# Investigating the relationship between the total of a three-step stair on a number grid.

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Introduction

Tom Brown Number Stairs June 2003

10 D

Introduction:

For this piece of GCSE math’s coursework I have been asked to investigate the relationship between the total of a three-step stair on a number grid and it’s position on the grid. The stair will be on a 10x10 number grid like this:

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 |

I would need to find the total for stairs on the grid. For example the one shaded in red:

45+35+25+36+26+27=194

I would then tabulate this result and move the stair one square to the right so then I would get.

46+36+26+37+27+28=200

I would then continue moving it to the right until I am unable to anymore.

Part 1:

I will start of in the bottom left hand corner and work myself to the right of the grid.

Position 1

21 | ||

11 | 12 | |

1 | 2 | 3 |

The total for this stair is:

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

Position 2

22 | ||

12 | 13 | |

2 | 3 | 4 |

The total for this stair is:

22+12+2+13+3+4=56

Position 3

23 | ||

13 | 14 | |

3 | 4 | 5 |

The total for this stair is:

23+13+3+14+4+5=62

I will now tabulate these results and others.

Position Number (Lowest number in grid) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Total | 50 | 56 | 62 | 68 | 74 | 80 | 86 | 92 |

From looking at my results I can see that when I move the stair shape left by 1 column the total increases by 6 each time. I am going to try to find a formula to explain the relationship between the position and the total.

Middle

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

Position 1

The total for this stair is:

1+2+3+4+11+12+13+21+22+31=120

Position 2

The total for this stair is:

2+3+4+5+12+13+14+22+23+32=130

Position 3

The total for this stair is:

3+4+5+6+13+14+15+23+24+33=140

Position 4

The total for this stair is:

4+5+6+7+14+15+16+24+25+34=150

I will now tabulate these results and those for other 4 step stairs.

Position Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

## StairTotal | 120 | 130 | 140 | 150 | 160 | 170 | 180 |

I will see whether 6n+44 will work to find the stair total for 4 steps. I will use position 1 as an example.

n=1

=6n+44

=(6x1)+44

=6+44

=50

As you can see the formula does not apply to four step stairs. Now I will try to find a formula that does apply to four steps.

Since this time there is a difference of 10 between the totals I now times n by 10 in my equation. I will see what 10n looks like compared to the totals.

10n | 10 | 20 | 30 | 40 | 50 | 60 | 70 |

## StairTotal | 120 | 130 | 140 | 150 | 160 | 170 | 180 |

Difference | 110 | 110 | 110 | 110 | 110 | 110 | 110 |

## The same way as with the three steps 10n does not equal the total it is larger. The difference is 110 so the formula for four step stairs is 10n+110. I will now test my new formula.

61+62+63+64+71+72+73+81+82+91=720

n=61

=10n+110

=(10x61)+110

=720

You can now see that my formula for the four step stairs is correct. I will now try five step stairs.

Position 1

The total for this stair is:

1+2+3+4+5+11+12+13+14+21+22+23+31+32+41=235

Position 2

The total for this stair is:

2+3+4+5+6+12+13+14+15+22+23+24+32+33+42=250

Position 3

The total for this stair is:

3+4+5+6+7+13+14+15+16+23+24+25+33+34+43=265

Position 4

The total for this stair:

4+5+6+7+8+14+15+16+17+24+25+26+34+35+44=280

Position 5

The total for this stair:

5+6+7+8+9+15+16+17+18+25+26+27+35+36+45=295

Position 6

The total for this stair:

6+7+8+9+10+16+17+18+19+26+27+28+36+37+46=310

Position Number | 1 | 2 | 3 | 4 | 5 | 6 |

## StairTotal | 235 | 250 | 265 | 280 | 295 | 310 |

Conclusion

1

2

3

4

5

6

7

+11h3

11

88

297

704

1375

2376

3773

-11h

-11

-22

-33

-44

-55

-66

-77

Total

0

66

264

660

1320

2310

3696

Divide by 6

0

11

44

110

220

Total

1n+0

2n+11

6n+44

10n+110

15+220

So every time when 11h³-11h is divided by 6 it gives the right number in the second part of each formula. So I now have a complete general formula to find out any sized stair anywhere on a 10x10 number grid. The formula is:

n+(h²+h) + (11h³-11h)

2 6

However I will need to test this formula:

T: Total of the stair

n: Stair number (the number in the bottom left had corner of the stair)

h: The height of the stair

n=72 Three Step Stair:

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

## T=n((h²+h)/2)+(11h³-11h)/6

## T=72((3²+3)/2)+(11x3³-11x3)/6

## T=72((12)/2)+(264)/6

T=72(6)+44

T=432+44

T=476

For this one the answer is correct now I will try another sized stair

n=55 Five Step Stair:

## T=55+56+57+58+59+65+66+67+68+75+76+77+85+86+95

=1045

T=n((h²+h)/2)+(11h³-11h)/6

## T=55((5²+5)/2)+(11x5³-11x5)/6

## T=55((30)/2)+(1320)/6

T=55(15)+220

T=825+220

T=1045

Now since I have tested it twice I am sure that my formula is correct and will work for any size stair anywhere on in the 10x10 grid. Now looking back I see what else I could have done to improve my investigation. Instead of changing stair height I could have instead changed the width of the grid or the height. I could also have changed the width and length simultaneously i.e. 5x5, 6x6, 7x7.

.

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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