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• Level: GCSE
• Subject: Maths
• Word count: 1849

# 3 step stair investigation

Extracts from this document...

Introduction

Mathematics Coursework

3 Step Stair Investigation

 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 N+20 N+10 N+11 N N+1 N+2

 82 83 84 85 86 87 88 89 90 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 17 18 1 2 3 4 5 6 7 8 9

 N+18 N+9 N+10 N N+1 N+2

 N+16 N+8 N+9 N N+1 N+2
 17 18 19 9 10 11 1 2 3

 N+14 N+7 N+8 N N+1 N+2
 15 16 17 8 9 10 1 2 3

 N+12 N+6 N+7 N N+1 N+2
 13 14 15 7 8 9 1 2 3

I can already observe a pattern therefore I should be able to write a final formula for a 3 by 3 step in any size grid so far the answers I have are:

10 by 10→ 6N + 44

9   by 9  → 6N + 40

8   by 8  → 6N + 36

7   by 7  → 6N + 32

6   by 6  → 6N + 28

 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

 N+30 N+20 N+21 N+10 N+11 N+12 N N+1 N+2 N+3

 82 83 84 85 86 87 88 89 90 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 17 18 1 2 3 4 5 6 7 8 9

 N+27 N+18 N+19 N+9 N+10 N+11 N N+1 N+2 N+3

 N+24 N+16 N+17 N+8 N+9 N+10 N N+1 N+2 N+3
 25 26 27 28 17 18 19 20 9 10 11 12 1 2 3 4

 N+21 N+14 N+15 N+7 N+8 N+9 N N+1 N+2 N+3
 22 23 24 25 15 16 17 18 8 9 10 11 1 2 3 4

There is an obvious relationship between the answers I have got for different size grids therefore I should now attempt to write a formula for a 4 by 4 step in any size grid.

10 by 10

Middle

1

2

3

4

 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

 N+40 N+30 N+31 N+20 N+21 N+22 N+10 N+11 N+12 N+13 N N+1 N+2 N+3 N+4

 82 83 84 85 86 87 88 89 90 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 17 18 1 2 3 4 5 6 7 8 9

 N+36 N+27 N+28 N+18 N+19 N+20 N+9 N+10 N+11 N+12 N N+1 N+2 N+3 N+4

 N+32 N+24 N+25 N+16 N+17 N+18 N+8 N+9 N+10 N+11 N N+1 N+2 N+3 N+4

 33 34 35 36 37 25 26 27 28 29 17 18 19 20 21 9 10 11 12 13 1 2 3 4 5

 N+28 N+21 N+22 N+14 N+15 N+16 N+7 N+8 N+9 N+10 N N+1 N+2 N+3 N+4
 29 30 31 32 33 22 23 24 25 26 15 16 17 19 20 8 9 10 11 12 1 2 3 4 5

There is an obvious relationship between the answers I have got again for different size grids therefore I should now attempt to write a formula for a 5 by 5 step in any size grid.

10 by 10 15N + 220

9   by 9   15N + 200

8   by 8   15N + 180

7   by 7   15N + 160

6   by 6   15N +?

? Should equal 140. To prove my formula I will check it works in a 6 by 6 grid.

 N+24 N+18 N+19 N+12 N+13

Conclusion

(∑Tn < X) (y+1)

This is the final formula for the second part of the investigation I already have the first so if I were to write out the complete formula it would be.

TnX+ (∑Tn < TnX)(y+1)  N = the of the shape and X = the corner number                                                       Y= size of the grid  (But there is another way)

By using the cubic formula I will be able to calculate the relationship between these numbers in red.

1 step stair = 1N   + 0(y+1)

2 step stair = 3N   + 1(y+1)

3 step stair = 6N   + 4(y+1)

4 step stair = 10N + 10 (y+1)

5 step stair = 15N + 20(y+1)

6 step stair = 21N + 35(y+1)

an3 +bn2 +cn+ d

0                        1                                4                                10

1                                3                                6

2                                3

1

a+b+c+d       8a+4b+2c+d              27a+9b+3c+d            64a+16b+4c+d

0                     1                                4                                  10

7a+3b+c                 19a+5b+c                   37a+7b+c

1                            3                                 6

12a+2b                      18a+2b

1. 3

6a

1

a= 1/6

b= 0

c= -1/6

d= 0

TnX +                                   (y+1)

I now have a final formula which hopefully works all I now need to do is test it. To begin with I will imagine I have a 3 by 3 step stair in a 10 by 10 grid. I know the answer for the formula should be 6x +44.

T3X= 6x

24

=     ----- =   4

6

6x+ 4 (10+1) = 6x + 44

If the corner number is one then the answer would be 50 which is the correct answer.

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

 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

Roy Vivasi                          Mr Clear

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