I am struggling to see and understand the principles in a way that I can understand them behind some of these problems like Endgame Enthusiasts post from 25 Mar and MaRRatigan's problem from 2019. I having trouble with understanding how you would determine the right circuit vs. wrong circuit. any help with the back to basic principles is appreciated.
Endgame: Knight vs Pawn

The idea in the ones I posted was essentially keeping the knight in a circuit where every move, if the losing side moves a pawn, the knight can checkmate next move. So the losing side is forced to shuffle the king back and forth every move, until the knight arrives on a square that prevents the king from moving, and also can move to the checkmating square next move.

this is also a draw but its only 1 correct line
The first two moves seem pretty obvious, but after that it seems like there are many correct lines.
this is also a draw but its only 1 correct line
Looks trivial.
The fiendish part is - the board's upside down.
this is also a draw but its only 1 correct line
The first two moves seem pretty obvious, but after that it seems like there are many correct lines.
To be accurate, the first one move seems obvious, but after that there are many correct lines (with the continuation in the post).
This is a tough one if you don't use tablebases.
The difficulty is it's likely to finish up in KQ(P(P))vKN and, while those endings are not difficult, evaluating the shortest distance to forced mate correctly is not so easy.
I copped out and consulted Nalimov.
Your 1...Ng6 is correct, but so is any king move.
Your second Black move (2...Nf8) is incorrect. Any king move delays mate by an extra two moves with accurate play by both players.
Krazy