Can we infer the next player in chess from the current board configuration?
There is, in fact, a subcategory of chess problems, the goal of which is to determine whose turn it is to move in the given position. Examples here. So while you need the "who moved flag" in general, for all the reasons given above, it seems to me that classification of all types of positions where you don't need the flag is a genuine research problem. Another question: 

Given a mover-determinable position in chess on an n by n board, what is the complexity of determining who moved? 

So you might want to look at things like that, if you want a theoretical project out of the problem.