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• Level: GCSE
• Subject: Maths
• Word count: 3997

# 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
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Prediction

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

Plan

During my investigation on stairs I will:

1. 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
2. test the formula to see if it works
3. repeat the same process for vertical and diagonal 3 – step stairs on the 10 x 10 grid and find a formula for both
4. test both formulae and see if they achieve their purpose
5. 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
6. repeat for 4 and 5 step stairs
7. 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
8. change the grid size for the 3 – step stairs and see if the outcomes are the same or different
9. find a formula using the grid size and the stair number for the 3 – step stair

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 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

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