Let's suppose you have a rectangular grid of cells and let's number the rows with numbers from 1 to n and columns with numbers from 1 to m. Now consider a closed path which begins from $(1, 1)$, ends in $(1, 1)$ and moves between two cells which share a side. A turn is a place where the path changes it's direction. For example if we have cells in the path $(1,1), (1,2), (1, 3), (2, 3)$ in this order then a turn occurs in $(1,3)$. It is known that each turn happens in a cell with odd coordinates. I need to prove that the number of enclosed cells(those that inside the area enclosed by the path, but not of it) is odd.
My attempt: Let's first prove it for a path with the shape of a rectangle. Because it's four corners have odd coordinates, then it's width and length are also odd. Let the width be $s$, and height be $t$. Then inside there are $(s - 2) * (t - 2)$ rectangles, this is an odd number. Now consider if you have a shape formed from a path, for which you know that there are odd number of cells inside, each turn is in odd coordinates, and you try to "combine" it with a rectangular group of cells, so that they "stick" together and form a new path of the same type. It's not hard to see that the new figure will also have an odd number of cells inside it. So I thought that I must find a way to prove that each shape obtained from a path with only odd-coordinate turns can be obtained by "merging" a smaller shape with a rectangle and then use something like induction. The problem is, I'm not sure if that's true. Or maybe there is a better approach, which I'll be happy to learn. Can anyone help me with the problem?