# To find any relationships and patterns in the total of all the numbers in 'Number Stairs', that might occur if this stair was in a different place on the grid.

Extracts from this document...

Introduction

Number Stairs Course Work

Aim:

To find any relationships and patterns in the total of all the numbers in ‘Number Stairs’, that might occur if this stair was in a different place on the grid.

The 10 by 10 grid below is where I am to get all the information I need from to enable me to carry out this investigation.

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 |

Part One

For other 3-stepped stairs, investigate the relationship between the stair total and the position of the stair shape on the 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 | 15 | 16 | 17 | 18 | 19 | 20 |

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

By looking at this grid, the stair total is:

25 + (25+1) + (25+2) + (25+3) + (25+10) + (25+11) + (25+20)

If we call 25, the number in the corner of the stair, ‘x’ then we get:

X + (x + 1) + (x + 2) + (x + 3) + (x + 10) + (x + 11) + (x + 20) =

6x + 1 + 2 + 3 + 10 + 11 + 20 =

6x + 44

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

The stair total for this three-step stair is 194

This formula should work with every number stair that can fit onto a 10 by 10 grid. To support my judgement I will repeat the formula another two times on the same three-stair grid, but in a different position to the previous three 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 |

By looking at this grid, the stair total is:

57 + (57+1) + (57+2) + (57+3) + (57+10) + (57+11) + (57+20)

If we call 57, the number in the corner of the stair, ‘x’ then we get:

X + (x + 1) + (x + 2) + (x + 3) + (x + 10) + (x + 11)

Middle

5

6

7

8

6x

6 56 62 68 74 80 86

12

18

24

30

36

42

48

6x doesn’t give me the correct answer so I add 44 to 6x and get the Stair Total:

Stairs number (x) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Stairs total (tx) | 50 | 56 | 62 | 68 | 74 | 80 | 86 | 92 |

6x | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 |

+44 | 44 | 44 | 44 | 44 | 44 | 44 | 44 | 44 |

6x+44 | 50 | 56 | 62 | 68 | 74 | 80 | 86 | 92 |

Testing the Formula

To ensure the formula works in every case, I tested it in five situations where

‘x’ was different each time.

Using the formula 6x + 44

If n = 1 then

· 6 x 1 + 44 = 6 + 44 = 50

If x = 12 then

· 6 x 12 + 44 = 72 + 44 = 116

If x = 23 then

· 6 x 23 + 44 = 138 + 44 = 182

If x = 34 then

· 6 x 45 + 44 = 270 + 44 = 314

All the results using the formula are correct, so I can come to the conclusion that the formula for the total of a stair:

· On a 10 by 10 grid

· Which travels downwards from left to right

· With a height of 3 squares

Is: 6n + 44

In every case of a three step, the difference in position has a different total of 6.

Part Two (Extension)

Investigate further the relationship between the stair total and other step

stairs on the number grids.

For past of this investigation I am going to find out the totals for each step, starting from 2 step, to a 10step. As I found a formula for part one, I will continue to use that, in this part of this investigation, to see if there is any initial relationship. To see if there are any relationships between the number of stairs and there total, I will then put the results into a table, to make it clearer.

Two and Three Step

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 |

Two Step

‘x’ =43

X + (x + 1) + (x + 10) =

3x+11

=43+44+53=140

#### The stairs total for this two-step stair is 140

##### Three Step

‘X’=17

X + (X + 1) + (X + 2) (X + 3) (X + 10) (X + 11) + (X + 20) =

6x+44

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

#### The stairs total for this two-step stair is 146

Four and Five Step

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 |

Four Step

‘X’=33

X + (X + 1) + (X + 2) (X + 3) (X + 10) (X + 11) + (X + 12) + (X + 20) + (X + 21) + (X + 30) =

10x+110

=33+34+35+36+43+44+45+53+54+63= 440

The stairs total for this two-step stair is 440

Step Five

‘X’=56

X + (X + 1) + (X + 2) (X + 3) (X + 4) (X + 10) + (X + 11) + (X + 12) + (X + 20) + (X + 21) + (X + 22) + (X + 30) + (X + 31) + (X + 40) =

15x+220

=56+57+58+59+60+66+67+68+69+76+77+78+86+87+96= 1060

#### The stairs total for this two-step stair is 1060

Six Step

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 |

Conclusion

=1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+21+22+23+24+25+26+27+28+31+32+33+34+35+36+36+37+41+42+43+44+45+46+51+52+53+54+55+61+62+63+64+71+72+73+81+82+91=1906

The stairs total for this two-step stair is 1906

Having found out the stairs totals and the formula’s for the steps of 2-10, I am then going to put my findings into a table, as I am not able to see any instant relationship between the group of results as they are scattered all about.

Stairs Number | Formulae | Stairs Total |

2 | 3x+11 | 140 |

3 | 6x+44 | 146 |

4 | 10x+110 | 440 |

5 | 15x+220 | 1060 |

6 | 21x+385 | 1330 |

7 | 28x+616 | 952 |

8 | 36x+924 | 1752 |

9 | 45x+1320 | 1814 |

10 | 55x+1815 | 1906 |

Having interpreted the results of the formulas and stairs totals into a table, I am now able to see a relationship between the formulas for working out each step. This being that the formulae’s for each step are triangular numbers.

The first 10 triangle numbers are: 3,6,10,15,21,28,36,45,55

As you can see, they all are in the formulas.

Starting from the basic two stairs, the formula (relationship) is the first triangular number, and continues through out the table, and beyond ten stairs.

As you carry on with the number stairs, for example working out the 11th stair formula, you will notice that it will be the 11th triangular number, this will be for all of the formulas.

The table below states this:

Stairs Number | Formulae | Stairs Total |

2 | 3x+11 | 140 |

3 | 6x+44 | 146 |

4 | 10x+110 | 440 |

5 | 15x+220 | 1060 |

6 | 21x+385 | 1330 |

7 | 28x+616 | 952 |

8 | 36x+924 | 1752 |

9 | 45x+1320 | 1814 |

10 | 55x+1815 | 1906 |

11 | 66x+{} | |

12 | 78x+{} | |

13 | 91x+{} | |

14 | 105x+{} | |

15 | 120x+{} | |

16 | 136x+{} | |

17 | 153x+{} | |

18 | 171x+{} | |

19 | 190x+{} | |

20 | 210x+{} |

Nadine Okyere 11Mi February 2003

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month