# In this investigation I will be trying to find an equation, which will allow me to work out the sum of numbers inside a 'T' shape on a number grid.

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Introduction

In this investigation I will be trying to find an equation, which will allow me to work out the sum of numbers inside a ‘T’ shape on a number grid.

Here is an example of what I am trying to find out:

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 |

So, I would like to find out an equation that will enable me to find out the total of all the numbers in red and also which will allow me to vary the size of the grid and the T.

The first comment that I must make is that there cannot be certain T numbers, for example there cannot be a T1 because there is no number to the left of T1, neither can there be any T down the right or left sides of the grid. Also there cannot be any T numbers past 19 as there are not enough boxes below the T number for the vertical part of the T to fit into the table.

First I will try to find out an equation for a T shape of the size above and also in the same size grid:

I will first use the middle number in the top row of the T; I will call this number N.

Middle

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

Tn = 8+9+10+24+39 Using equation: (5x9)+(3x15)

= 90 45+45

= 90

This new formula that includes two variables works.

For my third variable I am going to try and modify my formula to include a different width of the top bar in the T.

Before this I had also tried rotating the T however this failed as it involved making up a completely new equation which could not be entered into the equation containing width of the grid and the T of height 2 and with a top bar of width 3.

So the first thing that I must do is to draw a new T with a wider top bar, I am going to use a T with a top bar of width 5.

1 | 2 | 3 | 4 | 5 |

6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 |

16 | 17 | 18 | 19 | 20 |

This time:

T3 = 1+2+3+4+5+8+13

T3 = 35

So:

Tn = (N-2)+(N-1)+N+(N+1)+(N+2)+(N+5)+(N+10)

Tn = 7N+15

Note that 15 is 3W

Try with grid 10:

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 |

T6 = 72

Use formula:

T6 = (7x6)+(3x10)

T6 = 72

The formula Tn = 7N+3W would appear to work for any size grid.

This formula however now needs to be inserted into the original formula.

Again I will need a new letter, from my original formula I can see using B for the number of boxes along the top row the formulae for:

3B = 5N+3W

5B = 7N+3W

Conclusion

3(3+2)+10=25

The answer 25 is too small by 20, the answer should be 45. To get this to work the 1W would have to change back to 3W. I will try this again with the next height:

3(3+3)+30= 48

The answer 48 is again too small, this time too 30, the answer should be 78. For this too work the W number would have to be increased to 6.

However I have now spotted a pattern, which is needed to make each W number work. The numbers are 1,3,6… These are the triangular numbers. This makes the height variable easy to add into my final equation:

N(B+H)+∴W

Unfortunately I cannot leave a shape in my equation, however after doing some research the equation for a triangular number is:

T(T+1)

2

So if I insert this into my final equation it will look like this:

Tn = N(B+H)+(H+1)W

2

Here is a summary of all the letters:

N= T number

Tn = Total of all numbers inside the T

B = Total number of boxes along the top row of the T

W = The width of the grid

H = Height of the T

I will now check my final formula using my original grid.

T3 = 45 Use Formula: 3(3+2)+(2+2)x10

2

15+ 30 = 45

THE OVERALL FORMULA WORKS!

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

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