2024/01/14 DPA: "Square Dance and Diagonal Dilemma"

White to move.

This is obviously about checkmate, due to the position of the Black King and the promotable pawns.

The Knight is pinned.

1. Rc1# is prevented by the Queen.

There is no way for Black to check, as all squares are covered...except e3.  So, while trying to wrangle his Rook to the 1st rank, White must prevent ... Qe3+.

Or does he?  If 1. ... Qe3+  2. Kg3, Black has no checks except those that sac the Queen for the Knight, which means White will win.

So the obvious 1. Re2 to prevent 1. ... Qe3+ will fail:  Black plays 1. ... Qe3+ anyway and if 2. Rxe3 dxe3+  3. Kxe3, there's only a draw.

I'm pretty sure this puzzle is not about drawing.

So maybe we allow 1. ... Qe3+  2. Kg3.

So where does White move his Rook [he can't move his Knight and I don't think he wants to move his King]?

It would be great if he could move diagonally to b1.

Also, don't forget that checkmate could come with the Rook on the h file as opposed to the first rank:  the Knight and King  prevent the enemy King from moving whether the White King is on f2 or g3.

So White must find a way to generate a threat of moving the Rook to the 1st rank while simultaneously angling for the h file.  It will be like the "Zombies at both doors but neither one can be locked" scenario.

Once the Black Queen gets to a light square like c2, she can check the White King with ... Qg6 also.  If that happens, White won't have time to make the final Rook move; it will be a draw by perpetual check.

But along the c1-h6 diagonal, the White King is safe.

So I don't think White wants to allow 1. ... Qc1 as that will lead to 2. ... Qc2+ and 3. ... Qg6+.

So what about 1. Rc8 for starters?  It threatens 2. Rh8# but prevents 1. ... Qc1.

Black can play 1. ... Qe3+ but, as mentioned above, White calmly plays 2. Kg3 and he is now untouchable.

So it might not be worth it to play 1. ... Qe3+, as it might not change anything.  Let's assume for now he goes straight for defense:  1. ... Qh6 prevents 2. Rh8#.

2. Re8, hoping for 3. Re1#, is again dashed by 2. ... Qe3+.  So the Rook needs to attack elsewhere.

2. Rb8, threatening 3. Rb1#, is countered by 2. ... Qh7 and now Black has reached a light square which bodes ill for White.

2. Ra8, threatening 3. Ra1#, is better because the d4 pawn is blocking the a1-h8 diagonal.

The only way for Black to prevent 3. Ra1# is 2. ... Qc1 [analogous to 1. ... Qh6] but that leaves White free to play 3. Rh8+; it's a matter of style whether the puzzle creator will have Black interpose the Queen or just end the puzzle there.

If I'm right, the Zombies scenario was correct:  the Rook maneuvered to threaten checkmate on both edges [1st rank and h file].

Now:  what have I missed?

The obvious is 1. ... Qe3+  2. Kg3:  but does this change the subsequent moves?  I don't see how.

2. ... Qh6  3. Ra8 allows 3. ... Qg6+  4. Kf2 Qc2+ and Black draws.

OK, so 1. ... Qe3+ is relevant.

Does that change White's first move?  I don't think so.  1. Rc8 still threatens 2. Rh8# and protects c1.

1. Rc8 Qe3+  2. Kg3 Qh6

Ahh, 3. Rg8 prevents 3. ... Qg6+ and threatens 4. Rg1#.

Black can only check with 3. ... Qd6+  4. Kf2 but there are no more checks.

White threatens both 5. Rg1# and Rh8# [any Queen interposition on g1 or h2 will fail since the Knight controls these squares].

There's a problem:  White's King is on g3 which means 3. Rg8 does not threaten 4. Rg1# because the King is blocking the Rook's path.

Even so, preventing 3. ... Qg6+ seems critical so maybe 3. Rg8 is still the right move.

But Black can play 3. ... Qh7 and threaten the Rook and reach a light square from where he can then play 4. ... Qc2+.

White can't allow a Q/R trade since that will be a draw.

How about 1. Ra2:  it threatens 2. Ra1 so Black can defend with

• 1. ... Qe3+  2. Kg3
• 1. ... Qc1  2. Kg3 [threatening 3. Rh2#] Qc7+

Maybe 2. Kg3 is premature.  2. Ra7 would prevent 2. ... Qc7 but it would not prevent 2. ... Qc2+ and Qg6+.

Rook to c3 and c4 fail due to the pawns.

c5 is bad because the rank is blocked by the d5 pawn.

c7 is bad since the Queen controls it.

c6 is possible but it does not threaten check on the h file because h6 is controlled by the Queen.

So of all of the c file moves the Rook can make, c8 seems best.

Along the 2nd rank, only a2 holds any promise.

OK, here's another idea:  1. Rc8 Qe3+  2. Kg3 Qh6  3. Rg8 Qh7  4. Rc8, preventing 4. ... Qc2+ and threatening 5. Rc1+ while maintaining 5. Rh8+.

But Black has 4. ... Qg6/g7+  5. Kf2 Qg2+.

Now that the Queen is on a light square, she can't play ... Qe3+ so conceivably White could move his King back to f2 to escape the tempo-losing Queen check.  But he doesn't have time for that now.

Maybe the White King does not have to be on f2 or g3; maybe the Rook and Knight alone can deliver checkmate.  But then where would the King be safe?

Could White use the Knight as a decoy and checkmate with K+R?

1. Rc6:  even though it doesn't threaten check, since both c1 and h6 are controlled by the Queen, it forces Black to move and the Queen does not have any squares along the critical c1-h6 diagonal, as they are all covered by Rook or Knight except e3 so Black could play 1. ... Qe3+ but after 2. Kg3, what will he do?

Could this be a Zugzwang scenario?  It hadn't occurred to me until now.

The critical advantage of 1. Rc6 is that it prevents ... Qh6 which in my previous analysis allows Black to reach the g file and then Qg2+.

Here's at least one possibility:  1. Rc6 Qe3+  2. Kg3 d3  3. Ra6 [threatening 4. Ra1+] Qc1  5. Ra7, maintaining the threat of 6. Ra1 but now also 6. Rh7+.

Note that 5. Ra8 is bad because it allows 5. ... Qc3, which guards both a1 and h8, both dark squares.  5. Ra7 threatens a dark square check on a1 AND a light square check on h7.

Also:  the pawn on d3 prevents 5. ... Qb1, which would have guarded both a1 and h7.

I think this is it:  apart from 1. ... Qe3+, there is no good move for Black:  the King can't move, the Queen must stay on the c1-h6 diagonal and no square is available save for e3.  That leaves 1. ... d3.

But I just saw a defense:  after Ra7, Black can play ... d2 so if Ra1+, Black can block with either ... Qc1 or d1(=Q).

I think 1. ... d3 is the sturdier defense.  1. ... Qe3+ actually helps White by getting his King off of the 2nd rank where ... Qc2+ is possible.

Aah, but if 1. ... d3, that means he opts out of ... Qe3+ which means White's King is not on g3 anymore which means Rg8 ---> Rg1+ is once again a threat.

Two different variations.  I think the puzzle will opt for 1. ... Qe3+ because it looks scarier and because the pawn on d3 blocking control of h7 is very elegant.

I was right.

Total elapsed time:  2.5 hours.

After 1. Rc6:

Despite the Queen being in the open board, she is severely restricted due to the imperative of guarding both c1 and h6.  Since the King can't move and neither can the d5 pawn, the only choices are 1. ... Qe3+ and 1. ... d3.  We'll look at the puzzle solution 1. ... Qe3+ first.

2. Kg3 and now Black must play 2. ... d3.

Relying on my analysis that Rc8 ---> Rg8 ---> Rg1+ does not work since the King is on g3 and blocking the g file, White takes another tack with 2. Ra6, which threatens 3. Ra1+:

3. ... Qc1 is the only move which guards against both 4. Ra1+ and Rh6+.

4. Ra7 is essentially the end of the puzzle:  White threatens 5. Ra1+ and Rh7+.  4. ... Qb1 would have defended against both but the d3 pawn is blocking the Queen's defense of h7.

Note that this idea does NOT work if Black plays 1. ... d3 immediately because the King will still be on f2 so Black can gain a tempo with ... Qc2+ and then play ... d2, allowing the Queen to defend h7 AND indirectly defending against Ra1+ with d1(=Q).

1. ... d3  2. Ra6 Qc1  3. Ra7 Qc2+  4. Kg3 d2  5. Ra1+ d1(=Q)

Instead, White wins with 1. ... d3  2. Rc8 [threatening 3. Rh8+] Qh6  3. Rb8 [preventing 3. ... Qb7+] Qc1  4. Rh8+.

So there are two distinct solutions with common themes.

You should definitely engage the engine to explore the variations because they are subtle and numerous.