Here are my thoughts. Black has to stop white from playing Kf6 and Kxg6. Using the king to keep white out fails, for example 1. ... Kg7 2. h4 and either 2. ... h5 3. g5 or 2. ... g5 3. h5 and the black king has to move and allow white to play Kf6.

I think black's best try is to get in h5 and then hxg4 hxg4 g5 and when white takes on g5, black plays Kg7 and that position is a draw. But white can play g5 in response to h5, keeping both his pawns on the board.

So black has to stop white from playing g5 in response to h5. So here is my line:

1. ... g5 2. Kf6 h5 3. Kxg5 hxg4 4. hxg4 Kg7 achieving the drawn position.

When I first looked at the diagram, I found it hard to believe that black could draw with such an inactive king, as opposed to white's.

The only thing to add to comments is, nice analysis/explanation by Loomis.

black draws by playing g5 which has no value except to prepare the next black move.

it doesnt matter where white moves after g5. the next black move is genius h5!

then if white takes h5 the 2 h pawns are a draw and if white doesnt take then black takes and with one pawn on the g file its a draw for sure.

basically black sacs both pawns to either get 2 white pawns on the h file (=draw) or just one on the g file (-draw).

idosheepallnight, did you even read my post?

Also, you claim "with one pawn on the g file its a draw for sure." But actually with one pawn on the g-file, whether or not it's a draw or a win is highly dependent on the locations of the kings and who is to move.