Endgame Caveat #4 Rule of the Square

One of the first endgame rules people learn is the Rule of the Square. Take a passed pawn, draw a forward diagonal in the general direction of the enemy king. Then draw a square around this diagonal. If the king is in this square after his move, then he can stop the pawn from queening; if not, the pawn queens by sheer speed.

Here's an easy question. In the following diagram, if White is on move, why doesn't the Rule of the Square apply?

The pawn utilizes its two-square jump and leaves the Black King in the dust like the Roadrunner does to the Coyote.


Here's a study from Alburt's Just The Facts! There doesn't seem to be a credit for the creator of the study. I switched the colors for expediency. Black to move, White to win.

The Black King can get in the square of the passed h-pawn, but White has an extra trick.
1...Kd4 2.f6! exf6 White sacrifices one pawn to spring the other. By capturing on f6, Black blocks his own path to a8.
3.h5 Ke5 4.h6 and Black's King runs into his own pawn.
4...Ke6 5.h7 Kf7 6.h8Q



When I was trying to remember the study above, I tried to create my own study and came up with this: Black to move, White to win

1...Kc3 Like above, Black gets into the square of the h-pawn.
What should White play in the following position?

The diagonal blocking maneuver from the above study backfires. 2.e5?? dxe5 3.h4 (3.Ke2 Kc2 Black queens first and win.) 3...Kd3 4.h5 e4 5.h6 e3 6.h7 e2 mate.

2.h4! is the correct move.
2...Kd4
Now what should White play from this position?

3.h5! This indirectly protects the e-pawn, since 3...Kxe4 4.h6 and the king is out of the square.

3...Ke5
What's the only winning move for White in the following position?

4.Kxe2! is semi-obvious, but still, I was surprised it was the only winning move.
4...Kf6 5.Kd3 Kg5 6.Kd4 Kxh5 7.Kd5 Kg6 8.Kxd6 Kf7 9.e5 Ke8 10.Ke6 Kd8 11.Kf7 and the pawn marches in.