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move 3. why Rxe6 is not played?

I agree that is the move I would counter to throw a wrench in the proposed winning combo. This would force and end game that I didn't work out a combination for yet.

and in that second graphic. What about  a4 for the move 2?

hmmm...

Then what about 2...Bd3? My guess is you didn't look at the first graphic....

It's all right, this thread was about as lame as a three legged horse from the beginning.

My misfortune is that I actually tried to solve it for several seconds.

Don't you just hate that? There are several seconds of your life you can never get back. Thankfully I just hit the 'Solution' button on this one.

That's usually what I do right away too...and, rather than being prepared to say, "My, what a delightful continuation," I'm thinking:  "Geez, how unsound is this gonna be?" 

For some reason though I got hypnotized (by the near-mate) into thinking that this thing might have some actual bona fide solution...especially given the (as it turned out incomprehensible) title.

LOL, I can feel your pain! Fortunately(?) I've wasted enough time on some of these "puzzles" to realise that unless the opponent blunders there really isn't a solution. I saw the near-mate and Black lacking the means to either force the Rook off the d-file or force his Bishop onto the f1-a6 diagonal my finely honed instincts told me to just hit the "Solution" button and, like a box of Cracker Jacks, I would find the prize at the bottom!

You are just thin-skinned. Nobody said this was the worst "puzzle" in the forums. It's that you present a "Tricky Tactic" that only works with incorrect play by the opponent.

2....Ba2 in the second graphic does nothing to change the tide, again you present analysis based on incorrect play by the opponent. Instead of 3.Bf6? as you suggest, White plays 3.Bb2. Then 3...Re4 is met with 4.Ba3 and guess what, the pawns may have started to fall, but they don't finish. Now Black is faced with three connected passed pawns on the other side of the board. Black can only hope that the opposite coloured Bishops can bring about a draw. BTW if I were White I wouldn't be so eager to exchange my active Rook and be left with opposite coloured Bishops.

So if I were a hater I wouldn't spend so much time showing you the corrections to your analysis. If you were appreciative you'd say "Thanks". Jerkface.

Inaccurate diagrams typically provoke fewer harpoons than inaccurate puzzles.

Wait a minute. You never said your analysis was correct and you just presented the puzzle for feedback? Well, I guess I certainly got that all wrong, please accept my sincerest of apologies. Obviously I misunderstood "I did not see the right move.  Can you?" and when you put in "now the pawns start to fall" you must have only been checking to see if we were on our toes. And of course you just wanted some feedback (of which I still haven't seen a "thank you") just so you could try out using mature language like "haters" and "jerkfaces". Apparently the CDT doesn't have very high standards for its staff.

"I just don't want people to think I was trying to teach them something." I wouldn't lose any sleep on that one.

http://www.columbiatribune.com/news/2010/feb/21/chess-lets-kids-make-some-moves/