11

It is possible in a chess game to have a dead position in which both players have all eight pawns, one bishop, and a king--20 pieces total. If the two players' pawns are interlocked in "zig-zag" fashion and each player's bishop is the same color as his pawns, each player's army will be forever stuck behind his own wall of pawns, with no way to ever reach anything on the other side.

[FEN "4kb2/8/1p1p5/pPpPp1p1/P1P1PpPp/5P1P/8/4KB2 w - - 0 1"]

What is the largest number of pieces that can be on the board in a legally-reachable position such that either:

  1. The side on move has at least one legal move, but no sequence of legal moves would produce checkmate.

  2. It would be possible to play an arbitrary number of legal moves, but no sequence of legal moves would produce checkmate.

  3. No legal sequence of legal moves would produce checkmate or stalemate.

I would guess that the first of those might be possible with as many as 28 pieces on the board, but most escape-proof positions rely upon immobilized kings to block opposing pawns, making it hard to allow anything other than pawns to move safely. What are the actual limits?

Rewan Demontay
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supercat
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  • What exactly are you looking for? Are positions where the pieces are arranged such that each side can only move one piece back and forth good enough? Or do the pieces need more "freedom"? If so, how much more freedom? (Such almost-stalemates are pretty easy to construct, although maybe there would not be proof games if both sides have 16 pieces.) – TMM Sep 25 '17 at 01:20
  • @TMM: Positions where pieces are limited to moving back and forth would be fine, if they are reachable via sequence of legal moves. – supercat Sep 25 '17 at 01:38
  • You are asking three different questions, you have to delete it and post three separate questions. The questions are all so totally unrelated to each other that they must be posted on different sites in different languages at different times. The whole 1, 2, 3 thing makes me so very confused that we have to chat about it here for a long while now. I know that you know that feeling! – LocalFluff Oct 14 '17 at 13:29
  • @LocalFluff: Before asking the question, I didn't know whether the answers to #1 and #3 would be different. While it seems unlikely that #3 would be possible with all 32 pieces on the board, I still don't know if the best position for #2 would guarantee that no stalemate would ever be possible. If it wouldn't, then any answer for #3 would also hold for #2, making #2 redundant. – supercat Oct 14 '17 at 15:16

3 Answers3

11

I've come up with 23, in the following position, with promotions: 

[Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "?"]
[Black "?"]
[Result "*"]
[FEN "nb2k3/2p5/1pP5/1Pp5/2P1p1p1/3pPpPp/3P1P1B/4KNBN w - - 0 1"]

*

Herb
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11

Funny task. I think I'm still pretty far from the maximum, but here is a suggestion with

23 men:

[Event "?"]
[title "Challenge 3, 23 units"]
[FEN "NRN1k1bn/QRKpPp2/PPpP1Pp1/2P3Pp/7P/8/7B/8 w - - 0 1"]

With black to move, an almost-dishonest trick to reach 25:

[title "Challenge 3, BTM, 25 units"]
[FEN "BRN1Nkbn/QRKpPp2/PPpP1Pp1/2P3Pp/7P/8/1r6/B7 b - - 0 1"]

27 with promoted units:

[title "Challenge 3, Check, 27 unit, including promoted"]
[FEN "QNBk1bnr/RBpPpKbr/PpP1Pp1p/1P3P1q/6P1/8/1R6/b7 w - - 0 1"]

explanation: This is a dead draw because after the forced moves 1.gxh5 Bxb2, White could only play his king back and forth on f7 and g6 while Black's bishop explores the bottom half of the board. wRb2 could be replaced by a wB or a wN (or a wQ with obvious adjustment of the nature of the upper-left-corner stranded pieces), or even by a fourth bB while preserving legality. This is so many "degrees of freedom" that I strongly feel that 27 is not the maximum, and someone will soon come up with (at least) 28...

edit: indeed, Laska just scored a pretty 28 (or even 30 with "the trick").

Evargalo
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  • Your '24' needs promoted units too (Black has two black-square bishops) – AakashM Sep 25 '17 at 09:55
  • Oh, you're right of course. – Evargalo Sep 25 '17 at 09:56
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    I like the trick for this 24! – RemcoGerlich Sep 25 '17 at 10:21
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    Wow, it seems the 26 is even legal, black a-pawn captures twice to promote on a1 or c1, black d-pawn captures once, accounts for all white's missing pieces. The g-pawn may have promoted on g1 and got there because white's g-pawn captured twice on its way to g8, which happens to be white. Nice :-) – RemcoGerlich Sep 25 '17 at 10:29
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    The trick for 24 is fine. – supercat Sep 25 '17 at 13:42
  • What's happened to the 26 position? It seems to have become a non-solution since White is checkmated. – supercat Sep 25 '17 at 13:53
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    I think some of these violate the terms of the question: "It would be possible to play an arbitrary number of legal moves" and "No legal sequence of legal moves would produce checkmate or stalemate". – D M Sep 25 '17 at 16:30
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    @DM: There are three questions in one, which might or might not all have the same best answer. A situation where the player on move would have no legal move that didn't force stalemate would be a valid answer for the first. A situation where two players repeat the same sequence of four positions endlessly would qualify for any of them, since in the absence of a rule about draws by repetition, that sequence could be played arbitrarily many times, and in the presence of such a rule even the starting position would eventually lead to a draw by repetition if the game didn't by other means first. – supercat Sep 25 '17 at 17:10
  • @OlivierPucher: It might be helpful to add annotations showing how things would play out until the repeat. I got tripped up because your previous position had black to play, so I was looking for Black to do likewise on the last one. – supercat Sep 25 '17 at 17:12
  • @supercat I missed that it says "either". OK. – D M Sep 25 '17 at 17:13
8

The three dead position challenges are (1) general (2) no finite mandatory stalemate (3) definitely no stalemate possible ever. (3)=>(2)=>(1).

Retro enthusiasts often sub-divide Task Records into 3 types:
- Type A = no additional information in the stipulation
- Type B = you are told who is on move (e.g. White here), but there is no check.
- Type C = one player is in check.

The posts so far have focused on OP's challenge (3). Here are a couple for challenge (1): 

[title "Challenge 1, Type B, 31 units, Last move?"]
[fen "brnk1N1B/qnb1pBR1/rbKpP3/p1pP2NQ/P1P2p2/1p3P1p/1P5P/5R2 w - - 0 1"]

White to move. Last move?

Black's last move must have been b4-b3. If the prior position was dead, the game would have already terminated. So White's move before that was a2-a4 or c2-c4, and Black chose not to make the en passant capture, which would have kept the game alive. No en passant convention is required here: White's double pawn move is the only way the game could have reached the current position without dying already.

This position was built many years ago, to force a unique last move. Without that additional goal, it's easy to make a 32 unit dead position. 

[title "Challenge 1, Type B, 32 units"]  
[fen "bqn1KN2/rrk1pB2/nb1pPp1p/p1pP1PpP/PpP3P1/1P2N1R1/4Q3/1R4B1 w - - 0 1"]

White to move.

Moving on to challenge 3, Olivier Pucher had 27 pieces, Type C. I offer:

[title "Challenge 3, Type A, 29 units"]  
[fen "qrn1KRRB/brk1pPN1/1p1pPp1p/1P1P1P1P/1p6/1Pp5/2P5/N5bB w - - 0 1"]

and:

[title "Challenge 3, Type C, 30 units"]
[fen "brn1KRRB/brN1pPN1/1bkpPp1p/p1pP1P1P/q1P5/1P6/1Q6/b4B2 w - - 0 1"]
Laska
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    The difference between #2 and #3 is that a game of #3 could not end except by draw-by-repetition, draw-by-N-moves, or other such rule that is not based purely upon the position of pieces on the board, side on move, etc.. By contrast, #2 would allow for the possibility that players could end the game with a stalemated position. – supercat Oct 09 '17 at 20:13
  • Looking at the diagram, I can see five potential last moves for Black (b3, d6, f4, h3). I don't see any particular reason Black couldn't have had a pawn on b4, d7, f5, or h4 prior to the last move. If the f8 knight were on g6 and the f7 bishop on g8, Black could temporize indefinitely with Ke8/Kd8 and White with Ra1/Rf1 with pawns in any of the aforementioned positions. – supercat Oct 09 '17 at 20:27
  • Thanks supercat for clarification on 1,2,3 I will edit the post - I knew I was missing something. There are apparently three possibilities for last move (b3,f4,h3). (d6 had impossible prior check on wK). But the fact that we are in dead position.has retro implications. The answer is a bit sneaky. – Laska Oct 09 '17 at 22:57
  • I hadn't been asking for that retrograde wrinkle, but I must say it's quite cute if one adds the wrinkle that the position must have been rendered dead by the last move. I don't think the laws of chess require that play stop as soon as a dead position is reached, or would require that an arbiter recognize whether a position is dead except in cases where the game is over for other reasons (e.g. because a player's flag has fallen) and the question is whether to score a full or half point. Still, definite +1 for the retro puzzle and the 32-piece version. – supercat Oct 10 '17 at 05:58
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    Thanks very much! Actually the laws of chess do say that a dead position game ends immediately. It's as sudden as checkmate or stalemate. Francois Labelle on his website wismuth.com has a list of games where players carried on going for a couple of turns before they realized! (Nit: he incorrectly terms such impossible moves "illegal".) But for formal chess problems, a recent 2015 convention says dead position, like 50 moves, only applies by default to retros (to protect e.g. self-stalemates). Most dead position problems are retro, so it's no issue. In others, as here, the condition is implied. – Laska Oct 10 '17 at 06:37
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    If a position is dead, then *by definition* pretty much nothing the players could do would affect the outcome *whether or not the game was recognized as "dead"*. The only situation I can think of where there could be an issue would be if someone resigned a dead position and later realize it was dead, and could plausibly claim the resignation was not deliberate bad sportsmanship (since bad sportsmanship might justify a forfeit even if the game would otherwise have been drawn). Otherwise, I would think that the official scores for games would include all the moves that were played. – supercat Oct 10 '17 at 14:35
  • I agree. It replaced and generalized the old "draw by insufficient material" FIDE rule. I don't know why they felt this was necessary, but it was a gift to retro composers. It does make sense that no claim is needed. 50 moves & 3 rep are complicated and require arbiter verification. But deadness is usually simple to see. I have also heard that "no claim" simplifies the online world, where a game just ends immediately when e.g. KNvK is reached. – Laska Oct 11 '17 at 03:12
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    +1 for th beautiful 28-men dead draw. That will even be 29 if you force the move for black and add a wQb2... – Evargalo Oct 11 '17 at 09:31
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    I wanted to see if I could get at least 2 more units by moving to Type C. I have just found such a position with 30 units, where 3 forced captures take place at the beginning. For the first time, I am not confident that this can be improved. – Laska Oct 11 '17 at 23:37
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    Realized oh so much later that I could find a place for wN in the 3rd position. Missing pieces count is exactly what's needed to explain the pawn captures: very lucky – Laska Feb 18 '18 at 13:18