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I saw this question On an infinite board, which pieces are needed to checkmate? and was inspired by it. Many mating combinations won't suffice on a toroidal board. For example, queen vs king will be a draw in this scenario, as the king and queen will never cover all escape routes.

Which piece combinations will be able to checkmate in this scenario?

Laska
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blademan9999
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1 Answers1

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As a starting point: The following mates cannot work because, with no board edge, there is no way to cover all the squares for checkmate even if the opposing king cooperates:

  • A queen
  • A rook and a knight
  • A rook and a bishop
  • Any combination of three bishops or knights

All the mates that work on the infinite board will work on the 8x8 torodial board. From the other question, we have:

  • Two rooks (or queens)
  • A rook and two bishops
  • Four bishops.
  • Queen and knight
  • Queen and bishop

Just put two opposite-colored bishops side by side and the opposing king is corralled and cannot approach. (On the torodial board the enemy king is also very limited as to how far it can run before it actually starts getting closer to those bishops again.) The other pieces can then block enough remaining squares and deliver checkmate. This works for these mates:

  • Three bishops and a knight
  • Two bishops and two knights

Knights do much better on the torodial board than the infinite one. On an infinite board, even an infinite number of knights cannot checkmate if they are positioned wrong; on a torodial board, a finite number of knights can checkmate. A knight is never more than one hop away from controlling one of the 9 squares around the enemy king, and 4 knights can control half the squares on the board. But the exact checkmate procedure is unclear to me. So I'm unsure about these mates:

  • Four knights
  • Three knights and a bishop
  • Two knights and a rook
  • Rook, bishop, and knight
D M
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  • I don’t think 4 knights can force it. In the mating position, there must be 2 light & 2 dark, so prior to this, there are 3 and 1. So I think bK can always choose another square for his last move which is not near to wK – Laska Feb 26 '24 at 16:37
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    @Laska 8/8/3K4/2NN4/2k5/N7/8/N7 b - - 0 1 and Black has to move to the mating square, but then the question of course shifts to how you force that position. – D M Feb 26 '24 at 21:31
  • @DM there are setups like 8/8/3K4/2NN4/2k5/2N5/8/8 or 8/2NN4/8/2NN4/2k5/8/8/8 where KNNN or NNNN limit bK to two squares; the remaining N or K can then maneuver where it needs to be to complete the mating net. – Noam D. Elkies Feb 27 '24 at 05:35