Opposition of kings in endgame

Jeddah took on a grand teaching position for the afternoon, talking for over 4 hours and playing over the options for both white & black.  Hearing you has reminded me that that's the best way of teaching, or the one I can handle, anyway: relaxed discussion, one to one.

Gathered about my laptop, we tried also a couple of videos from chess.com, where I have recently bought a membership which permits that.  I thought that wasn't so successful though:  too much background noise in cafe; not enuf oomf in computer; no PAUSE button; not responsive to individual needs.  Got any thoughts?  More videos, done a bit differently?

While Daisy moved towards delivering Jeddah a coup de grace, Marcus & I sat down to look over several things.  We played out one tiny part of that most fundamental of chess-positions: king vs. king and one pawn.

(I've made a mistake, here, on white's move 3. See comment below.)  This position introduces a term you will find useful in late endgame positions: opposition of kings.  By this opposition, kings are both limited in their options of legal moves.  Neither can move toward the other.  When one king has his back to a wall, he has only two moves.  So "opposition" helps you chivvy an opponent's king backwards in the following position, and then to give checkmate:

On this blue board, black's aim is to keep out of "opposition" with white to move.  On moves 1 & 8, being in opposition first reduces mobility for the black king, and then reduces his height by a head.

The same is partly true of the position on the brown board. White uses opposition to reduce black's options.  White wants the black king  one side of the pawn's advance, so that black cannot stop that advance.  Opposition of kings enables this - right until the very end. This means that the difference in which side is to move on the brown board above is crucial.  But in such positions, black can also use opposition to get a draw.  Let's see what happens if it's still white to move in that position on the brown board, not black. Because at the very last move below, black wants to be in opposition with the black king required to move, but with nowhere to go, at all:

This is a draw by stalemate. 

Look at how carefully black must play on the 3rd move!  Only one move will do; the others lose.  By accurate play, black has saved half a point.  Thus, in endgames the options become fewer and fewer, until we reach positions such as the ones here.  Technique and knowledge become important to the chess-player in ways that are different from earlier in a game.