Taylor The Roof Tiler

Probability Level 4

Taylor the roof tiler wants to know how many different perimeters can be obtained by tiling N N tiles on a grid. The tiles are squares with side length 1 meter and are placed in such a way so that each tile is joined with at least one other tile.

There are 4 different ways of tiling N = 2 N=2 tiles as shown above. The sum of the possible distinct perimeters for N = 2 N=2 tiles is 8+6=14 (meters).

What is the sum of the possible distinct perimeters for N = 7 N=7 tiles?

Photo credit: 123rf.com


The answer is 180.

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Thaddeus Abiy by Thaddeus Abiy , 25, Ethiopia

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2 solutions

Nitin Pranami
Apr 7, 2014

max. value of perimeter = 28

min. value of perimeter = 12

possible values of perimeter = 12, 14, 16 ... 24, 26 , 28

sum = 12 + ..+28 = 180

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oh wait your right

karan khosla - 7 years, 1 month ago

Plz expl how u got the possible values

Chandrachur Banerjee - 6 years, 11 months ago
Anton Shkrunin
Aug 31, 2015

To get the sum, we should first observe that perimeter is never an odd number .

To prove, we add a new tile to configurations with one, two, three and four vertically and/or horizontally adjacent tiles. Every time the perimeter is changed by an even number of tiles. Adding any new diagonally adjacent tile, too, adds an even number of sides to the perimeter. Since every tile has 4 sides, the perimeter can only be even.

Next we look for any constraints on perimeter.

First, we build the minimal perimeter from 7 tiles. It can only be a configuration with maximum number of tiles with adjacent borders: a square. But since 7 is only 2 tiles away from a square of 3x3, we can experiment with taking out 2 tiles from 3x3 for the minimal perimeter.

Enumerating across several possibilities, we land on the number 12 for minumum perimeter.

Finding maximum perimeter is easy, since tiles can be adjacent diagonally for 0 tile border adjacency. The configuration is when all tiles are lined up diagonally. Its perimeter is 7 * 4 = 28.

Final step, notice how you can build new even perimeters from 28 down to 16 by adding tiles one after another to eventually form a line.

You can then shift one tile in the minimal configuration to get perimeter 14.

Summing up from 12 to 28 you get 180.

The answer is 180.

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