# 3 by 3 step stair

Extracts from this document...

Introduction

Andrew Belcher

91 | 92 | 93 | 94 | 95 | 96 | 97 | 97 | 99 | 100 |

81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |

61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |

51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |

41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |

31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

This is a 3 by 3 step stair, this is because the step stair goes up by three squares and down by three squares. We will call this step stair ‘S-number 1’ as that is the number in the bottom left of the step stair. The step stair total (which we will call the S-total) is all the numbers within the step stair added together. The step stair total for this step stair for example is 50, as 1+2+3+11+12+21=50

I am now going to try and find an algebraic equation for finding the S-total from the S-number. I am now going to find a link between the S-number (which we will call N) and the rest of the numbers in the grid:

N+20

Middle

68

69

70

51

52

53

54

55

56

57

58

59

60

41

42

43

44

45

46

47

48

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50

31

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34

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36

37

38

39

40

21

22

23

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25

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27

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29

30

11

12

13

14

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20

1

2

3

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7

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9

10

In this example the S-total is 14, as 1+2+11=14. I am now going to find how the S-number links with the rest of the numbers in the step stair (we will call the S-number N).

N+10

N N+1

I am going to find the general rule as I did previously:

This will give me 3N+11.

I can prove this by using the example S-number 1. Using the rule the S-total comes to 14, by adding all of the numbers within the step stair it also comes to 14. Therefore the rule works.

I can prove this again by using the S-number 2.

Conclusion

N+30

N+20 N+21

N+10 N+11 N+12

N N+1 N+2 N+3

Using the previous method I can determine that the rule is

10N+110

I can prove this by using the first example. Using the rule the S-total is 120, adding all the numbers within the step stair also gives you 120. So the rule works.

I am now going to find a relationship between grid size and the S-total. I am going to use a standard 3x3 grid.

11x11 Grid

111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 |

100 | 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 |

89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 |

78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 |

67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 |

56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 |

45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 |

34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 |

23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 |

12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

I am going to find the rule like I have done in the previous exercises for several grid sizes. I am now going to draw a table of my results:

Grid size | Rule |

9x9 | 6N+41 |

10x10 | 6N+44 |

11x11 | 6N+48 |

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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