# My investigation will be on 3 - step stairs where I will be: Trying to investigate the relationship between the stair total and the position

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Introduction

Investigation on Step Stairs

Introduction

My investigation will be on 3 – step stairs where I will be:

- Trying to investigate the relationship between the stair total and the position of the stair shape on the grid

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

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

71 | 72 | 73 | 73 | 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 |

- Further investigating the relationship between the stair totals and other step stair on other number grids e.g. 5 x 5 grid and 4 – step stairs

21 | 22 | 23 | 24 | 25 |

16 | 17 | 18 | 19 | 20 |

11 | 12 | 13 | 14 | 15 |

6 | 7 | 8 | 9 | 10 |

1 | 2 | 3 | 4 | 5 |

31 | ||||

21 | 22 | |||

11 | 12 | 13 | ||

1 | 2 | 3 | 4 |

Prediction

As my investigation unfolds I believe I will find a pattern connecting the all

Plan

During my investigation on stairs I will:

- do the horizontal 3 – step stairs and find out whether there is a formula to locate another 3 – step stair horizontally on the 10 x 10 grid
- test the formula to see if it works
- repeat the same process for vertical and diagonal 3 – step stairs on the 10 x 10 grid and find a formula for both
- test both formulae and see if they achieve their purpose
- compare the formula found for horizontal, vertical, and diagonal 3 – step stairs on the 10 x 10 grid and see if there is a pattern
- repeat for 4 and 5 step stairs
- compare the formulae for 3, 4 and 5 step stair to uncover an overall formula linking them together with the ability to find the total for any stair no matter step stair or stair number
- change the grid size for the 3 – step stairs and see if the outcomes are the same or different
- find a formula using the grid size and the stair number for the 3 – step stair

Middle

71

72

73

73

75

76

77

78

79

80

61

62

63

64

65

66

67

68

69

70

51

52

53

54

55

56

57

58

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60

41

42

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50

31

32

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40

21

22

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30

11

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1

2

3

4

5

6

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8

9

10

31 | |||

21 | 22 | ||

11 | 12 | 13 | |

1 | 2 | 3 | 4 |

32 | |||

22 | 23 | ||

12 | 13 | 14 | |

2 | 3 | 4 | 5 |

33 | |||

23 | 24 | ||

13 | 14 | 15 | |

3 | 4 | 5 | 6 |

34 | |||

24 | 25 | ||

14 | 15 | 16 | |

4 | 5 | 6 | 7 |

35 | |||

25 | 26 | ||

15 | 16 | 17 | |

5 | 6 | 7 | 8 |

Stair Number | Addition | Total | Pattern |

1 | 1 + 2 + 3 + 4 + 11 + 12 + 13 + 21 + 22 + 31 | 120 | 10*1 + 110 |

2 | 2 + 3 + 4 + 5 + 12 + 13 + 14 + 22 + 23 + 32 | 130 | 10*2 + 110 |

3 | 3 + 4 + 5 + 6 + 13 + 14 + 15 + 23 + 24 + 33 | 140 | 10*3 + 110 |

4 | 4 + 5 + 6 + 7 + 14 + 15 + 16 + 24 + 25 + 34 | 150 | 10*4 + 110 |

5 | 5 + 6 + 7 + 8 + 15 + 16 + 17 + 25 + 26 + 35 | 160 | 10*5 + 110 |

Un = 10n + 110

n = stair number

If I’m correct I believe this formula will be able to find any other 4 – step stair horizontally, vertically and diagonally on the 10 x 10 grid.

Therefore to find the total for stair number 10 I will simply use the formula, which will be U10 = 10*10 + 110 and the answer is 210.

5 – Step stairs horizontally

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

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

71 | 72 | 73 | 73 | 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 |

41 | ||||

31 | 32 | |||

21 | 22 | 23 | ||

11 | 12 | 13 | 14 | |

1 | 2 | 3 | 4 | 5 |

42 | ||||

32 | 33 | |||

22 | 23 | 24 | ||

12 | 13 | 14 | 15 | |

2 | 3 | 4 | 5 | 6 |

43 | ||||

33 | 34 | |||

23 | 24 | 25 | ||

13 | 14 | 15 | 16 | |

3 | 4 | 5 | 6 | 7 |

44 | ||||

34 | 35 | |||

24 | 25 | 26 | ||

14 | 15 | 16 | 17 | |

4 | 5 | 6 | 7 | 8 |

45 | ||||

35 | 36 | |||

25 | 26 | 27 | ||

15 | 16 | 17 | 18 | |

5 | 6 | 7 | 8 | 9 |

Stair Number | Addition | Total | Pattern |

1 | 1 + 2 + 3 + 4 + 5 + 11 + 12 + 13 + 14 + 21 + 22 + 23 + 31 + 32 + 41 | 235 | 15*1 + 220 |

2 | 2 + 3 + 4 + 5 + 6 + 12 + 13 + 14 + 15 + 22 + 23 + 24 + 32 + 33 + 42 | 250 | 15*2 + 220 |

3 | 3 + 4 + 5 + 6 + 7 + 13 + 14 + 15 + 17 + 23 + 24 + 25 + 33 + 34 + 43 | 265 | 15*3 + 220 |

4 | 4 + 5 + 6 + 7 + 8 + 14 + 15 + 16 + 17 + 24 + 25 + 26 + 34 + 35 + 44 | 280 | 15*4 + 220 |

5 | 5 + 6 + 7 + 8 + 9 + 15 + 16 + 17 + 18 + 25 + 26 + 27 + 35 + 36 + 45 | 295 | 15*5 + 220 |

Un = 15n + 220

n = stair number

If I’m correct I believe this formula will be able to find any other 5 – step stair horizontally, vertically and diagonally on the 10 x 10 grid.

Therefore to find the total for stair number 10 I will simply use the formula, which will be U10 = 15*10 + 220 and the answer is 370.

Step Stair | Formulae | Pattern |

3 | 6n + 44 | [3(3+1)/2]n +[(3-1)*3*(3+1)/6]*11 |

4 | 10n + 110 | [4(4+1)/2]n + [(4-1)*4*(4+1)/6]*11 |

5 | 15n + 220 | [5(5+1)/2]n + [(4-1)*4*(4+1)/6]*11 |

Un = [s(s+1)/2]n + [(s-1)*s*s(+1)/6]*11

If I’m correct I believe this formula will be able to find any other step stair horizontally, vertically and diagonally on the 10 x 10 grid.

Therefore to find the total for stair number 10 for a 10 – step stair I will use the formula, which will be U10 = [10(10+1)/2]n + [(10-1)*10*(10+1)/6]*11 and the answer is 715.

Grid size for 3 – step stairs

Using g as the grid size I will see whether there is a formula that links the 3 – step stairs together with the size of the grid.

9 x 9 grid

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 |

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

37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 |

28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 |

19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 |

10 | 11 | 12 | 13 | 14 | 15 | 16 | 16 | 18 |

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

19 | ||

10 | 11 | |

1 | 2 | 3 |

20 | ||

11 | 12 | |

2 | 3 | 4 |

21 | ||

12 | 13 | |

3 | 4 | 5 |

Stair Number | Addition | Total | Pattern |

1 | 1 + 2 + 3 + 10 + 11 + 19 | 46 | 6*1 + 40 |

2 | 2 + 3 + 4 + 11 + 12 + 20 | 52 | 6*2 + 40 |

3 | 3 + 4 + 5 + 12 + 13 + 21 | 58 | 6*3 + 40 |

The formula for the 9 x 9 grid for 3 – step stairs is Un = 6n + 40

n = stair number

Therefore to find the total for stair number 9 I will simply use the formula, which will be U9 = 6*9 + 40 and the answer is 94.

8 x 8 grid

57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 |

49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 |

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

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

25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |

17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

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

17 | ||

9 | 10 | |

1 | 2 | 3 |

18 | ||

10 | 11 | |

2 | 3 | 4 |

19 | ||

11 | 12 | |

3 | 4 | 5 |

Stair Number | Addition | Total | Pattern |

1 | 1 + 2 + 3 + 9 + 10 + 17 | 42 | 6*1 + 36 |

2 | 2 + 3 + 4 + 10 + 11 + 18 | 48 | 6*2 + 36 |

3 | 3 + 4 + 5 + 11 + 12 + 19 | 54 | 6*3 + 36 |

The formula for the 8 x 8 grid for 3 – step stairs is Un = 6n + 36

n = stair number

Therefore to find the total for stair number 8 I will simply use the formula which will be U8 = 6*8 + 36 and the answer is 84.

I will now compare the 8 x 8 grid to the 9 x 9 and 10 x 10 to see whether there is a formula connecting them with the 3 – step stairs.

Grid size | Formula | Pattern |

8 x 8 | 6n + 36 | 6n + 4*8 + 4 |

9 x 9 | 6n + 40 | 6n + 4*9 + 4 |

10 x 10 | 6n + 44 | 6n + 4*10 + 4 |

The overall formula for the grid size and the 3 step stairs horizontally, vertically and diagonally is Un = 6n + 4g + 4

n = stair number

g = grid number

Therefore to find the total for stair number 5 on a 5 x 5 grid I will simply use the formula U5 = 6*5 + 4*5 + 4 and the answer is 54.

From the comparison between the grid size and position of 3 - step stair, I have found a new and easier formula to use using the 3 – step stair shape which when added makes the same formula as 6n + 4g + 4.

n+2g | ||

n+g | n+g+1 | |

n | n+1 | n+2 |

Conclusion

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 |

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

37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 |

28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 |

19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 |

10 | 11 | 12 | 13 | 14 | 15 | 16 | 16 | 18 |

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

37 | ||||

28 | 29 | |||

19 | 20 | 21 | ||

10 | 11 | 12 | 13 | |

1 | 2 | 3 | 4 | 5 |

38 | ||||

29 | 30 | |||

20 | 21 | 22 | ||

11 | 12 | 13 | 14 | |

2 | 3 | 4 | 5 | 6 |

39 | ||||

30 | 31 | |||

21 | 22 | 23 | ||

12 | 13 | 14 | 15 | |

3 | 4 | 5 | 6 | 7 |

Stair Number | Addition | Total | Pattern |

1 | 1 + 2 + 3 + 4 + 5 + 10 + 11 + 12 + 13 + 19 + 20 + 21 + 28 + 29 + 37 | 215 | 15*1 + 200 |

2 | 2 + 3 + 4 + 5 + 6 + 11 + 12 + 13 + 14 + 20 + 21 + 22 + 29 + 30 +38 | 230 | 15*2 + 200 |

3 | 3 + 4 + 5 + 6 + 7 + 12 + 13 + 14 + 15 + 21 + 22 + 23 + 30 + 31 + 39 | 245 | 15*3 + 200 |

The formula for the 9 x 9 grid for 5 – step stairs is Un = 15n + 200

n = stair number

Therefore to find the total for stair number 9 I will simply use the formula, which will be U9 = 15*9 + 200 and the answer is 335.

8 x 8 grid

57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 |

49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 |

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

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

25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |

17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

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

33 | ||||

25 | 26 | |||

17 | 18 | 19 | ||

9 | 10 | 11 | 12 | |

1 | 2 | 3 | 4 | 5 |

34 | ||||

26 | 27 | |||

18 | 19 | 20 | ||

10 | 11 | 12 | 13 | |

2 | 3 | 4 | 5 | 6 |

35 | ||||

27 | 28 | |||

19 | 20 | 21 | ||

11 | 12 | 13 | 14 | |

3 | 4 | 5 | 6 | 7 |

Stair Number | Addition | Total | Pattern |

1 | 1 + 2 + 3 + 4 + 5 + 9 + 10 + 11 + 12 + 17 + 18 + 19 + 25 + 26 + 33 | 195 | 15*1 + 180 |

2 | 2 + 3 + 4 + 5 + 6 + 10 + 11 + 12 + 13 + 18 + 19 + 20 + 26 + 27 + 34 | 210 | 15*2 + 180 |

3 | 3 + 4 + 5 + 6 + 7 + 11 + 12 + 13 + 14 + 19 + 20 + 21 + 27 + 28 + 35 | 225 | 15*3 + 180 |

The formula for the 8 x 8 grid for 5 – step stairs is Un = 15n + 180

n = stair number

Therefore to find the total for stair number 8 I will simply use the formula, which will be U8 = 15*8 + 180 and the answer is 300.

Grid size | Formula | Pattern |

8 x 8 | 15n + 180 | 15n + 20*8 + 20 |

9 x 9 | 15n + 200 | 15n + 20*9 + 20 |

10 x 10 | 15n + 220 | 15n + 20*10 + 20 |

The overall formula for the grid size and the 5 - step stairs horizontally, vertically and diagonally is Un = 15n + 20g + 20

n = stair number

g = grid number

Therefore to find the total for stair number 7 on a 7 x 7 grid I will simply use the formula U7 = 15*7 + 20*7 + 20 and the answer is 265.

Step stair | Formulae | Pattern |

3 | 6n + 4g + 4 | [3(3+1)/2]n + [(3- 1)*3*(3+1)/6]g + [(3-1)*3*(3+1)/6] |

4 | 10n + 10g +10 | [4(4+1)/2]n + [(4-1)*4*(4+1)/6]g + [(4-1)*4*(4+1)/6] |

5 | 15n + 20g + 20 | [5(5+1)/2]n + [(5-1)*5*(5+1)/6]g + [(5-1)*5*(5+1)/6] |

Un = [s(s+1)/2]n + [(s-1)*s*(s+1)/6]g + [(s-1)*s*(s+1)/6]

Pascal's Triangle

1 | ||||||||||||||||

1 | 1 | |||||||||||||||

1 | 2 | 1 | ||||||||||||||

1 | 3 | 3 | 1 | |||||||||||||

1 | 4 | 6 | 4 | 1 | ||||||||||||

1 | 5 | 10 | 10 | 5 | 1 | |||||||||||

1 | 6 | 15 | 20 | 15 | 6 | 1 | ||||||||||

1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 | |||||||||

1 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 |

Triangular numbers – 1 , 3 , 6 , 10 , 21 , 28 ……

Un = n*(n+1)/2

Tetrahedral numbers – 1 , 4 , 10 , 15 , 35 , 56 ……

Un = n*(n+1)*(n+2)/6

Hyper tetrahedral numbers – 1 , 5 , 15 , 35 , 70 ……

Un = n*(n+1)*(n+2)*(n+3)/24

I have realised that our formula involves triangular and tetrahedral numbers along with hyper tetrahedral number, which can be found on Parcel’s triangle. Their relationship with one another is triangle number have a difference of square numbers, tetrahedral number have a difference of triangle numbers and hyper tetrahedral number have a difference of tetrahedral numbers.

Derrick Gachiri

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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