# Number Stairs

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Introduction

Year 11 Algebra coursework – Number Stairs Laura Bird

I am going to investigate into the stair total of different sized stair shapes on different sized number grids, the stair total being the total of the numbers inside the stair shape. I will then see if I can produce an algebraic formula to calculate this total for any sized stair shape on any sized grid. I shall start by finding a formula for the total of numbers in a 3-step stair on a 10×10 grid (for example see fig. 1), based on one of the numbers inside the stair shape.

Fig. 1

This is a 3-step stair. The stair total for this stair shape would be:

25 + 26 + 27 + 35 + 36 + 45 = 194

I started simple to form some simple equations to calculate the stair total.

Eg 1. For a 3-step stair on a 10×10 grid I came up with the following formula to calculate stair total, based on one of the numbers in the stair shape:

The formula for finding the stair total for a 3-step stair on a 10×10 grid would be:

6x + 44 x being the number in the bottom left of the stair.

I then calculated similar formulas for different sized steps (see Egs) on different sized grids and displayed them in the Table 1 to see if I could notice any patterns:

Egs:

3-step stair on a 9×9 grid

6x + 40

Middle

3-step stair

3-step stair = 6x + 4g +4

I then went on to calculate more of these formulas for other size step stairs on any size grids. I showed these formulas in Table 2:

Table 2

Step size | Formula - any size grid |

1 | x |

2 | 3x + 1g + 1 |

3 | 6x + 4g + 4 |

4 | 10x + 10g +10 |

5 | 15x + 20g + 20 |

From Table 1 and Table 2 I can see that the co-efficient of x is always the same for each step size no matter what size grid it is on, and that it is only the end part of the formula (the + something bit) that changes on each size grid. The co-efficient of x is always a triangular number as the stair shapes make a triangle shape (Fig. 2), therefore the formula for finding the coefficient of x will be the same as the formula for finding triangular numbers: n(n+1) n being the step size.

2

Fig. 2

For example for a 2-step stair the formula is always 3x plus something, therefore to find the co-efficient of x, in this case 3, I would substitute the step size, in this case 2, into the triangular formula:

2(2+1) = 4+2 = 6 = 3

2 2 2

Therefore this equation works for a 2-step stair as the end result was 3. I then went on to check if this formula worked with other sized stairs.

Conclusion

Step size (n) | 1 | 2 | 3 | 4 | 5 |

y | 0 | 10 | 40 | 100 | 200 |

1st difference +10 +30 +60 +100

2nd difference +20 +30 +40

3rd difference +10+10

This was the case, therefore the 3rd difference, which is also the number divided by 6 in my equation is always grid size plus 1, therefore in my equation would be:

[ n(n+1) ] x + [ ( g + 1 ) n³ - ( g + 1 )n ]

[ 2 ] [ ( 6 ) ( 6 ) ]

Where n = step size, x = number in the bottom corner of the stair and g = grid size

I then simplified this equation:

[ n(n+1) ] x + [ g + 1( n³ - n ) ]

[ 2 ] [ 6 ]

This is the formula for any sized step stair on any size grid.

I then went on to try this formula on a different size step stair on a different size grid to prove and check that it works. The example I used was a 3-step stair on a 10×10 grid with the number in the bottom left of the step shape being 25. The stair total for this step stair should equal 194. I substituted my example into my formula to calculate the stair total to see if this answer was given:

[ n(n+1) ] x + [ g + 1( n³ - n ) ] = Stair Total

[ 2 ] [ 6 ]

[ 3(3+1) ] 25 + [ 10 + 1( 3³ - 3 ) ] = Stair Total

[ 2 ] [ 6 ]

[ 9+3 ] 25 + [ 11 (27 - 3 ) ] = Stair Total

[ 2 ] [ 6 ]

[ 12 ] 25 + [ ( 11 )24 ] = Stair Total

[ 2 ] [ ( 6 ) ]

150 + 44 = Stair Total

194 = Stair Total

This proves that my formula works as it produced the correct stair total for a 3-step stair on a 10×10 grid.

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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