Am I missing something in this puzzle?

So I've just done

and unsurprisingly I failed! But I've looked at it for quite some time and I can't see how the lichess answer is better than mine (perhaps on some ludicrously high GM level it leaves you in a better position than my solution?)

white: Nh6 (check), Black Kh8 (forced as f5 cannot move to capture, and the king cannot go to G7 as it creates a fork for the knight)
White NxG6 (check) Black f7xg6 (or h7xg6)
White Rxf5 (rook is safe from capture as black queen now undefended) Black Qxg5
White Rxg5

I'm not suggesting my move is better, but it seems to bring about exactly the same result. Perhaps I'm missing something though?

Any thoughts?

In the computer's solution, white has an extra piece than in you solution. So materially speaking, the score is 4 vs 1.

Can you explain how you come to that?

From what I can see:
In the computers solution he takes a queen and a knight, for a queen, so that's 12 v 9 points
In my solution it's queen knight bishop for a queen knight, so 15 v 12 points.

Both are a gain of 3. Arguably as mine also forces an additional exchange when you are leading you are better off no? Particularly as bishops are stronger in the end game, so a knight for a bishop is probably marginally better than 3 points in reality.

Unless I'm missing something?

I'm fairly sure that whatever chess supercomputer figures out the answers is correct that the end position must in some way be better, but I can't figure out if it's just a nuance of being a computing megabrain, or if I've just made a simple mistake and actually my material gain isn't as good.

I just looked at it on a board for 10 minutes, and I'm mistaken, and you are right. Your end position is clearer and some of the computer moves appear completely illogical.

What I can see just glancing is that your solution allows some counterplay with the Rook, which is able to get to the 2nd rank at the end of your solution while the original solution, they are still inactive.

Your solution:
1.Nh6+ Kh8 2.Nxg6 fxg6! (...2 hxg6 is worse since fxg6 at least opens a file for a rook. ) 3.Rxf5 Qxg5 4.Rxg5 Rf2!(counter play!) 5.b3 Re8(more counter play) etc.

Also look at the Black King shelter, its undamaged at the end of this.

yes that makes sense, the position at the end does look better now that you've described that.

I'm not sure what you mean by the black king shelter though, as it seems more broken my way, although to be fair it's probably better defended as I've developed his rooks as you say.

Thanks for the clarification. I think you're certainly right that his move was better... though it does still seem harsh for it to be considered a fail though! But still, that is often the way with these puzzles :)

In your solution fxg6 allows black to reconnect his king shelter, that's what I mean by that, but the main thing to worry about in this position is the counterplay. The counterplay plus those black queenside pawns gives Black some chances for a draw

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