Pandolfini's Puzzler #21 - The 3-4-5 Problem

“It’s good to see the both of you,” the Professor began. “And because you’ve become my two favorite students, I’ve got an intriguing problem for you to consider today. Working as a team, I’m hoping you can apply your wonderful analytic skills to solving it.”
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“We love intriguing problems,” Zephyr offered with lots of energy. 
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“That’s right,” Lucian agreed. “Please show us what you have planned for today.”
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“OK,” the Professor interjected.  “Why don’t you start by taking a look at the following position?”
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The Professor continued: “I saw a position like this one the other day. As soon as I saw it, I knew it would be just the right stimulus to get our session going.”
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“The first thing I’d like to know is whose move it is,” Lucian made clear.
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“It’s Black’s move,” the Professor cleared up. 
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“That’s good for Black,” Zephyr pointed out. “Otherwise, White’s b-pawn could capture on a7, and that might be a problem for Black.”
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“No kidding,” Lucian shot back. “The a-pawn would then be ready to queen.”
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“Quite so,” the Professor remarked, “though Black does have a tactical resource to stop that pawn. But let’s go to the actual question.”
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Question: How can Black play and mate in three moves? 
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Zephyr and Lucian wasted no time. They both knew to review the position for a few minutes before committing to exchanging their thoughts, and that’s what they did. Suddenly, after 10 minutes, and several failed tries, both students seemed to come upon an apparent solution. Excited, they shouted it out in unison.
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“You’re right,” the Professor replied. “But you’re also wrong. That is, I’m wrong. I gave you the wrong position. There’s no black knight on d8. What’s more, it’s not mate in three moves. Now, it’s mate in four moves. Let’s see you solve it now!”
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Once again, the brilliant girl and brilliant boy went about the process of solving the Professor’s problem. It took a little longer, this time almost 15 minutes, but their mutual efforts wound up working it out once again.
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“We think we have it,” Zephyr said emphatically.
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“Here’s the variation we think works,” Lucian clarified.
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“Wow, once again you’re right,” the Professor admitted. “But, once again, you’re also wrong, and so am I. Maybe I stayed up too late last night. You see, there is no black pawn on a7. What’s more, because of that, it’s not mate in four moves anymore. Now it’s mate in five moves.”
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“Oh, Professor,” Zephyr cried out. “How could you do that to us?” – this she asked while fighting back a secret smile.
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The truth is, both Lucian and Zephyr loved such surprises. And with their usual methodical approach, they immediately set out to solve the newly changed position. It took 20 full minutes, but their determination and cooperation produced what seemed to be the correct answer.
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“Well done,” said the Professor. “I’m very proud of both of you - so much so, I’m not even going to take anything else off the board. No, I won’t ask you to solve the mate in six.”
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Can you come up with the variations Zephyr and Lucian found that solved the three positions and their tasks?
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Answers Below - Try to solve ProfessorPando's Puzzle first!
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ANSWER #21
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Here are the main variations:
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1) The mate in three moves begins with 1…Rxb6. 
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If 2. Bxb6, then 2…Nc6. After the bishop moves, it’s over with 3…b5 mate! 
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And if instead White plays 2. Bxd8, then Black wins by 2…Ra6+ 3. Ba5 b5 mate.
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2) The mate in four moves (with the knight no longer on the board) begins with 1…axb6. The main continuation thereafter is 2. Bd6 Rb5 3. Bb4 Ra5+ 4. Bxa5 b5 mate!
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3) The mate in five moves (with the a7-pawn also gone) begins with 1…Re2. The main continuation then is 2. Bd6 Re8 3. Bb4 Ra8+ 4. Ba5 Ra7! 5. bxa7 b5 mate!
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Take note
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The above problem appears in various chess sources, though not with the same story behind it. In its most famous presentation by Irving Chernev, one of the game’s greatest writers, the narrative revolves around a game played between two kings. They continue playing with a battle raging outside, as if nothing other than the chess game mattered. Stray bullets (or arrows) knock pieces from the board, comparable to what happens in the above account, and they keep thinking. A similar scene is set up by Alexander Kostyev in one of his chess books. It’s a charming, though totally fabricated story. 

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