Queen/King Checkmate

There are many ways to checkmate with a queen here is one:

12. Kb6 Kc8 13. Rg8 Kd7.

You meant 12. Kc6 Ka8 13. Kb6 Kb8 14. Rg8#

Yes that's true, going into detail for checkmating with a rook, you should "waste" a move for black's king to move to the square you need it to by forcing it, by moving the rook. I'll create a forum post about it too.

the king can waste a move by moving in a triangle

but not is effective as the rook. sometimes it doesn't work, the triangle only works when the king is cornered.

If you can get the queen a "knights move" away from their king like this and copy every move - if he moves left you do, if he moves diagonally forward and right you do etc...

You'll always eventually corner the king and you can run your king in to finish off the mate. Just dont put your queen a knights move away when he goes in to the corner!

I think it's a good idea to start as you mean to go on. Once you have a method for mating look for the fastest way.

Why did I play 4.Qg7 in the first one?

And various other places. I think Black wanted to get home for his dinner. 

If you want to drive the king to the edge using just the queen, it's not so efficient, but you're less likely to stumble into stalemate. Even so there's no need to be slow about it. Below I play Qh5 instead of Qf4, which immediately deprives the black king of an extra rank. Notice also that there's no virtue in restricting the black king to two squares in the corner because there are mates anywhere at the edge.

Why does the black king leave his third rank on move 2?

Anything's eassy when you know how.

Your position is impossible for someone learning to checkmate with a queen. Your position is impossible for Syzygy (and he's pretty good).

This might be more appropriate:

(Don't worry if you get "incorrect" it probably isn't - just click on the lightbulb.)

Or that.

But an explanation shouldn't be necessary. Anyone of average intelligence can work out this mate over the board having just learned the moves.

Getting to be perfectly efficient with the rook might take longer, but is still best done under one's own steam, if only because published explanations usually aren't.

Just set up a random legal position, play it against Stockfish (which is probably perfect on this endgame, as opposed to many others) then check the number of moves it took to mate against the number of moves shown here: http://tb7.chessok.com/probe.

It should come out the same whichever side you play, if you played it right.

In fact you can bypass the tablebase check by playing both sides from each position and scoring yourself (number of moves Stockfish took to mate - number of moves you took to mate). If that comes out positive then congratulations, you outplayed Stockfish.

Tarrasch/Stockfish can be downloaded free here https://www.triplehappy.com/.

Lewis's "explanations" consist of well-selected positions and solutions. This one, for instance, has two solutions. The second is quicker. That's the sort of explanation missing from most books.

This book (https://www.amazon.com/Five-Days-Better-Chess-Essential/dp/1521987289/) offers two solutions to the rook checkmate--the slower one posted in this thread, and a faster more elegant one.

I like this one from Lewis. It is mate in five.

And this challenge from Five Days to Better Chess:

Create a position that requires four moves to checkmate with two queens and a king against a lone king. It's not easy (and yet I've seen many 1900+ who need six moves with two queens from a position that should require two).

The challenge is pretty easy so long as the lone king gets first move (and I think I have a position in the other case).

I practice a range of endgames from time to time. I have a program called Wilhelm together with the 3-4-5 man Nalimov EGTBs. I can ask Wilhelm for mates that take a particular range of numbers of moves to mate for a given endgame (e.g. I generally practice random positions that need at least 18 moves in KBNK). I then keep a running account of the the percentage of successes (mate in the number of moves according to Nalimov - this works whichever side you play).

I do practice KQK, KRK and KRRK from time to time, so I actually saw the quickest mate in your first diagram at a glance. I think this method is better than books, but for some endings (even KRK) you need to do some analysis yourself if you want to get up to 100%. If you do your own analysis it's much easier to remember than an analysis from a book.

I also think that how well you play basic endgames of this sort has nothing to do with chess talent or rating (though I seem to be a bit isolated on that point). For endgames harder than KBBK nobody really works out what to do over the board. Witness Epishin and Ushenina failing to mate in KBNK and Topolov and Karpov simulating rank beginners as soon as they got into KNNKP. It all depends on prior analysis and practice. 

White to move. Mate in four. That's the exercise.