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A somewhat odd puzzle (SPOILER ALERT)

http://en.lichess.org/training/26602

This puzzle doesn't quite fit the category of checkmate or material advantage. Instead the accepted solution here is to have a position that is the best for Black. This is strange because you are solving as white but thinking about the best of the possible outcomes for black.

Black has the option of giving up just one pawn but the resultant position is worse (-~5.5 eval) as compared to the solution where material is lost to achieve a better (-~3 eval) position.
What are you talking about saying black has an advantage? White wins material and has 3-4 point advantage
I just tried it. I ruled out Rxh7 because I saw the fork but thought Black doesn't need to take and can play Kg8, and then you can threaten mate with Rah1, but Black can survive, and after that I didn't see anything clear. When I analyzed it to understand why Kg8 is not possible, after Kg8 the engine gives 6.8+ for White, but White is just up two pawns. I kept playing the best moves to understand, but nothing much happens in the following moves, so I guess the advantage is just positional and not based on anything very clear?
#3 That's exactly what I'm talking about.

#2 Relative advantage if the Queen is sacrificed. Keep in mind that you are solving as white and as #2 said, its very easy to see the fork. Rxh7 (if black wishes) is just one pawn lost and there is no immediate mate. But the position is very bad, so in this puzzle you have to find out that Black has to sack the Q to get a better position.
This is what I got with black declining the rook:

http://en.lichess.org/ZXAwDCni/white#0

Lost for black

This is what I got for black accepting the rook:

http://en.lichess.org/w5icb2WT#0

Lost for black as well

The above was done using Stockfish 8 on lichess.

Using Rybka this is what it comes up with:

http://en.lichess.org/gruSaAdR

Still lost for black, perhaps a bit more lengthy. Takes the route of attacking with the rook to free up a queen move after RaH1 - allowing the king to sidestep the check.

The position is so murky that it's probably worth offering up the rook and hoping they take the trade - knowing that either way you're winning but it results in a shorter, more concise game.

What makes this puzzle a bit unique is that black is not forced to take the rook but the correct solution is where it does.
It's not forced, but they come from real games, and taking the rook looks just as tasty as not taking the rook - especially from a players perspective.

The goal is to find the best moves for white. Can you find any moves for white that are just as good as moving the rook into position? Making one or two or three outcomes that all result in a win is better than moving a piece.

Rybka always recommends moving that rook like that in the start. There is no better move for white, seemingly.

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