How much material is needed to force the capture of a lone queen (for example in horde chess)?

Question

In horde chess, the following starting position is used:

[FEN "rnbqkbnr/pppppppp/8/1PP2PP1/PPPPPPPP/PPPPPPPP/PPPPPPPP/PPPPPPPP w - - 0 1"]

The goal of White is to checkmate Black; the goal of Black is to capture all White's pieces. The main strategy for White is to try to promote a pawn; the main strategy for Black is to get their queen or rook behind White's pawns, and destroy White's forces from behind.

It naturally comes to mind that, with good play from both sides, a situation may arise in which Black manages to capture all but one of White's pawns, and the remaining pawn promotes to a queen. In this kind of endgame, White cannot win because a lone queen cannot realistically deliver checkmate (except in very special cases which are not bound to arise if Black knows what they are doing).

Therefore, Black has to force the capture of White's queen, or accept a draw. This endgame practically becomes a game of "checkmating a queen" before the 50-move rule kicks in.

So how much material is needed to force capture of a lone queen?

I guess Black's plan will begin with promoting all their remaining pawns. To start somewhere, let us assume that Black's material after promoting everything is one king plus N queens.

With a king plus 5 queens, it seems that Black has an easy win. They just need to set up a domination position, where the 5 queens attack every square on the board:

[FEN "6qk/8/4q3/3q4/2q5/6Q1/q7/8 w - - 0 1"]

No matter where the White queen moves, it will be captured on the next move. I do not have a proof, but it seems easy for Black to achieve this position while blocking checks by White.

With a king plus 4 queens, I have not found a way for Black to force the capture of White's queen. While it seems overkill to have to attack every single square of the board in order to trap a queen, maybe the queen's mobility is so large that there is no other way.

So the question is

What is the smallest N such that a Black king with N Black queens can force the capture of a lone White queen?

Answer 1

This is not quite definitive, but I claim that three queens plus any other piece is enough, while it appears that three queens alone is a draw. To determine this, I let Stockfish run on lichess for a while with various amounts of material for black until either it found a forced win or I ran out of patience. I tried to choose starting positions with all of black's pieces placed as passively as possible, so that black would only win if it's winning by virtue of its material. The lines shown below are what Stockfish eventually claims as examples of perfect play.

With four queens:

[Variant "Horde"]
[FEN "kqq5/qq6/8/8/6Q1/8/8/8 w - - 0 1"]

1. Qe2 Qd6 2. Qh5 Qac5 3. Qh4 Qcd4 4. Qe1 Q6e5 5. Qf1 Qee6 6. Qa1+ Qxa1#

This one seems winnable even in practice: once black reaches the configuration after move 4 (with the king on b8 if necessary to get out of check), you can trap white's queen in one move regardless of where she is--unless she's on a2, in which case (for example) 1... Qdf4 2. Qa3 Qfc4 wins.

Three queens and a rook:

[Variant "Horde"]
[FEN "kqr5/qq6/8/8/6Q1/8/8/8 w - - 0 1"]

1. Qf5 Qa1 2. Qh3 Qbe5 3. Qg4 Rh8 4. Qc4 Qbe4 5. Qb3 Qab1 6. Qf7 Qed3 7. Qf2 Qdd5 8. Qe1 Qexe1#

Three queens and a bishop:

[Variant "Horde"]
[FEN "kqb5/qq6/8/8/6Q1/8/8/8 w - - 0 1"]

1. Qd1 Qf4 2. Qe1 Qa3 3. Qe2 Qbe4 4. Qh5 Qc3 5. Qd1 Kb7 6. Qd8 Qa1 7. Qg8 Kc7 8. Qb3 Qc4 9. Qb1 Qxb1#

Three queens and a knight:

[Variant "Horde"]
[FEN "kqn5/qq6/8/8/6Q1/8/8/8 w - - 0 1"]

1. Qe2 Qg3 2. Qe8 Qae3 3. Qf8 Kb8 4. Qf1 Qge5 5. Qd1 Ne7 6. Qa4 Qbb3 7. Qh4 Qf3 8. Qh6 Qa2 9. Qb6+ Kc8 10. Qh6 Kb7 11. Qh4 Qfh5 12. Qb4+ Ka6 13. Qb1 Qxb1#

Three queens and a pawn is also a win, since (as you say) black can promote the pawn.

Besides four queens, I can't make sense of black's strategy in the other positions beyond "put your pieces on reasonable squares and poke around until there's a forced win". It seems like Stockfish can't either, based on its evaluations--e.g. for QQQN, it evaluates basically every move as -5.8 for several minutes until it finds the win.

I can't rule out a forced win with only three queens, since there are some plausible positions where white's queen is trapped, like below. But it seems hard to reach such a position by force. I also haven't been able to find a forced win with QQRR.

[Variant "Horde"]
[FEN "3q4/8/8/4q3/4q1Q1/8/7k/8 w - - 0 1"]

Answer 2

Four pieces is the lowest possible amount needed to trap the White queen.

[FEN "Q6r/8/8/8/8/8/5k1p/r7 b - - 0 1"]

1... h1=Q