## Chequered Floor

#### Problem

A room measuring 5x5 is tiled with a chequered design.

If someone stood in the top left corner of the room and walked around the outside edge, they would step on 8 grey tiles in total.

If they walked around the outside edge of a larger square shaped room, with the same chequered tile floor design, and stood on 148 grey tiles in total, what are the dimensions of the room?

#### Solution

By working out the number of grey tiles on the perimeter of different sized square rooms.

Room Size | Grey Tiles(around perimeter) | |

2x2 | 2 | |

3x3 | 4 | |

4x4 | 6 | |

5x5 | 8 |

Leading to an `n` x `n` room having 2(`n` 1) grey tiles around outside edge.

Solving 2(`n` 1) = 148, we get `n` = 75.

So the room measures 75x75 tiles.

How many grey tiles in total are there on the floor of a room measuring 75x75?

Can you prove that the number of grey tiles on the perimeter of an `n` x `n` room will be 2(`n` 1)?

What about a room measuring `m` by `n`?