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Jeremy Lennert wrote on Mon, Feb 6, 2012 07:47 AM UTC:
If you suppose your king will always maintain opposition to the enemy king,
then it can block the rank with no help at all.  But if you don't maintain
opposition, then the enemy king necessarily has a head start towards one
side or the other, and can break your line there before your king can get
close enough to defend, since you are vulnerable on both sides.  And since
maintaining opposition at all times will require all of your moves, it
cannot be part of any plan to force mate.

But a pair of DD (with leaping power) plus a king CAN force the enemy king
back using zugzwang, unless I am much mistaken.  For example, suppose white
DD on a2 and b2, king on c3, black king f3.

1. Kd3 Kg3 (black tries to stay on rank 3)
2. Ke3 Kh3
3. Kf3 Kh4 (black is forced back a rank; only legal move)
4. Kg2 Kg4
5. DDb4 Kg5 (King threatens f3, g3, and h3, DD threatens h4 and f4)
6. DDa4

For thoroughness, we should also advance our own king to rank 5 without
breaking the line, so we can repeat the process...

6. ... Kf5
7. Kg3 Kg5 (black tries to maintain opposition)
8. DDc4 Kf5 (white loses a tempo to break opposition)
9. Kh4 Kg5
10. Kh5

The final mate is only slightly more complex.  Suppose white DD a6 & b6,
white king c7, black king f7.

1. Kd7 Kg7
2. Ke7 Kh7
3. Kf7 Kh8
4. Kg6 Kg8
5. DDf6 Kh8 (f8 is threatened; black confined to corner)
6. DDa4 Kh7 (white loses a tempo)
7. DDa8+ Kh8
8. DDf8# (or DDh6#)

I trust it's easy to see why this doesn't work if the DDs block each
other.

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