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• Level: GCSE
• Subject: Maths
• Word count: 1703

# 3 Step Stairs

Extracts from this document...

Introduction

Part 1

3 Step Stairs

Structures

This is a 3 step stair on a 10 x 10 number grid:

 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

All 3 step stairs consist of 6 squares and its total is found by adding all the numbers inside the squares.

Total = (25+26+27+35+36+45)

Total = 194

This 3 step stair is in a certain position on the grid. This position is called P1 (see below)

This position is called P1

This position is called P2

This position is called P3

This position is called P4

Formulas

There are 4 different formulas to find the total of the 3 step stairs, each one depending on the position of the stairs.

e.g. total = number added + ((number of squares x squares right) + (10number of squares x squares up))

 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

Number of squares = 6

Squares right = 0

Squares up = 0

Abbreviations:

Total = n

Number of squares = s

Middle

This position is called P1

This position is called P2

This position is called P3

This position is called P4

Formulas

There are 4 different formulas to find the total of the 4 step stairs, each one depending on the position of the stairs.

 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

Number of squares = 10

Squares right = 0

Squares up = 0

For this position, in 4 step stairs, the formula would be:

n = 120 + ( ( s x r ) + ( 10s x u ) )

For this position, in 4 step stairs, the formula would be:

n = 220 + ( ( s x r ) + ( 10s x u ) )

For this position, in 4 step stairs, the formula would be:

n = 230 + ( ( s x r ) + ( 10s x u ) )

For this position, in 4 step stairs, the formula would be:

n = 130 + ( ( s x r ) + ( 10s x u ) )

2 Step Stairs

Conclusion

For this position, in 2 step stairs, the formula would be:

n = 24 + ( ( s x r ) + ( 10s x u ) )

For this position, in 2 step stairs, the formula would be:

n = 24 + ( ( s x r ) + ( 10s x u ) )

For this position, in 2 step stairs, the formula would be:

n = 24 + ( ( s x r ) + ( 10s x u ) )

Proof

This is a randomly selected 3 step stair

 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 can work out the total by using my formula:

n = 50 + ( ( s x r ) + ( 10s x u ) )

= 50 + ( ( 42 ) + ( 360 ) )

N = 452

Simple total = 68+69+70+78+79+88

= 452

This is a randomly selected 4 step stair

 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 can work out the total by using my formula:

n = 120 + ( ( s x r ) + ( 10s x u ) )

= 120 + ( ( 60 ) + ( 600 )

N = 780

Simple total = 67+68+69+70+77+78+79+87+88+97

= 780

This is a randomly selected 2 step stair

 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 can work out the total by using my formula:

n = 14 + ( ( s x r ) + ( 10s x u ) )

= 14 + ( ( 18 ) + ( 180 )

N = 212

Simple total = 67+68+77

= 212

Maths Investigation

Number Stairs

Alex O'Carroll 10a

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