# Tiles investigation

Extracts from this document...

Introduction

Tim Buxton Page Mathematics Coursework

Investigation

‘Tiles’

A builder is asked to arrange some tiles on a wall in a 4 by 4 array as shown below

To help him he has variously shaped ‘spacers’ which he places between the tiles.

He uses the spacers to separate the tiles evenly as in the following arrangement.

He has used 4 spacers like this

He uses 9 spacers shaped like this

He uses 12 spacers shaped like this

He has 25 spacers altogether

Aim of Investigation:

The aim of this investigation is to investigate the number of different types of spacers required for other arrangements of tiles.

Result Table 1:

Arrangement (A) | C | S | N |

1 by 1 | 4 | 0 | 0 |

2 by 2 | 4 | 1 | 4 |

3 by 3 | 4 | 4 | 8 |

4 by 4 | 4 | 9 | 12 |

By observing the results from this table after drawing the arrangements, as shown above, I have discovered a

Middle

8 by 8

4

49

28

The drawing below is drawn to express that my formulae is correct. I must state that this formula is used so that the person will not have to draw large diagrams.

Using the results from the table, I want to see if my formulae work. I decide to use the 6 by 6 arrangement as shown below,

This is what I discovered,

Arrangement (A) | C | S | N |

6 by 6 | 4 | 25 | 20 |

By referring back to Result Table 2, I can conclude that my formulae are accurate and do work.

This can be proven below,

C = 4 = (This is correct)

S = (A – 1) 2 = 6 – 1 = 5 = 25 (This is correct)

N = 4(A – 1) = 6 – 1 = 5 × 4 = 20 (This is correct)

Knowing that my formulae is correct, the builder if given a square arrangement will know how many different spacers he will need, Below is a table to demonstrate this,

Arrangement (A) | C | S | N |

Conclusion

4

6

10

I started to notice patterns. Obviously we know that (C) = 4. However (N) is increasing by 4 every time.

I carried on drawing the rectangular tile arrangement, as shown on the rough diagrams sheet.

These are the results that I discovered

Arrangement (A) | C | S | N |

4 by 5 | 4 | 12 | 14 |

5 by 6 | 4 | 20 | 18 |

6 by 7 | 4 | 30 | 22 |

7 by 8 | 4 | 42 | 26 |

I have discovered the formula to work out S

I must state like I did earlier on in the investigation that the letter (A) represents the arrangements. However when this letter is in formulae, you must take the first number of the arrangement in order to work out.

A - A = S

However this formula only applies in certain arrangements, for example, if the first number in the arrangement is 12 it has to by 13, one added on to the first number.

Using this formula I can know how many spacers I need to fill between the tiles.

Arrangement (A) | S |

8 by 9 | 56 |

32 by 33 | 992 |

128 by 129 | 16256 |

300 by 301 | 89700 |

This student written piece of work is one of many that can be found in our GCSE Sliding Puzzle section.

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