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# Pandolfini's Puzzler #47 - The Fortress and the Cage, Part 2

| 12 | Scholastics

Professor: Hello, class. I see you’re back for more after last week's discussion.

Zephyr: We certainly are.

Hale: You had me thinking of fortresses all week.

Professor: Did I? I’m sorry about that.

Hale: Don’t be. True, I thought a couple of your problems last session were moot, though the concept was neat.

Rachel: This week, you promised to continue talking about positional draws, especially the idea of the cage.

Thomas: Can you tell us again what a cage is?

Lucian: I can tell you what it is.

Zephyr: Okay. What is it?

Lucian: It’s what they have at a zoo.

Ryan: Thanks for being so informative.

Professor: Zoos, game reserves and whatever other places there are to keep wildlife aside, let’s move ahead to our first problem.

Question 1: Can White create a cage?

Professor: You can see that if it were Black’s turn, White could resign.

Thomas: But it’s White’s turn, and it’s obvious how White can avert the trouble.

Rachel: So that’s what you mean by a cage. In the end, Black’s king is unable to participate.

Hale: Not from the inside of that cage.

Professor: Precisely. Let’s reinforce the concept with our next problem.

Question 2: Can White create a cage?

Thomas: It’s nice, though it’s practically the same problem.

Zephyr: Actually, it’s different enough to be included in the lineup.

Hale: A cage is a cage, bar none.

Ryan: Do you have another problem?

Professor: I do. And I imagine it's somewhat different.

Question 3: Can White create a cage?

Zephyr: That’s a curious brainstorm. The black king really does get trapped in a cage.

Ryan: It works, all right, as long as White doesn’t get zugzwanged at some point.

Professor: Well, if Black does have a winning line, I’m sure this group will find it.

Hale: Eventually.

Ryan: While we’re waiting for that eventuality, how about another problem?

Professor: Okay. Here it is.

Question 4: Can White create a cage?

Lucian: Now this is a harder problem.

Zephyr: But we see what you’re getting at.

Thomas: Yeah, another cage.

Sure enough, the answer did involve the fashioning of a cage, and the super six were able to design one. It was part of a trend, and the clamor was for more.

Professor: This group is so suited to problem-solving, and so well-dressed, I think I can accommodate you.

Ryan: So well-dressed? That doesn’t sound like this group or any chess class, for that matter.

Professor: Well, this class isn’t any old chess class.

Ryan: True, but I still don’t think we dress that well, either.

Hale: Can we skip the discussion of sartorial splendor and just see the next problem?

Professor: You win. Here it is, in all its revealing cloth.

Question 5: Can White create a cage?

Once again the group was able to piece together an answer. It didn’t take long, but it took long enough to exhaust the period.

Professor: I’m afraid we’ve come to the end of the present engagement, yet there’s still more to say about cages.

Zephyr: You mean there’s something germane we haven’t covered yet?

Hale: Are you kidding?

Lucian: What pertinent stuff haven’t we covered yet?

Professor: That’s for me to know and for you to see in part 3.

Answers below - Try to solve NM brucepandolfini's puzzles first!

Answer 1: With Black threatening Nh3-g1+, blocking out the white rook, White’s only salvation is 1. Rh1!. After 1…Kxh1, White has 2. Kf1, and Black’s king is trapped in the corner, unable to escape its cage.

Meanwhile, Black’s knight can’t help out, since it can’t gain a tempo on its own. That is, every time a knight moves, it winds up guarding different squares from the ones it guarded on the previous move. Knights always go from light to dark to light ad nauseam — they are compelled to maneuver in color-coordinated blocks of two moves.

After 2. Kf1, however, if somehow it were White’s turn (it’s not), White would be lost. The white king would have to play to e1 or e2, allowing the black king to extricate itself from h1. The h-pawn could then promote on the next move.

As a practical matter, it's good to imagine possibilities and alternatives that could materialize if something were slightly different. It’s a way to stir your powers of perception, attuning them for those moments in future games and situations when such contingencies might suddenly arise.

Answer 2: With Black threatening Na4-b2+, creating an obstacle for the bishop, White’s only resource is 1. Ba1!. The game is then drawn whether or not Black takes the bishop immediately.

If 1…Kxa1, White draws with 2. Kc2, and Black’s king is stuck inside its cage. The zookeeper (White’s king) is not going to let the black king out of the cage, nor can the knight provide meaningful assistance since it’s restricted by the color rule that shackles it to operations of two moves.

Note that 1…Nb2+ (instead of immediately capturing the bishop) doesn’t change much. After 2. Kd2 Kxa1 White locks the door of the cage with 3. Kc1! (but not 3. Kc2?? Nd3). After 3. Kc1! the knight cannot come to the aid of its king, and the position is drawn.

Here we see an application of a color rule (chess has a number of such pragmatic color rules, not just this one). Here the white king can set up the cage by moving to a square of the same color as that occupied by the knight. Moving to the opposite color, on the other hand, loses.

Answer 3: By 1. Ndf6!, White ensnares the black king in an Alcatraz-like cage. Accordingly, after Black promotes with check, Black’s queen can’t rely on the needed support of its own king. Moreover, the knights protect each other, so neither can be won by the queen’s marauding attacks.

Overall, the position is drawn as long as White remains vigilant and doesn’t stumble into zugzwang, constraining one of the knights to move. With careful and alert play, White can avoid such missteps, and Black can make no progress.

Answer 4: Here the cage is secured by tactics. White first drives the black king to the corner by 1. Nh6+, and after 1…Kh8, the lockdown is completed by 2. Bd6!. That reduces Black essentially to two options: moving the knight or moving the queen.

Clearly, moving the knight doesn’t work because of Bd6-e5+, forcing the knight to go back to g7. So that leaves Black with queen moves.

Let’s do a thought experiment, à la Henri Poincaré. That is, let’s assume that eventually the black queen is able to fork the white king and bishop. As long the white king doesn’t move to a square allowing the bishop to be captured with check, the bishop remains immune.

If the queen ever takes the bishop without also giving check, White wins the queen with a forking knight check at f7. Then, after taking the queen, White’s knight has the time to get into position to sacrifice itself for Black’s h-pawn, and that’s a draw. But don’t take my word for it. Try it and see.

Answer 5: Here we have another tactical cage in the making. The bars of the prison are set up by 1. Ne6+ Kg8  2. Be8!. Thereafter White has threats to play Be8-f7+ and g6-g7+.

Not surprisingly, the black queen can win the knight at once (or later), here by 2…Qa2+. But after, let’s say, 2…Qa2+ 3. Kg3, the line 3…Qxe6 4. Bf7+ Qxf7 5. gxf7+ Kxf7 6. Kf3 creates the distant opposition; White can hold the fort, or keep the cage door shut.

So, with the black king tactically trapped in a cage, and the queen unable to capture the white knight satisfactorily, the situation essentially reduces to a positional draw.

Take note

As a reminder, fortresses prevent enemy pieces from entering. Cages do the opposite. They prevent enemy pieces from getting out. This lesson’s examples focused on stopping the opposing king from participating.

Next week we’ll see how cages can prevent the involvement of other key forces; not just the enemy king, but also opposing rooks and queens. Tune in next Friday to find out more.

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