# Investingating L-totals

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Introduction

Practice coursework

INVESTIGATING “L” TOTALS

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 |

An 8x8 grid is a grid that is 8 rows across by 8 rows down. The numbers go up in sequences of 1.

The “L” total is the total of all of the numbers in the L shape. The “L” shape is the shape below. Its name comes from the shape that it is.

1 | |

9 | |

17 | 18 |

=45

2 | |

10 | |

18 | 19 |

=49

3 | |

11 | |

19 | 20 |

=53

4 | |

12 | |

20 | 21 |

=57

I have found that my numbers go up in sequences of 4 if add them together

To identify where I placed my “L” on the grid I am going to use the bottom left number

E.g.

34 | |

42 | |

50 | 51 |

=177

I am going to come up with a table of results for different numbers

THESE ARE MY RESULTS FOR THE L NUMBER BEING 17 GOING UP TO 24.

L NUMBER | L TOTAL |

17 | 45 |

18 | 49 |

19 | 53 |

20 | 57 |

21 | 61 |

22 | 65 |

23 | 69 |

24 | 73 |

I have found that my totals increase in sequences of 4

I am now going to miss a few out and carry on at number 33

First off I am going to predict that number 33’s L total is 109

17 | |

25 | |

33 | 34 |

33+34+17+25 = 109

I have found that however much further up the table that you go then they still go up in 4’s. It is also always going to be an odd number.

L-16 | |

L-8 | |

L | L+1 |

L+1+L+L-8+L-16

4L-24+1

4L-23

Middle

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71

I am going to add 1+2+11+20

=This total equals 34

I am now going to add 2+3+12+21

This total equals 38

I am now going to add 3+4+13+22

This total equals 42

The formula for this is 4l+3g-1

I am now going to flip the “L” shape so that it is the other way around – this “L” shape will be coloured blue. The “L” number is the one in the top right. It is highlighted blue

I am now going to add 2+11+20+19

The total of this is 52

I am now going to add 3 +12+21+20

This total is 56

I am concluding that this is increases in sequences of four

The formula for this box is 4L+3g-1. This is the same as the sequence above. This means that their totals will be the same

I am now going to see what happens to the sequences if I increase the size of the table so that it is 9X9. This means that it will be nine squares across and 9 squares down

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 |

I am now going to add up the numbers in the “L” shape. The “L” number is the one highlighted in yellow

1+10+19+20 = 50

I am now going to add the next lot of numbers

2+11+20+21 = 54

I am now going to, again add the next sequence of numbers

3+12+21+22 = 58

I am going to make a table of results to show you what I mean

“L” number | Total |

19 | 50 |

20 | 54 |

21 | 58 |

I am concluding that this sequence also goes up in fours. This surprised me however because I would expect it to go up in fives because I have added one more column of numbers onto the table. The total is always an even number.

The algebra of this sequence is:

L-18 | |

L-9 | |

L | L+1 |

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 | 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 |

L-25 | L-24 | ||

L-16 | L-17 | ||

L-8 | L-7 | L-6 | L-5 |

L | L+1 | L+2 | L+3 |

I have worked out that this formula is:

12L-107+6

12L-101

12L-6G+6

I am going to try to work out the total of the numbers in the box

I have added 1+2+9+10+17+18+19+20+25+26+27+28

This total equals 202

I am now going to add 2+3+10+11+18+19+20+21+26+27+28+29

This total equals 214

I am now going to add 3+4+11+12+19+20+21+22+27+28+29+30

This total equals 226

I have concluded that it goes up in sequences of 12

I am now going to try to adjust the size of my “L” so that it is 5 down and 5 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 |

I am going to try to work out the total of the numbers in the box

I am going to add 1+2+9+10+17+18+25+26+27+28+29+33+34+35+36+37

The total of this is 367

I am now going to move the “L” shape along one space so now I am going to add 2+3+10+11+18+19+26+27+28+29+30+34+35+36+37+38

The total of this is 383

I am now going to move the “L” shape along one space so now I am going to add 3+4+11+12+19+20+27+28+29+30+31+35+36+37+38+39

The total of this is 399

I have concluded that the totals go up in sequences of 16

If I have an L, shape like this:

L-32 | L-31 | |||

L-24 | L-23 | |||

L-16 | L-15 | |||

L-8 | L-7 | L-6 | L-5 | L-4 |

L | L+1 | L+2 | L+3 | L+4 |

I have worked out a formula for this:

16L-168+4

16L-164

Or:

16L-8g+6

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