Cat and Mice puzzle!

Question

I was studying an Iranian olympiad book problems then I faced a very intersting puzzle, I wanted to solve this but faced lots of problems. I'd like to share this with you and know your mind.

We have divided the following figure garden into two areas with fences.We have drawn the fences on a simple and closed curve. Only three parts of the garden are known and the rest of the garden is covered with trees. Three cats are waiting for the mice. All three are in the outer area and can not cross the fence. How many mice might be trapped? (Iranan Math Olympiad).

Answer 1

Since the fence is a simple closed curve, and the cats are on the outside, we can tell which mice are on the inside by counting how fences a nearby cat has to cross to reach a particular mouse.

Every time you cross a fence, you go from the inside of the curve to the outside, or the reverse. If a cat and a mouse are separated by an odd number of fence sections, the mouse is safe; if by an even number the mouse may be caught.

By inspection, only one mouse is vulnerable.

Answer 2

Hint: The walls form a closed curve, so every time you cross a wall you go from the outside area to the inside area, or vice versa.