Tracks: new parity-based deduction. This is another deduction I've known about in principle for ages but the game didn't implement. In the simplest case, it's obvious: if you can draw a line across the grid that separates the track endpoints from each other, and the track doesn't yet cross that line at all, then it's going to have to cross it at _some_ point. So when you've narrowed down to only one possible crossing place, you can fill it in as definite. IF the track already crosses your line and goes back again, the same rule still applies: _some_ part of your track is on one side of the line, and needs to get to the other. A more sensible way of expressing this is to say that the track must cross the boundary an _odd_ number of times if the two endpoints are on opposite sides of it. And conversely, if you've drawn a line across the grid that both endpoints are on the _same_ side of, then the track must cross it an _even_ number of times - every time it goes to the 'wrong' side (where the endpoints aren't), it will have to come back again eventually. But this doesn't just apply to a _line_ across the grid. You can pick any subset you like of the squares of the grid, and imagine the closed loop bounding that subset as your 'line'. (Or the _set_ of closed loops, in the most general case, because your subset doesn't _have_ to be simply connected - or even connected at all - to make this argument valid.) If your boundary is a closed loop, then both endpoints are always on the same side of that boundary - namely, the outside - and so the track has to cross the boundary an even number of times. So any time you can identify such a subset in which all but one boundary edge is already filled in, you can fill in the last one by parity. (This most general boundary-based system takes in all the previous cases as special cases. In the original case where it looks as if you need odd parity for a line across the grid separating the endpoints, what you're really doing is drawing a closed loop around one half of the grid, and considering the actual endpoint itself to be the place where the track leaves that region again - so, looked at that way, the parity is back to even.) The tricky part of implementing this is to avoid having to iterate over all subsets of the grid looking for one whose boundary has the right property. Luckily, we don't have to: a nice way to look at it is to define a graph whose vertices are grid squares, with neighbouring squares joined by a _graph_ edge if the _grid_ edge between those squares is not yet in a known state. Then we're looking for an edge of that graph which if removed would break it up into more separate components than it's already in. That is, we want a _bridge_ in the graph - which we can find all of in linear time using Tarjan's bridge-finding algorithm, conveniently implemented already in this collection in findloop.c. Having found a bridge edge of that graph, you imagine removing it, and find one of the two connected components it's just broken its previous component up into. That's your subset of grid squares, and now you can count track crossings around the boundary and fill in the bridge edge by parity. When I actually came to implement this, it turned out that the very first puzzle it generated was actually hard for me to solve, because as it turns out, this general analysis is much better at identifying opportunities to use this deduction than I am. A straight line right across the grid is often obvious: a few squares tucked into a complicated half-solved piece of the worldl, not so much. So I'm introducing a new Hard difficulty level, and putting this solution technique in there.