This feels almost like a metaphor for teaching young people about anything... step 1:learn to play a game and the rules. Step 2: learn the implied rules for winning, and how to know when you are winning.

That's why game with victory points can be easier to teach.

It's a vague outline for anyone learning anything, really.

I still don't know how to checkmate with two bishops. I mean, I've read how to do it, but if you stuck me in an endgame situation, I'd probably draw. Luckily, I have never actually had an endgame with a king and two bishops versus a king. Same thing with king, knight, and bishop.

I have a humble level and did checkmate both with two bishops and with bishop and knight... yes, I've found both finals. I had learned the mecanics years before, didn't remember exactly how, but it wasn't so difficult to find the path with enough time.

What I don't want to find is queen vs. rook. I played once queen and pawn vs. rook and won, but it was painful. It seems there are some masters that are specialists in drawing in this positions, even if it should always be a win. Computers with end tables have not this problem.

The mate King-Bishop-Night vs. King is a bit elaborate.

From a random position on the keyboard, you typically need around 20-25 moves to achieve it, and there is a rule that draws a game after 50 moves without any pawn moves.

I've seen an International Master offer draw to his opponent in KNB + K game because he didn't remember what the winning method was (I suposo that after 5+ hours playing your eroded mindset also plays a role.

I did remember the general strategy: take the opponent's king to a corner of the same colour as the bishop. There are a couple of moves that must be thought carefully. But the rest comes very naturally, provided you've practiced it a few times.

It's always about creating a "box" where the king cannot get out of, and make the box smaller as you force the king onto the edge or corner, where you can finally get the mate move.

You have to move the king into a corner and block his way out with your king.

Which is also a good recipe for a stalemate.

The principle is the same as king and rook - the two bishops can trap the enemy king just like a rook can, and you can position your king opposite him and then advance one and force him back. You can work that one out reasonably well from first principles.

Bishop and knight is nontrivial to do in the 50 moves you have, but almost never comes up.

More interesting is checkmate with two knights which is only possible if the other king makes a mistake. You cannot force it so if two good players end up in this situation, they mostly agree on a draw.

There's absolutely nothing interesting about a position which is game-theoretically a draw and actually needs a fairly big blunder to be winnable which is why no-one even bothers in practice.

More interesting is that the position is actually winnable if the losing side has an extra piece (a pawn that isn't too advanced up to board).

For those wondering about that: the pawn allows one to move through a position that otherwise would be a stalemate.

The general trick is to block the pawn with one knight, force the king into a corner with king and other knight, and then bring in the second knight for the checkmate so fast that the pawn cannot promote to a queen and then make another move (promotion to a queen is OK if the checkmate immediately follows)

regarding the teaching thing, there's a book that wonderfully describes part of the process behind the "incremental learning", it's not a chess book btw. The book is "The Art of Learning" of Josh Waitzkin.

Really wonderful and insightful book. There's a small part about his vision in how to teach chess and in that part he also shares why he believes it's better to avoid the "learn all the rules than play" but instead to learn the rule incrementally by using actual pieces on the chess table in the most simple situations (2 or 3 pieces max). Then increase the complexity by adding pieces.

Obviously I'm going by memory so I might be a little incorrect into the explanation, but if you love the subject of learning and the subject of chess, the book is worth a look (in fact I'd advise to read the book to almost anyone).