# Investigation to find out the number of matchsticks on the perimeter in a matchstick staircase using the GENERAL RULE.

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Introduction

Mary Anitha Edward Antony

## Matchsticks Coursework

Introduction

This investigation is based on the ‘number sequence’ and I am going to make further more matchstick staircases for this investigation.

Investigation to find out the number of matchsticks on the perimeter in a matchstick staircase using the GENERAL RULE.

I have drawn 6 matchstick staircases on the graph paper and I am going to put the number of matchsticks on the base, number of matchsticks on the perimeter, total number of matchsticks in a table based on the 6 matchstick staircases.

Table to show the number of matchsticks on the base, on the perimeter and the total number of matchsticks.

Number of matchsticks on the base | Number of matchsticks on the perimeter | Total number of matchsticks |

1 | 4 | 4 |

2 | 8 | 10 |

3 | 12 | 18 |

4 | 16 | 28 |

5 | 20 | 40 |

6 | 24 | 54 |

And I’m going to make another table to find out the differenceonperimeter from the number of matchsticks on the perimeter.

Number of matchsticks on base | 1 | 2 | 3 | 4 | 5 | 6 |

Number of matchsticks on perimeter | 4 8 12 16 20 24 4 4 4 4 4 | |||||

Perimeter difference |

From this table I’m going to make a general rule, in terms of letters.

Number of matchsticks on perimeter = P

Number of base = b

Perimeter difference = 4 (always)

Middle

1st difference

2nd difference

When the 1st difference is not the same but the 2nd difference is, the formula will follow the quadratic pattern:

t = an2 + bn + c

But I am going to use different letters.

t = total; b = base

t = xb2 + yb + c

x is the coefficient of the 1st term and is always ½ the 2nd difference.

So in my example x is ½ of 2 = 1.

∴My formula will begin with 1b2 = b2.

To find ‘c’ I have to find the value of t, when b = 0. So I am going draw another table.

B | 0 | 1 | 2 | 3 | 4 | 5 | 6 |

t | 0 4 10 18 28 40 54 4 6 8 10 12 14 2 2 2 2 2 | ||||||

1st difference | |||||||

2nd difference |

In my table when b = 0, t = 0 so c = 0.

Now I have to find the value of ‘y’.

x + y + c = the value of t when b = 1 (from the table)

x + y + 0 = 4

1 + y + 0 = 4

∴ y = 3

So my formula should be:

b2 + 3b = t

Now I am going to test my formula:

Example 1: when b = 6; t = 54

b2 + 3b = t

62 + 3(6) = 54

36 + 18 = 54

I am going to try another example to make sure whether my formula is right or not.

Example 2: when b = 2; t = 10

b2 + 3b = t

22 + 3(2) = 10

4 + 6 = 10

Therefore I would say my formula is right and using this formula I will predict that the total number of matchsticks in the diagram when the base is 9 is:

b2 + 3b = t

92 + 3(9) = t

81 + 27 = 108.

## Introduction

I am going to do another investigation to find out the matchsticks on the perimeter and the total number of matchsticks in a double matchstick staircase. I have drawn four diagrams (see the graph attached).

## Investigation to find out the number of matchsticks on the perimeter in a double matchstick staircase using the GENERAL RULE.

Table to show the number of rows, number of matchsticks on perimeter and the total number of matchsticks.

r = number of rows; P = number of matchsticks on perimeter; t = total number of matchsticks

r | P | t |

1 | 4 | 4 |

2 | 10 | 13 |

3 | 16 | 26 |

4 | 22 | 43 |

Conclusion

t = ar2 + br + c

But I am going to use different letters.

t = total; r = rows

t = xr2 + yr + c

x is the coefficient of the 1st term and is always ½ the 2nd difference.

So in my example x is ½ of 4 = 2.

∴My formula will begin with 2r2.

To find ‘c’ I have to find the value of t, when r = 0. So I am going draw another table.

R | 0 | 1 | 2 | 3 | 4 |

t | -1 4 13 26 43 5 9 13 17 4 4 4 | ||||

1st difference | |||||

2nd difference |

In my table when r = 0, t = -1 so c = -1.

Now I have to find the value of ‘y’.

x + y + c = the value of t when r = 1 (from the table)

x + y + -1 = 4

2 + y + -1 = 4

∴ y = 3

So my formula should be:

2r2 + 3r -1 = t

Now I am going to test my formula:

Example 1: when r = 3; t = 26

2r2 + 3r -1 = t

2(3)2 + 3(3) -1 = 26

2(9) + 9 -1 = 26

18 + 9 -1 = 26

I am going to try another example to make sure whether my formula is right or not.

Example 2: when r = 4; t = 43

2r2 + 3r -1 = t

2(4)2 + 3(4) -1 = 43

2(16) + 12 -1 = 43

Therefore I would say my formula is right and using this formula I will predict that the total number of matchsticks in the diagram when the rows are 9 is:

2r2 + 3r -1 = t

2(9)2 + 3(9) -1 = t

2(81) + 27 -1 = 188.

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