## 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?

Problem ID: 87 (Nov 2002)     Difficulty: 2 Star

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