# Number Stairs

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Introduction

Number Stairs

1+2+3+11+12+21=50

8+9+10+18+19+28=92

48+49+50+58+59+68=332

71+72+73+81+82+91=470

15+16+17+25+26+35=161

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 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 the stairs numbers and I add all the stair number and every time I got different totals. E.g. 50, 92, 332, 470 and 161

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 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 |

71+72+73+81+82+91=470

72+73+74+82+83+92=476

73+74+75+83+84+93=482

74+75+76+84+85+94=488

I am finding out different totals, but going up by set amount.

Every time it goes up by 6.

Stair Number: 71 72 73 74

### Stair total: 470 476 482 488

Difference: 6 6 6

Now I know this pattern I can predict that the stair total for stair 75 will be 492 as 488+6.

To make sure that is right I have to test prediction.

75+76+77+85+86+95=494 my predict is right

We can therefore say that every time you move the stair shape one square to the right you increase the stair total by 6 and every time you move the stair shape one square to the left you decrease the stair total by 6.

72 + 73 + 74 + 82 + 83 + 92 = 476

72 72+1 72+2 72+ 10 72+11 72+20

1 + 2 + 10 + 11 + 22 = 44

72x6+44=476

This is the formula

Middle

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

31+32+33+41+42+51=230

32+33+34+42+43+52=236

33+34+35343+44+53=242

I am predicting the next one that start with 34 the answer is going to be 248 to find out I am right I have to work it out.

34x6+44=248

N=34

34 + 35 + 36 + 44 + 45 + 54 =248

N n+1 n+2 n+10 n+11 n+20=6n+44 change 34 to N

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 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 |

8+9+10+18+19+28=92

18+19+20+28+29+38=152

28+29+39+38+39+48=212

38+39+40+48+49+85=272

Every time it goes up by 60.

Conclusion

We can now say that in order to find the equation for an 11 by 11 number grid all we have to do is add 4 to the equation of a 10 by 10 grid. If we do this then the equation will be 6x + 48 = stair total.

I am now going to check this equation with stair 12 on an 11 by 11 number grid.

34 | ||

23 | 24 | |

12 | 13 | 14 |

6x + 48 = stair total

(6 x 12) + 48 = 120

12 + 13 + 14 + 23 + 24 + 34 = 120

This is true so I conclude that the stair total for any 3-step stair on an 11 by 11 number grid is 6x + 48= stair total.

We can now say that every time you increase the size of the grid by one (going from a 9by9 number grid to a 10by10 number grid) you have to add four more to the equation. Therefore if you decrease the size of the grid by one you must also decrease the equation by 4.

With this knowledge I can now find a general equation for any 3-step stair on any size grid.

#### Grid Size Formula

7 by 7 6x + 32

8 by 8 6x + 36

9 by 9 6x + 40

10 by 10 6x + 44

11 by 11 6x + 48

Masood Akbari 11.6HA page

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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