# Number Stairs

Extracts from this document...

Introduction

I have been given a number grid that counts in ascending order from one to a hundred, beginning at the bottom left hand corner to end at the top right corner with the number one hundred. With this grid I have been given the task of investigating the relationship/s between the grid size, the stair size, the stair total and the ‘n’ number.

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 | 35 | 16 | 17 | 18 | 19 | 20 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

The step shape given above is a 3-step stair, simply because it consists of 3 steps.

The stair total is labelled as being the sum of all of the numbers in the stair shape:

24 + 25 + 26 + 34 + 35 + 44 = 212

The Stair total for this 3-step shape is 212

The ‘n’ number is defined as being the smallest number of the stair shape, in the grid above it is specified as ‘ 24’.

I will systematically work my way through this problem to find appropriate algebraic solutions to simplify the workings of the stair totals. In doing so, I will use different size grids and use different size stairs. Again, investigating relationships and discovering formulae for each problem I encounter.

I am going to start with a 10 by 10 grid with a 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 | 35 | 16 | 17 | 18 | 19 | 20 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

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 | 35 | 16 | 17 | 18 | 19 | 20 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

n | T |

1 | 56 |

2 | 62 |

3 | 68 |

4 | 74 |

5 | 80 |

6 | 86 |

Middle

Again, you can see that the ‘T’ column is increasing by 6 each time the ‘n’ number is increased by 1.

The ‘T’ column increases by 6 each time, so we multiply the ‘n’ number by the increase:

- 6n, and then add the remaining amount to end up with the ‘T’ number.

- 6n + 36 = T

To check this formula I will calculate another sum to which I know the solution to:

n | T |

1 | 42 |

2 | 48 |

3 | 54 |

4 | 60 |

5 | 66 |

37 | 258 |

Using another method, I will ensure that my formula is correct and if possible simplify the expression given.

n+3G | n+3G+1 | n+3G+2 |

n+2G | n+2G+1 | n+2G+2 |

n+G | n+G+1 | n+G+2 |

n | n+1 | n+2 |

n+n+1+n+2+ n+G+n+G+1+n+2G=T

Simplifying the above equation using means of collecting up will give me:

6n+4G+4=T

For this grid, it would be: 6n+4*8+4=T

I have developed formulas for each grid (8 by 8, 9 by 9 and 10 by 10), I have come to the simple conclusion that the general formula 6n+4G+4=T applies to all grids as long as the stair number remains the same at 3 (3-step stair).

Developing further on the generic subject matter, I will find solutions on moving the ‘n’ number either along the x

Conclusion

n+5G+2

n+5G+3

n+5G+4

n+5G+5

n+5G+6

n+5G+7

n+5G+8

n+5G+9

n+4G

n+4G+1

n+4G+2

n+4G+3

n+4G+4

n+4G+5

n+4G+6

n+4G+7

n+4G+8

n+4G+9

n+3G

n+3G+1

N+3G+2

n+3G+3

n+3G+4

n+3G+5

n+3G+6

n+3G+7

n+3G+8

n+3G+9

n+2G

n+2G+1

N+2G+2

N+2G+3

n+2G+4

n+2G+5

n+2G+6

n+2G+7

n+2G+8

n+2G+9

n+G

n+G+1

N+G+2

N+G+3

n+G+4

n+G+5

n+G+6

n+G+7

n+G+8

n+G+9

n

n+1

n+2

N+3

n+4

n+5

n+6

n+7

n+8

n+9

Above I will develop a 6-step stair shape formula that will work with any grid:

N+n+1+n+2+ n+3+n+4++n+5+ n+G+ n+G+1+ n+G+2+ n+G+3+n+G+4+n+2G+ n+2G+1+n+2G+2+n+2G+3+ n+3G+n+3G+1+n+3G+2+n+4G+n+4G+1+n+5G=T

Using means of collecting up I can simplify this formula to: 21n+35G+35 = T

I will check that this formula is correct using the 10 by 10 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 | 35 | 16 | 17 | 18 | 19 | 20 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

n | T |

1 | 406 |

2 | 427 |

3 | 448 |

4 | 467 |

5 | 486 |

6 | 505 |

Here I am using the formula to see whether it works…

- 21*1+35*10+35=406

- 21*2+35*10+35=427

- 21*3+35*10+35=448

From the sums I have calculated above you can see that the formula works and is accurate as it follows my table completely.

I assume and believe that the formula will work with all problems that use a 6-step stair.

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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