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A pure checkmate is a checkmate where for each square the king might go to, there’s exactly one reason it can’t. The famous Légal mate is one example, so is a KR vs K checkmate.

What’s the minimum number of pieces in a position, reachable from the starting position by a sequence of legal moves, that’s a doubly pure checkmate, that is, the king can move in neither of the eight directions nor can it remain in place, where each of the nine options is forbidden for exactly two reasons?

A square being attacked by one enemy piece counts as one reason, so is being occupied by a friendly piece. No special treatment for pinned pieces, which (if adjacent to the king) contribute one reason for being there and another for being attacked. For the purposes of the puzzle, let’s consider being unable to move off the edge of the board one reason, so the king being mated can’t be located on the edge.

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I can do

5 pieces (2K,3Q)

and this is clearly optimal, because

1 of the 5 pieces must be the black king. One must be the white which can only cover 3 squares leaving at least 15 for the remaining pieces. As obviously no piece can block more than 6 squares (a queen next to the opponent's king does 6) no fewer than 3 more pieces are needed.

Picture of solution.


enter image description here
[FEN "Q2Q2/4k1Q1/8/3K4/8/8/8/8 w - - 0 1"]
The mating move would have been a pawn promotion: f7-f8=Q#

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    $\begingroup$ Nice pattern. And it can actually be forced in a game on someone who stubbornly doesn’t resign, starting with something like 8/3kNPQ1/8/2QK4/8/8/8/8. $\endgroup$
    – Roman Odaisky
    Commented Jan 24, 2021 at 14:20
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    $\begingroup$ This is not a valid position. Consider black's last move--what was it? All places the king could have come from are under fire and white's last move wasn't a capture so black's last move has to be of the king. $\endgroup$
    – Loren Pechtel
    Commented Jan 24, 2021 at 19:24
  • $\begingroup$ @RomanOdaisky Nice, Didn't even think about that. $\endgroup$
    – Paul Panzer
    Commented Jan 24, 2021 at 19:41
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    $\begingroup$ @LorenPechtel "Consider black's last move--what was it? All places the king could have come from are under fire" What you are describing is the black king being in check and moving out of it. Which is perfectly legal. $\endgroup$
    – Paul Panzer
    Commented Jan 24, 2021 at 19:43
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    $\begingroup$ @LorenPechtel Take the FEN I posted above and play Qc8+ then f8=Q#. $\endgroup$
    – Roman Odaisky
    Commented Jan 24, 2021 at 19:44

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