# Investigating Number Stairs

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Introduction

Maths Coursework©

– Investigating Number Stairs

## Part 1

46 | ||

36 | 37 | |

26 | 27 | 28 |

To investigate the relationship of other 3-step stairs, their stair total and their position on the grid, we can move it one space to the right. We then get the following in the 10x10grid:

26+27+28+36+37+46=200

Therefore total = 200

If we continue moving the shape to the right, and continue this for each row on the grid, we get a pattern like this:

1 | 2 | 3 | 4 | 5 |

194 | 200 | 206 | 212 | 218 |

S+20 | ||||

S+10 | S+11 | |||

S | S+1 | S+2 |

We can say that it increases in a linear fashion (the increase is constant). To use a formula to find out the stair total every time, we can make use of the bottom left had corner of the stair and change it to S. Every other number can be changed so that it is related to S

So the total for any 3-step stair on the 10x10 grid= 6S+44

Middle

S+2

S+3

S+32

S+22

S+23

S+12

S+13

S+14

S

S+1

S+2

S+3

Formula is 10S +10G + 10

Test: (10x25) + (10x10) + 10= 360, 25+26+27+28+35+36+37+45+46+55=360 So OK!

5x5 step stair:

For 10x10 For 11x11 For 12x12

S+40 | ||||

S+30 | S+31 | |||

S+20 | S+21 | S+22 | ||

S+10 | S+11 | S+12 | S+13 | |

S | S+1 | S+2 | S+3 | S+4 |

S+41 | ||||

S+31 | S+32 | |||

S+21 | S+22 | S+23 | ||

S+11 | S+12 | S+13 | S+14 | |

S | S+1 | S+2 | S+3 | S+4 |

S+42 | ||||

S+32 | S+33 | |||

S+22 | S+23 | S+24 | ||

S+12 | S+13 | S+14 | S+15 | |

S | S+1 | S+2 | S+3 | S+4 |

Formula is 15S + 20G + 15

Test: (15x25) + (20x10) +20 = 595

25+26+27+28+29+35+36+37+38+45+46+47+55+56+65=595 so OK!

6x6 step stair:

For 10x10 For 11x11

S+50 | |||||

Conclusion

1 | 4 | 10 | 20 | 35 |

3 | 6 | 10 | 15 |

3 | 4 | 5 |

1 | 1 |

With the table above, we find the differences between each number, so that we can find the formula easier (since it involves many calculations, it will be shown on the next page).

Stair Size | Stair Number (S) | Grid Number (G) |

1 | 1 | 0 |

2 | 3 | 1 |

3 | 6 | 4 |

4 | 10 | 10 |

5 | 15 | 20 |

6 | 21 | 35 |

As you can see, the Stair number and the grid number before the current grid number that you want to find.

That rule works for every other grid number. Somehow, the extra added on number seems to work this way also. We try to find a formula:

G1 = n

G2 = n + n

G3 = n + n + [n+(n+1]

G4 = n + n + [n+(n+1)] + [n+(n+1)+(n+2)]

G5 = n + n + [n+(n+1)] + [n+(n+1)+(n+2)] + [n+(n+1)+(n+2)+(n+3)]

. .

. .

. .

Gn = Gn-1 +[n+(n+1)+…+(n+(n-2)]

So therefore, the equation is:

[(S2 + S)/2] + 2[ Gn-1 +[n+(n+1)+…+(n+(n-2)] ]

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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