What is Réti's study?
White has a king on h8 and a pawn on c6. Black has a king on a6 and a pawn on h5. It is White's move, and the task is to draw.
At first sight there is nothing to play for. The black pawn needs four moves to reach h1. The white king needs seven moves to reach h1.
The white pawn looks just as hopeless. It needs two moves to reach c8, and the black king on a6 gets to b7 in one move to stop it.
Yet White draws, and there is exactly one first move that does it.
Why can the white king not simply chase the pawn?
Use the rule of the square. Draw a square from the black pawn on h5 down to h1. Its corners are h5, h1, d1 and d5.
The white king stands on h8, well outside that square. Chasing straight down the h-file does not work either. The king would go h7, h6 and then run into the pawn's path too late.
So a straight chase fails by one move. That is the whole problem, and one move is exactly what the solution finds.
Why is a diagonal walk not slower?
This is the idea worth taking away from the page.
A king moving diagonally changes its rank and its file on the same move. Moving from h8 to g7 brings the king one rank closer to h1 and one file closer to the c-pawn.
So the diagonal walk h8, g7, f6, e5 is not a detour. It reaches the h-pawn's path just as fast as walking straight down the h-file. On the way it also arrives next to the c-pawn.
The king is doing two jobs with one set of moves. That is the trick.
What are the two goals?
White is not trying to win. White is trying to reach one of two drawing positions, and Black cannot stop both.
1. Catch the black pawn. If the white king gets inside the pawn's square, it takes the pawn and only kings are left.
2. Push the c-pawn. If Black spends time on the h-pawn, White pushes c7 and the black king has to give it attention.
After 1.Kg7 h4 2.Kf6 Kb6 3.Ke5 the king stands on the fork of both roads. From e5 it can step to d6, next to its own pawn, or to f4, inside the black pawn's square.
What happens if Black takes the c-pawn?
After 3...Kxc6 Black has won a pawn, and it changes nothing. White plays 4.Kf4.
Count the race again. The black pawn is on h4 and needs three moves to reach h1. The white king on f4 is inside the square h4-h1-e1-e4, so it arrives in time.
The finish is 4...h3 5.Kg3 h2 6.Kxh2. Two kings and nothing else on the board is a draw.
Why does 1.Kg8 not work?
1.Kg8 looks like the same idea, and it loses.
From g8 the king walks f7, e6 and so on. That road is one square further from the black pawn than the road through g7, f6 and e5.
The difference shows up at the end: 1.Kg8 Kb6 2.Kf7 h4 3.Ke6 Kxc6 4.Kf5 h3 5.Kg4 h2 6.Kg3 h1=Q. Black queens with one move to spare and wins.
This is why the study is famous. Two moves look alike, and only one of them is a draw.