Logical Continuation

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FEN: r3k3/1br2p2/2pR1R2/2b2Q2/4P2p/6nP/p1PB2PK/1q6 w q - 0 1
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This is a repost of the Sunday Puzzle, which appeared yesterday on Susan Polgar's chessdailynews.com site.  I'm reposting it as a separate thread because it has some value as a teaching aid, and I don't want it buried behind 200 "nice puzzle" comments.
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I'm not posting the  solution in the puzzle here, as the point in reposting this puzzle is to  show the logic behind each move.  This would actually be a very simple mate to find over the board, mainly because each  move is logical, from the position it occurs in (in other words, without the use of much "think-ahead".
Below is a slightly edited version of my my comment/analysis made on Polgar's site.  Hopefully it will help whoever looks at it.
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White must deal with the threat of mate at h1 or g1. There's no way to take Black off of these threats by direct defense other than 1 Qf1, which loses the Queen to Nxf1+, with mate not far behind.
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Therefore, if White is to have any chance, she needs to find a way to checkmate Black, with every move needing to be a check.
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White has five ways to check here.  Examining three of the obviously bad ones:
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1 Qe6+ fxg6 simply gives up the Queen.
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1 Qe5+ Re7 is also useless, as White must either give up the Queen by Qxe7+ or the Rook with Rd8+, with neither leading to mate.
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1 Rd8+ Rxd8 gives up the Rook without any good continuation.
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Therefore, we have to look at the two Rooks giving check on e6. 
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1 Rde6+ Re7, and now, White only has one check: 2 Rxe7+ Bxe7. At this point, White has no more checks of any value.
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Therefore, the winning move, if it exists, must be:
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1 Rfe6+!
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Of course, we know that this wins, since there would be no reason to post the position if that were not true. 
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In live play, if White can see that all of the other first-moves automatically lose (and that doesn't take deep analysis here) to simple direct replies, then 1 Rfe6+ can be played even if the player can't see all of the continuations that follow, simply because there's nothing to lose by playing it. Uusally the position is easier to analyze after a move is played on each side.  White must play the given move or lose quickly, so why not try it?
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Analyzing the position just on its merits, it can be seen that 1 Rfe6+ is superior to the same check by the other Rook because leaving the other Rook on d6 keeps Black's Bishop blocked off from e7 and also blocks the Black King's escape to the safety of the Queenside. So given that White must check here, 1 Rfe6+ is logical on at least two counts.
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1.... fxe6
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It's easy for Black to see that the other two legal moves lose quickly: 1 .... Kf8, 2 Bh6+ Kg8, 3 Qg5+ Kh8, Qg7#; or 1... Re7, 2 Rxe7+ Kxe7, 3 Rd7+ Ke1, 4 Qxf7#. So the given move is mandatory, win or lose.
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2 Qxe6+
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This move is natural and obvious, since it gets the Queen a rank closer to the action. The silly 2 Rxe6+ allows Black many ways to escape, the simplest being -
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2... Re7. While the move might not be as obvious from the initial position, it's fairly logical from the position it occurs in.
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2.... Kf8
One of the points of 2 Qxe6+, as Black can no longer interpose the Rook, e.g. 2... Re7, 3 Qg8#
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3 Bh6+ 
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Another obvious move from this position. The Bishop gets into action. A superficial analysis shows that anything else allows Black to hold on, i.e. 3 Qg6+? Kg8, 4 Qg6+ Rg7!, and now White can no longer get the Bishop into play. and must settle for perpetual check.
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3.... Rg7
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Black's only move.
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4 Bxg7+
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Yes, 4 Qf6+ mates a move sooner. However, the continuation to this move is easy for a less talented player to see.
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4 .... Kxg7
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5 Rd7+ 
There are other ways to go, but this one is simplest to see: Rook checks on d7, King moves to 8th rank, Queen checks in opposition to King, King moves, Queen mates.
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5.... Kh8
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The Bishop can interpose and delay mate by a move, but only a computer, or a player whose opponent was very short on time, would bother doing it.
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6 Qh6+ Kg8
7 Qg7#
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Even though a player of modest skill might not be able to see this mate from the initial position, the same player could easily play this line, simply because every move was necessary and/or natural from each successive position. In the initial position, after White eliminates all other moves as worthless, the given move is necessary. As the game is played out from the original position, every move White makes is sensible from each position faced at the moment.

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Well, that was certainly popular, wasn't it?  Thanks for all the feedback.

Here's one that might be more to your liking  No fair using the computer to help!

 

 

White to move and draw.