An upper bound is 27 queens, which can threaten all squares of a 3D chessboard hyperplane (and the two adjacent ones), which sweeps through the hypercube and smashes the king against a hyperwall. This assumes that the game doesn’t draw after 50 turns.
I’m unlikely to try to solve it, but are you looking for an answer like “if the king starts here, you can do it with N queens placed at...”, or “no matter where the pieces start, you can do it with N queens”? Are you limiting positions to those which could theoretically be achieved in a legal game of 4D chess?
(By that last one, I mean that on a 2D board, you could have a king in the corner and a queen directly adjacent above and beside it, and that would be mate. But you can’t ever have that position in a legal chess game. If something like that turns out to be the optimal, would you accept it?)
It’s the worst case scenario for queens, of course. Just as you ask how to mate the solitary black king with the white king and a white rook in 2D chess. The mating method should always work.
If it doesn’t always work, which means that there is a position from where the mate isn’t possible … then that number of queens isn’t the answer we are looking for.
Just to be sure, does it mean that a king can move by a non-zero vector (a, b, c, d) where a, b, c, d in {-1, 0, 1}, and a queen can move by a non-zero vector (a, b, c, d) where a, b, c, d in {-n, 0, n} for some n?
Kindly invited to solve this.
An upper bound is 27 queens, which can threaten all squares of a 3D chessboard hyperplane (and the two adjacent ones), which sweeps through the hypercube and smashes the king against a hyperwall. This assumes that the game doesn’t draw after 50 turns.
50 moves rule is totally inappropriate in 4D. Let us dismiss that rule here, yes.
An upper bound is 17 queens: 16 threaten all 6^4 inner squares, then the 17th moves to the inner square closest to the king.
Edit: Nevermind, this amounts to the 17th queen checkmating the king on a 3D board with warp sides.
I’m unlikely to try to solve it, but are you looking for an answer like “if the king starts here, you can do it with N queens placed at...”, or “no matter where the pieces start, you can do it with N queens”? Are you limiting positions to those which could theoretically be achieved in a legal game of 4D chess?
(By that last one, I mean that on a 2D board, you could have a king in the corner and a queen directly adjacent above and beside it, and that would be mate. But you can’t ever have that position in a legal chess game. If something like that turns out to be the optimal, would you accept it?)
No, unless the queen is defended by some other piece, otherwise the king could just capture it. Or am I missing something?
Ah, I was unclear: I meant two queens, one each above and beside.
It’s the worst case scenario for queens, of course. Just as you ask how to mate the solitary black king with the white king and a white rook in 2D chess. The mating method should always work.
If it doesn’t always work, which means that there is a position from where the mate isn’t possible … then that number of queens isn’t the answer we are looking for.
To answer the other question: there exists a checkmate with two queens. Just pin the king into a corner with one, and guard that queen with another.
But can you actually pin a noncooperative king there with only 2 queens? You can in 2D, but hardly in 3D and even less in 4D.
Just to be sure, does it mean that a king can move by a non-zero vector (a, b, c, d) where a, b, c, d in {-1, 0, 1}, and a queen can move by a non-zero vector (a, b, c, d) where a, b, c, d in {-n, 0, n} for some n?
Affirmative! ;)