# Number Stairs.

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Introduction

## Number 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 |

On the 10 by 10 number grid above I have drawn a stair shape. This is known as a three-step stair.

The total of the numbers inside the stair is:

55+45+46+35+36+37=254

So the stair total for this shape is 254.

The stair number for this shape is stair 35 as the number in the bottom left hand corner is 35.

Now we can investigate the relationship between the position of the stair shape on the grid and the stair total. The diagram below is stair 1 as the number in the bottom left hand corner is 1.

21 | ||

11 | 12 | |

1 | 2 | 3 |

As we see, the stair total for this 3-step stair1 on this 10 by 10-number grid is 50 as 1 + 2 + 3 + 11 + 12 + 21 = 50.

Middle

23 | ||

13 | 14 | |

3 | 4 | 5 |

The stair total for this stair shape is 3 + 4 + 5 + 13 + 14 + 23 = 62.

Now we can find the stair total for stair4 (which is a translation of stair 3 one square to the right).

24 | ||

14 | 15 | |

4 | 5 | 6 |

The stair total for this stair shape is 4 + 5 + 6 + 14 + 15 + 24 = 68.

Now we can compare the stair totals of stairs 1, 2, 3, and 4 in order to find a pattern. If we look at their stair totals it is easy to see that as the stair number moves one square to the right the stair total increases by 6.

Stair Number: 1 2 3 4

### Stair total: 50 56 62 68

Difference: 6 6 6

Now I know this pattern I can predict that the stair total for stair 5 will be 74 as 68+6=74.

Here is the stair to prove my prediction.

25 | ||

15 | 16 | |

5 | 6 | 7 |

The stair total for this stair shape is 5 + 6 + 7 + 15 + 16 + 25 = 74

We can therefore say that every time you move the stair shape one square to the right you increase the stair total by 6 and every time you move the stair shape one square to the left you decrease the stair total by 6.

Now we are going to see what happens when we move the stair shape up one square on the grid and if there is a pattern. As we already know that the stair total for stair1 is 50, we are now going to find the stair total for stair11 (a translation of stair1 one square up).

31 | ||

21 | 22 | |

11 | 12 | 13 |

Conclusion

With this knowledge I can now find a general equation for any 3-step stair on any size grid.

#### Grid Size Formula

7 by 7 6x + 32

8 by 8 6x + 36

9 by 9 6x + 40

10 by 10 6x + 44

11 by 11 6x + 48

So from the results table I can see a linear pattern which confirms my theory. Every time you increase the grid size by 1 you increase the formula by 4.

Here is the diagram that illustrates the equation for any 3-step stair on any size grid:

g = grid size x = stair number

x+2g | ||

x+g | x+g+1 | |

x | x+1 | x+2 |

x+x+x+x+x+x = 6x

g+g+2g = 4g

1+1+2 = 4

stair total = 6x + 4g + 4

In order to prove that my formula is correct I am going to test it by using it to find the stair total for stair17 on a 10 by 10 size grid.

X=17 g=10

6x + 4g + 4 = stair total

6(17) + 4(10) + 4 = 146

17 + 18 + 19 + 27 + 28 + 37 = 146

This is correct. Therefore I conclude that the general equation for finding a 3-step stair on any sized grid is 6x + 4g + 4 = stair total.

I have 6x as there are always 6 x in any 3 step stair

I add 4g as this is the number found when we increase the grid size by one.

Finally I add 4 to these two numbers because 1 + 1 + 2 equals 4.

Also every time you move one square to the right you increase the stair total by 1 and every time you move one square down up you increase the total by g.

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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