Do not be alarmed, this is only a test.

If I can persuade Calum to delete his answer (which I think can't be a bad idea because then the question will appear under the "unanswered" questions) then you might have to re-advertise your chat room.

He actually posted the link a moment ago. I apparently just missed the definite article "The" in the description, and assumed it was part of the Question title! I'll look for that 2x2 board you mentioned, I only started paying attention to the number of moves a few boards in.

The reason I find it hard to model this game is that at first glance you think "aah, a*b switches, and they commute, so there's some (Z/2Z)^{ab} living in some group and we just have to figure out what the question is". But I've somehow never found the (Z/2Z)^{ab}. Certainly the symmetric group on a+b symbols does not contain a (Z/2Z)^{ab} in general (not even for a=2 and b=1).

I think what's a little misleading is that the permutations don't, in general, permute - but the setup forces them to. For example, using your example, you posted a group of permutations, and only some of them can be arranged in either order (the first pair on separate "strands" of the group of 2 can be rearranged, but must be performed before any permutations "lower" on the strand

Rather, I shouldn't say forces them to permute, but forces them to be performed in a certain order; there's a partial order on the grid (A, B), and "smaller" permutations need to be performed later.

So, using the image as an example: Yes, we can add the switches in any order. But they'll still be performed based on that hierarchy. The "game states" seem to be parameterized by $2^{|A| \times |B|}$, the set of switches we've toggled.

In fact the set of game states can be thought of as a subset of 2^{ab} but the actual output, the stuff we're interested in, is typically of much smaller size (unless a or b is 1). What is confusing is that one switch can give more than one permutation of the output depending on what state other switches are in.

In the 2x2 case there are 24 output permutations, 16 game states, and 14 accessible outputs: if both the left-most and the right-most switches are switched then the output depends only on the sum of the top and bottom switch mod 2.

Hmm, I see. I'll have to think about that more; I've only started by figuring out how I want to formalize things, I haven't played around with small games yet. I will definitely have to, clearly!