Question

A 1x1 square and an n-tiling is a group of n tiles placed so that every tile is contiguous along at least one side to another tile. (The tiles cannot be offset. That is, they must be placed into a grid like Scrabble tiles.) The edge length of a tiling is the total length of edges that are not touching other tiles. For instance, a 20-tiling, the edge length is 36.
Required:
Prove that for all n-tilings where n >= 1, the edge length is even.

Answer 1

Answer:

Step-by-step explanation:

The first picture below shows the missing image in the question.

There are 4 edges of each tile;

This clearly explains to us that the total number of edges present in the n tiles is 4n.

Now, if a tile "X" touches and comes in contact with another tile "Y", then edges touching each other are one edge of a tile X and one edge of a tile Y.

From the second diagram below, the edge, X_2, and Y_4 are touching each other.

Now, the total number of edges that touch some edges is always even.

We can now say that:

The edge length of filling = No. of edges that do not touch another file

= Total no. of edges  -  No. of edges that touch another edge.

However, the total number of edges is even.

The number of edges that touch another edge is also even.

Thus, the difference is also even and the number of edges that do not touch another is even.

So, the edge length of filling is always even.