"Parity" of "insuficient mating material" What is sufficient mating material?

I found this question and must pose my own as a response.

I you have a chess computer on easy (or the Kasparov Team Mate - anyone who has played one will know the "why did it do that?" -> "Oh, if we don't trade knights, then bishops, then our other bishops and a rook, I'll lose!" it loves to swap and the above has actually happened) you will probably end up with most of the pieces off the board.

Sometimes you can see a checkmate, but you can never get to it because of the square the king is currently on. BUT you may only checkmate if black skips a turn. 

The algebra of this behavior coupled with intuition suggests that if you think of a move as a 2 element tuple (white's move followed by black) that these are from groups with an LCM of their orders of 2. Which is why no number of moves will give a checkmate, thus meeting the 50 move draw at most.

My question is this, what is the algebra behind this? I'm pretty sure it deals with cosets of orbits of board layouts. My Algebra isn't strong enough to analyse this but there is SOMETHING at play.

Secondly, imagine you could "bump" the board, so your knight shifted without anyone noticing one square horizontally. Now it is on a different colour, this should adjust the parity (illegally, albeit) and now you should be able to mate. 

Now this raises the question of "if this happens a lot, play to protect the other knight, as it starts on a different colour (thus is the opposite parity), then you should be able to win)" This would make which knight you have left the deciding tool between stalemate (accidentally making it so they cannot move their king (with nothing else left) but not checking him) and checkmate. 

Chess is a hobby of mine, I am quite new to the established realm of it, I sort of learned myself from electronic boards (which I still do, I have some strong ones! (Kasparov Turbo s24k, the one after this famously played Kasparov, a GK2000, a GK2200, a Kasparov Cougar (ELO estimated by enthusiasts to be 2100, manufacturer claimed 2250 IIRC) -I say this to try and show I'm not a noob).

I hope somebody can parse what I'm trying to say, I do apologise for the poor quality of this question. I'll try and play an example tonight and edit it in if no one can understand this.
It is a little unclear exactly what you are after, but the following portion of the question points to an area where something directly mathematical is going on:


  Sometimes you can see a checkmate, but you can never get to it because of the square the king is currently on. BUT you may only checkmate if black skips a turn.


This describes a situation known as mutual zugzwang (or, reciprocal zugzwang), which is where we have a position in which it is detrimental for either side to have a move. A prototypical example comes from K+P vs. K.

[fen "3k4/8/3KP3/8/8/8/8/8 w - - 0 1"]

1.e7+ Ke8 2.Ke6


White has no better than to allow this stalemate, whereas if Black had the move, White wins by 1...Ke8 2.e7 Kf7 3.Kd7 and queens.

Much more complicated examples exist, of course. In the following position, for instance, Black to move will lose, while White to move has no better than a draw:

[fen "8/q7/5q2/8/8/8/6QQ/k6K w - - 0 1"]


The above example was discovered by computer, actually. It is mentioned in Lewis Stillers' "Multilinear algebra and chess endgames" (PDF link), and that article gives a fair amount of background on the group-theoretic considerations involved. For further reading on the matter, you might try to get your hands on "Chess art in the computer age," American Chess Journal, 1(2):48--52, September 1993. I haven't seen the text myself, but this piece is referenced in Stillers' article, and it is by Noam Elkies, who is a world-class mathematician and a chess master (not to mention the 1996 World Chess Solving Champion). I know for sure that the article at least touches on the topic at hand, because it includes a composed study that is based on the mutual zugzwang above:

[fen "5Q2/5P1b/8/7K/8/1q4k1/1p4B1/8 w - - 0 1"]
[White "Elkies"]
[Black "Study"]


White to play and win (solution indicated in Stillers' article).