# Number stairs

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Introduction

Asad Rajabali Number Stairs

Number stairs

Introduction:

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 have drawn a number grid above with the stair below (ie the 25 one inserted)

45 | ||

35 | 36 | |

25 | 26 | 27 |

The above is a stair shape. This is the numbers I am going to use to analyze for future workings.

The following stair shape is a 3 step stair. The total of the numbers in the stair shape can also be expressed by a formula.

45 | ||

35 | 36 | |

25 | 26 | 27 |

The stair number 25 can be expressed as X=25

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

St(25) = 194

St(25) is the stair total for this particular stair shape. For other stair shapes for you to get the stair total then you would look for the number in the bottom left hand corner of the stair shape.

The following terms that I will use throughout my project are:

Grid: This is the grid that you are going to use to find out the numbers i.e. the original grid is with 10 by 10 numbers.

Stair Shape: This is the shape found in the grid.

Stair Number: This is the number in the bottom left hand corner of the stair shape.

Stair Total: This is total of the numbers in the stair shape.

Blocks: This is how many blocks there are in the stair shape.

Steps: This is how many steps there are in the stair shape.

Finding a formula the Stair Total

Middle

St(15)=134

In the two examples above I have used the stair number formula to check if the numbers in the stair number agrees with the stair total.

Changing the grid width

When changing the grid width then that means that the numbers you are going to pick will also change. This is because you are compressing the grid to make it smaller or larger. If changing the grid width then that will change the position of the numbers it generates in the grid.

49 | 50 | 51 | 52 | 53 | 54 |

43 | 44 | 45 | 46 | 47 | 48 |

37 | 38 | 39 | 40 | 41 | 42 |

31 | 32 | 33 | 34 | 35 | 36 |

25 | 26 | 27 | 28 | 29 | 30 |

19 | 20 | 21 | 22 | 23 | 24 |

13 | 14 | 15 | 16 | 17 | 18 |

7 | 8 | 9 | 10 | 11 | 12 |

1 | 2 | 3 | 4 | 5 | 6 |

The formula will change if changing the grid width because when expanding or compressing the grid of numbers. You are choosing different numbers each time. So this in turn will give a different formula.

X+20 | ||

X+10 | X+11 | |

X | X+1 | X+2 |

St(X) = X+X+1+X+2+X+10+X+11+X+20

= 6X+44

Finding a Stair Total Formula for any grid width

I am now going to show what will happen if I change the grid width.

I arrived at the below stair shape by noticing that every time you go up a row an extra W is added and every time going across a column an extra 1 is added.

X+2W | ||

X+W | X+W+1 | |

X | X+1 | X+2 |

St(X,W,3) = X+X+1+X+2+X+W+X+W+1+X+2W

= 6X+6+4W

St(X,W,3) = 6X+4+4W

Changing the number of steps, p

Conclusion

X+3W+3

X+3W+4

The translation is.

St(X,W,3,2,3)=6X+22W+16

There is an extra 18W+12

X+2W | ||

X+W | X+W+1 | |

X | X+1 | X+2 |

X+2W+a | ||

X+W+a | X+W+1a+1 | |

X+a | X+a+1 | X+a+2 |

The translation is.

St(x,w,3,a,0)=6X+4W+6a+4

There is an extra 6a

When translating by (a,0) each block moves across by ‘a’ units

therefore you have to add an ‘a’ to the stair shape.

As the formula is different by 6a this has appeared as you are moving each block by ‘a’ then there will be 6 blocks you are moving.

X+2W | ||

X+W | X+W+1 | |

X | X+1 | X+2 |

X+2W+b | ||

X+W+b | X+W+1b+1 | |

X+b | X+b+1 | X+b+2 |

The translation is.

St(X,W,3,b,0)=6X+4bW+4W+4

There is an extra 6bw

Every time the block moves up by 1 unit you add 1W. As you are moving up by ‘b’ units then you add bW. If you go up by ‘b’ then you are going up by bW.

X+2W | ||

X+W | X+W+1 | |

X | X+1 | X+2 |

X+bW+2w+a | ||

X+bW+w+a | X+bW+w+1+a | |

X+bw+a | X+bw+1+a | X+bw+2+a |

The translation is.

St(X,W,3,a,b)=6X+6bW+4W+6a+4

There is a difference of 6bW and 6a

The reason for the difference is that as you move up by 1 unit upwards then you have an extra bW, as you are going across by 1 unit 6 times then there is an extra 6a’s

November 2005 GCSE Mathematics

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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