2025/10/01 DPA: "Squaring The Circle"

Black to move:

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White's King is frozen.

What about 1. ... Qe1+:  if 2. Bf1 Rh1+. 3. Kh1 Qxf1#.

Oh, but that's not checkmate:  the White Queen is guarding f1.

If the Bishop wasn't there, this idea would work:  is there any way of luring the Bishop away?  It doesn't appear so, at least not now.

1. ... Qe1 is the most direct path:  is there an indirect one [ie look for LQMs and BQMs].

The Queen can't move backward to any great effect.

It could slide laterally if the Rook wasn't there.

Aah, I see it:  1. ... Rh1+. 2. Kxh1 Qh5+. 3. Kg1 Qh2+. 4. Kf1 Qh1#.

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The key was recognizing how constrained the White King was and that 1. ... Qe1+ did not work so there had to be another way.

This pattern has come up before in the DP so it's worth remembering.

1. ... Rh1+ Tractor Beams the King to h1.  This is critical because it then allows

2. ... Qh5+ to come with check and thus no loss of tempo.

3. ... Qh2+ forces the King away from g1 where it was guarding h1.

4. ... Qh1#, made possible by White's Bishop.

This can be visualized as a more complicated back rank checkmate where, instead of 3 pawns in front of the enemy King, there's one opponent pawn blocking one square, an enemy pawn that controls the next square, and a Bishop that blocks the 3rd square.

The Bishop was critical to stopping the idea beginning with 1. ... Qe1+ but also to enabling the idea starting with 1. ... Rh1+.