Sorry, but I absolutely disagree that you need to memorize move order in theoretical positions. You only need to memorize the ideas.
Once you know the ideas, you can calculate two, three, or more moves to attain the ideal position, because those calculations are based on not making the wrong move.
Outflanking, Counting, and the Opposition
8/5K2/8/4pk2/4R3/8/8/8 w - - 0 1
I have not memorized a single move in how the winning variation ends, but one thing's for certain:
- White needs to get on the opposite side of the king if he's going to capture that pawn.
- Black's king is like a bodyguard, preventing the white king from getting past him.
- Black will control the queening square in seven moves. Black moves after me, so I have seven moves to capture the pawn, or the game will be drawn.
These ideas are known as outflanking, counting, and the opposition. Unless you have studied endgames, you won't have a clue about them (see point #3 further below).
Now I can ask myself which moves accomplish these ideas. After 1. Ra4?? or (1. Rh4??) 1...e4! The black king remains where he's at, and he better: He's got the opposition.
- Re1?? Yep, this doesn't do it. The black king also keeps the opposition: 1...e4! 2. Rf1 Kg4 and the white king is late for his appointment. We can try to outflank: 2. Ke7 Ke5!! Opposition.
The only way to force the king to lose the opposition is to entice him forward: 1. Re3!! (or 1. Re2!) The black king will lose the opposition, despite attacking the rook. If he tries 1. Kf4 2. Re1 Kf5 (after 1. Re3!!), you'll notice that we're back to the original starting position, but with the rook on e1 now. With that extra tempo, we can now outflank: 3. Ke7 e4 4. Kd6 Kf4 5. Kd5 (The pawn is now under attack) e3 4. Kd4 and the pawn has been won.
Black's only option is to give up the opposition, unfortunately. 1. Re3!! (or 1. Re2!) e4 2. Re1 Kf4 (2... Ke5 3. Kg6 leads to the above variation with kings on opposite sides) 3. Ke6! Kf3 4. Kd5 (The pawn is now under attack) e3 5. Kd4 e2 6. Kd3 We have done it! We have prevented black from controlling e1 in seven moves or less.
And notice a key difference between the variations where the black king was on f4 and g4: The rook was on f1 in the latter. White loses a move when the black pawn hits e2, because his rook is attacked. So in reality, white only had not six, but FIVE! (because of Re1-Rf1+ and Rf1-Re1) instead of seven moves in that losing (drawing) variation.
King Centralization, Counting
7K/7p/k1P5/8/8/8/8/8 w - - 0 1
Well, clearly it's a draw, just capture the black pawn, get some lunch at Subway, and go on with my day...except that's not the case. Black to move.
Now this is interesting. Most amateurs would say it's lost despite the problem stating "White draws".
At the end, they succumb to an engine and suddenly change their minds...I cannot stand the lack of emotional control of some people (ironically), but I digress...
But then I remembered the famous grandmaster Richard Réti's words.
‘I once heard a story where a reporter asked the famous master Jose Capablanca, “How many moves do you see ahead?” He jokingly said, “Fifty!” The reporter, not knowing much about chess, wrote that down in his notes and then proceeded to ask another famous master, Richard Réti, the same question. Réti said, “One! But always the right one.” There is something to be learned from this: You only need to see as deep as necessary to establish the best move for the position.’ - Page 39 of Winning Chess Tournaments by Robert M. Snyder (Lincoln, 2007)
Well, the white king is threatening the pawn if he goes down, but he's also making an attempt to promote his if he goes sideways.
He can't do both, he must choose one or the other...
WAIT! He can just move diagonally, and move down and sideways at the same time.
It doesn't even matter if we can make it in time or not. We have no choice.
2. Kg7! Kb6 3. Kf6!! f4 (Kxc6 4. Ke5 draws) We're getting closer to not just one, but both pawns. Black can also try 2... f4 3. Kf6 f3, but the choices are the same:
- This is the easier variation. Black promotes in three moves; if we promote in four (because White moves first) we are drawn. We are also only one tempo behind the pawn. If the king moves at some point, the pawn is ours! Thus 4. Ke5!! and now 4... f3 Kd6 promotes in time whereas 4... Kxc6 Kf4 gives us the required tempo.
- It's clear that the king cannot reach the pawn anymore, but he is within the distance to help his pawn safely promote. 4. Ke6!! Kb6 (4...f2 5. c7 h1=Q 6. c8=Q is the same) Kd7 5. c7 h1=Q 6. Qc8+ 1/2-1/2
These are my favorite positions: Once you know the theory, everything is easy. Most people will overthink it by analyzing, but the endgame is not about analysis. It's about ideas. Analysis paralysis!
A Means to an End
Both of these positions have taught me how to play correctly. (They're also studies by Réti, and you can find them here and here.) Because of the second one, I know it is absolutely ideal to centralize my king, because I have more options. (I wish I could bold that, but that's overdoing it.) King centralization is essential in the endgame, and I wouldn't have realized it without that study; I used it in one game so far, and won it without a sweat.
On a final note, of course he's never encountered those "theoretical" positions - he hasn't tried to create them. He doesn't know how, or he's trying to force a different position:
- Attacking when it is better to wait, because the opponent would be in zugzwang.
- Winning material when simplifying would be easier. Perhaps even trading the rook - AND a pawn - for just the opponent's bishop, but creating a won pawn endgame, would be better than material gain with complications (but it would certainly be more fun, which it looks like what your student wants).
- Making a simple mistake, because you don't know what you're doing. Sorry, but if you don't know the ideas behind these positions, it will simply cost you points in tournament games.
Learning the Hard Way
At the Chicago Open in 2017, all (ten) of my games were won (according to later analysis by Stockfish).
I had trained really rigorously for this tournament by reading up on positional play and tactics. I also gave some, but not too much, attention at the endgame.
I gave perhaps thirty minutes to the opening over a matter of weeks. It really isn't important at all; the opening is just a subset of the middle game, and its study is just important to avoid traps. That's it.
Back on topic, my opponents could not touch me to save their life in the middle game. That training had really paid off, and I was able to convert my imbalances (space, pawn structure, bad bishops to good, etc.) easily. If you had given me my opponent's position, I'd say they're simply lost, because I wouldn't know what to do with them.
But in six of those ten games, I did one of the three above points. I attacked prematurely. I complicated things. And I simply didn't know what to do in two of games, which I vividly remember LOSING: Rook and pawn endgames.
Coincidentally, those endgames are the ones I hated the most. I did not know the Philidor position (LOL).
Now I've picked up Secrets of Rook Endings by John Nunn and have devoted more time to Jesus De La Villa's 100 Endgames You Must Know. You couldn't pay me to get me to read a middle game book, because I'm simply not interested.
You can know one endgame idea really well and force a draw - even win on your opponent's negligence - or you can know one opening really well and hope he doesn't survive the middle game.
And hope chess is ALWAYS a fast road to nowhere.