Cross Diagonal Cover – II

I will continue the previous post. Now the question to  be answered is that:

What percentage (or how many) of given grid can be covered by following Cross Diagonal Cover Algorithm?

I did some calculations, for example:

 

As per my calculations, I have following conjecture:

The following 3 cases are possible:

1. If , then squares will be covered.
2. If and both are odd, then squares will be covered.
3. If at least one of is even then squares will be covered.

As you can see that Case – 1 is quite trivial. I tried to prove other two cases for one week, but failed, so decided to post it as conjecture.