Arranging bricks safely

Question

The goal is to arrange bricks safely so that there is no line that can be cut through.

For example

This arrangement is considered unsafe

because you can cut through the red line.

This is a safe arrangement because you cannot cut through any line.

So the question is, What is the condition for the length and the width so that there is at least one possible way to arrange bricks safely?

Obviously, one of the length and the width should be even number, for obvious reason.

I also figured out that the length and width should be bigger than 3

because it will be one of these cases, which already creates line.

Also $ 1\times{2} $, which is just a single brick will also be one of the possible way.

I also know that $ 6\times{11} $ is possible.

So what would the general condition be?

Answer 1

These are called fault-free rectangles. Mathworld has a short article that gives the conditions you seek. For $1 \times 2$ tiles you can tile a $p \times q$ rectangle if

• At least one of $p$ and $q$ is even
• $x +2y=p$ and $x+2y=q$ each have two distinct solutions in positive integers
• $p$ and $q$ are not both $6$