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• Level: GCSE
• Subject: Maths
• Word count: 2022

# Number Stairs

Extracts from this document...

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

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88

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90

71

72

73

74

75

76

77

78

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80

61

62

63

64

65

66

67

68

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70

51

52

53

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55

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59

60

41

42

43

44

45

46

47

48

49

50

31

32

33

34

35

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37

38

39

40

21

22

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24

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28

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30

11

12

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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

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