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Timothy and Kimothy were playing a rather intense game of Kriegspiel the other day. For those of you who don't know...

Kriegspiel is a version of chess in which the players are not allowed to see the moves of the opponent. Instead, there is a "referee" who keeps track of the each move on his own private board and tells the players if an attempted move is legal. If an attempted move is legal, the player must play it; otherwise, the player may try again. In addition, the referee announces all captures and whether the captured object is a piece or a pawn. In the RAND rules, the referee also announces checks, and possible captures by a pawn.

Blind Uncle Jimothy seemed to quite enjoy listening to them to play. Which led me to wonder...

Can a blind person, hearing only the announcements of the referee, deduce a mate for one of the players?

Clarification:

A referee would say "No" following an attempt by a player to make a move that appears legal only taking into consideration his pieces on his own board, but is in fact illegal according to the actual position as seen by the referee.

The RAND rules, consider this stricter definition of a ``No'' to be in force if the present announcement is a check of some sort.

(Note: This is not an original puzzle, It comes from UCLA professor Tom Ferguson's web page)

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  • $\begingroup$ If an attempted move is not legal, is this announced in a way Uncle Jimothy can hear? $\endgroup$ – hexomino Aug 24 '16 at 9:00
  • $\begingroup$ I don't see why he wouldn't be able to deduce such a thing considering that no one but the ref is seeing the board anyhow. $\endgroup$ – dcfyj Aug 24 '16 at 12:11
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    $\begingroup$ @Acerfire37 As I understand it, there is an important difference in that in Kriegspiel, each player does not know the other player's move so would not necessarily be able to picture the board. $\endgroup$ – hexomino Aug 24 '16 at 13:44
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    $\begingroup$ @Acerfire37 Each player could have their own copy of the board, containing only their pieces. They "announce" a move by playing it on their board, and the judge says "legal" or "illegal," followed by any other information listed, if it were legal $\endgroup$ – Sconibulus Aug 24 '16 at 13:51
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    $\begingroup$ In a game of kriegspiel, as Sconibulus says, the moves are not stated out loud. Our hypothetical blind person does not get told what the moves are; otherwise the answer to the question would be an obvious and boring "yes". The question is whether the very limited information "leaked" by the referee could ever be enough to deduce that one or other player could give mate. $\endgroup$ – Gareth McCaughan Aug 24 '16 at 14:03
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Influenced by Gareth's answer I believe the answer is:

Yes - in some situations. For example: During the course of a game, the referee announces the capture of all white pieces and pawns except for the queen, and all black pieces and pawns except for one (any) remaining black piece/pawn. Then the referee announces capture of the final black piece/pawn. Then referee announces that it is White's move or that Black has just played a legal move (the question doesn't explicitly mention if the referee says anything after a legal move but presumably he/she does or the other player would not know it was their move). At this point Uncle Jimothy knows B has just a king, W has a king and queen and B has not already been stalemated or checkmated. At that point he could announce "checkmate in 10 moves or less" (see GM Fine, Basic Chess Endings, p. 1).

Note that this doesn't rely on any assumption that White can win from the starting position.

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  • $\begingroup$ The famous boring but technically correct answer $\endgroup$ – ffao Aug 25 '16 at 4:17
  • $\begingroup$ Wait, how does our blind listener know that it's a queen white has? What exactly should we assume is being announced about captures? (I thought the rule was that when a capture occurs the capturing player knows s/he has taken something but not what, and the captured player knows that a particular one of their pieces is gone but not what took it.) $\endgroup$ – Gareth McCaughan Aug 25 '16 at 12:16
  • $\begingroup$ Ah, but I do see a way to know that someone has a queen. First of all everything gets captured except one of their pawns. Then the other player gets checked "on the horizontal", which can only be done by a rook or queen. Then they get checked "on the (long/short) diagonal". Now we know that the other player has a queen. $\endgroup$ – Gareth McCaughan Aug 25 '16 at 12:17
  • $\begingroup$ I don't think KQ v K is as quick as 10 moves when you're playing kriegspiel rather than chess. (KR v K is, I think, 41 moves in the worst case, versus 16 for chess.) $\endgroup$ – Gareth McCaughan Aug 25 '16 at 12:22
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This is a cheeky cheaty answer, but

if this puzzle is known to be solvable, then the answer must be that it is possible to "deduce a mate"

because

for all anyone knows, the initial position of a chess game may permit White to force mate with best play on both sides. A proof that this can't be done would be a staggering feat of chess calculation and we would certainly have heard about it by now.

But this may be addressing the wrong question. What does "deduce a mate for one of the players" actually mean in the question?

  • If it means "deduce that one of the players, in an ordinary game of chess, could force mate" then the argument above works (though, again, it's a bit of a cheat) but having a forced mate is not necessarily of any actual use in a kriegspiel game, because executing the forced mate may depend on knowing what moves your opponent makes.
  • If it means "deduce that one of the players can force mate in a kriegspiel game, without knowing what moves the other player is making", then the argument above doesn't work.
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  • $\begingroup$ Are you saying that the initial position isn't the standard black on one side and white on the other side? $\endgroup$ – Acerfire37 Aug 24 '16 at 15:12
  • $\begingroup$ No. The initial position in kriegspiel is the initial position for a standard game of chess. Therefore, if in a standard game of chess white always wins with perfect play on both sides (which, for all anyone knows, might be true, though I think most people would guess it probably isn't), then our hypothetical blind person could say right at the start "White has a forced mate in 63 moves" or whatever. The kriegspielness of the game wouldn't come into it at all. $\endgroup$ – Gareth McCaughan Aug 24 '16 at 15:16
  • $\begingroup$ Though ... actually I'm not sure what counts as a forced mate in kriegspiel, now I come to think of it. I might be able to prove that I could deliver mate in 5 if we were playing ordinary chess, while still not being able to win in kriegspiel except by good luck. $\endgroup$ – Gareth McCaughan Aug 24 '16 at 15:17
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My answer is:

Generally, no.
In some cases (see @Penguino's answer) the mate is obvious and can be deduced. But in the other cases it can be completely unpredictable. Take the famous Scholar's mate, for example. Three legal moves by both sides followed by a pawn capture by white queen that leads to mate. The same announcement chain ("yes[x6] - yes, pawn capture, check") can be produced in a lot of different ways, but only some of them lead to mate, so when the referee follows "check" by "mate", it couldn't be deduced from the announcements.
Or we even can look at Fool's mate, though it's highly unlikely. "yes[x3] - yes, checkmate". Boom! Totally didn't see it coming.

Actually,

I don't really see any positions when the mate can be logically deduced. Even in some "really obvious" situations, for example, if whites managed to capture all black pawns and figures without losing a single pawn, it's still theoretically possible (will never happen in any real game, but doesn't break the rules) to make a draw by "feeding" figures one by one to the black king until whites also have only the king left.

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  • $\begingroup$ Yes, the player can throw away the game and not capitalize on the mate. But to me, deducing mate means recognizing the opportunity is unavoidably there. $\endgroup$ – knrumsey Aug 25 '16 at 20:23

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