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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 minus 1) grey tiles around outside edge.

Solving 2(n minus 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 minus 1)?
What about a room measuring m by n?

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

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