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Sometimes my pupil may play the game having always 2 moves. (I of course play only 1.) The exact rules vary due to strength, especially regarding checks, let's try worst case: Black may capture and run through check in the 1st move. I.e. White's king must keep a distance of 2. Can you mate with 2Q? (A mating position is Kg5 Qa8 Qa7 - Kh8.)

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Looks like two Queens can force checkmate even without their King's help. Mutually protecting Queens are safe and can restrict the KK to an ever smaller sector of the board. Eventually mate happens. For example, Qd6+Qe6 vs KKe8 forces KKg7. Then Qd6-e5+ and if KKf8 then Qe5-f6+ etc., while if KKh7 then Qe6-f6 with the threat of Qe5-f5+ KKg8; Qf5-g6#, and if the KK goes to g8 then symmetrically Qe5-e6+, KKh7; Qe6-f7#. Note that the KK can never be stalemated (unless boxed in by its own pieces) because it can always move into check and back.

Here's a procedure that works on a board of any size. We'll push the KK back row by row. Start with the Queens on squares 1 and 2 of the same file, say c1 and c2. Then KK is on one side or the other. If it's on the third rank, play Qcd2, Qcd1, Qde2, Qde1, etc. until the KK leaves the third rank. If that happens when the Queens are staggered, just move the rear Queen up two squares and move the other Queen behind it, say Qe1-e3, Qd2-e2. Otherwise move the rear Queen where the front Queen would normally go, say Qd1-e2. If the KK returns to the third rank, it must be on the same side as before, so play Qd2-e1 and keep pushing to the right. Otherwise realign the Queens on rows 2 and 3 (either Qe2-d3 or Qd2-e3), and the barrier has advanced one row. Once it advances all the way to the opposite side of the board, the KK gets mated in the corner.

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  • How do you force the KK to the edge of the board if it just stays in the center as much as possible? It can stay in place if not in check, and can take one Q even if protected if it is one step away, and can run through check if there's not a width 2 barrier or if it's already in check, and can even threaten to take the white K (if I understand correctly). If two Qs are diagonally adjacent it can stay knight's distance away from both, if they are orthogonally adjacent it can stay a knight's distance from one of them or even further or even run through the barrier Commented Jun 12 at 17:23
  • The same procedure should work; instead of being checkmated the KK is pushed one row back, then repeat until the edge. Commented Jun 12 at 17:37
  • Sorry I don't get it, if Qd6+Qe6 vs KKe4, suppose KKg5 (it could go to c3 too), then Qde7+ right? then KKf4, Q7f6+ right? Suppose KKd3, then what to do? Commented Jun 12 at 17:52
  • See the new paragraph at the end of my answer. Choose a direction, right or left, and push the KK there until it has to give ground. For example, if Qd6,Qe6 vs KKc3 we push left with 1 Qe6-d5 KKa4 (else Qd6-c5 or Qd6-e5 and we've gained a row) 2 Qd5-c5 KKb3 (if KKa4 3 Qd6-c6+ and the barrier advances -- or better yet Qd6-b6, Qc5-c6+ and it's already mating season down the a-file) 3 Qd6-d4 and 4 Qc5-d5 and tbe KK has been pushed back a row. Commented Jun 12 at 21:23
  • A request @NoamD.Elkies: I can understand this much better with the replayer used. Declare a bQ to be the K (to avoid run-through-check issue), place wK and bK where they don't interfere and then you can demonstrate your algorithm "on board". Commented Jun 13 at 7:13

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