within my table of results. One of the patterns I have noticed is, as the T-Shape translates down by 1, the T-Number increases by 8, which is also the size of the grid. Another noticeable pattern is the T-Total difference is 40 each time.

To prove that the patterns work anywhere within the grid I am testing it at a different point within the grid. The T-numbers I will test it with is 50 and 58. If my patterns are work at any point in the grid, than the difference between both results should be 45.

Conclusion

The results above of the predictions made earlier as to what the T-Total should be if the

T-Number was 62 and 72 prove that the pattern follows anywhere within the grid.

The values inside the T of any 10x10 grid can be expressed in an algebraic way. This is as follows with n being the value of the T-Number.

Algebraic expression for a 10x10 grid:

Using the algebraic expressions a formula can be found so the

T-Total for any T-Number inside an 10x10 grid could be found straight away. The rule is:

T-Total = n+(n-10)+(n-21)+(n-20)+(n-19)

This can be simplified to:

T-total = 5n – 70

For example if n=82:

T-Total=(5x82)-70

T-Total=410-70

T-Total=340

To see if the formula is correct for this I will find the T-Total using the original method.

61 | 62 | 63 |

72 | ||

82 |

T-Total=82+72+61+62+63=340

T-Number=82

To establish that this expression is correct I will present further examples to show that it is correct

If n=92:

T-Total=(5x92)-70

T-Total=460-70

T-Total=390

71 | 72 | 73 |

82 | ||

92 |

T-Total=71+72+73+82+92=390

T-Number=92

As the other example shows, my formula is correct as the same T-Total was found for each T-Number when using both methods.

9x9 Grid Rotations:

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

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