14

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?

8
  • 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, 2017 at 1:20
  • @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, 2017 at 1:38
  • 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, 2017 at 13:29
  • 1
    @Laska: Sounds good to me.
    – supercat
    Dec 14, 2023 at 16:08
  • 1
    @Laska: It took me a moment to understand what the parentheses meant in the table. Instead of saying "implied by inequality" say "Implied by challenge 3". Also, am I missing something or would your first diagram satisfy R(3,A) if the white rooks were moved to a2 and c2, and the queen to g2? The last two moves could have been 1. Ke8 g6 2. Bf8 g5, with Black having been able to move the king back and forth as many times as needed prior to that. Because white's last move could not have been a double pawn push, the only way the game could be live would be if Black had moved last.
    – supercat
    Dec 14, 2023 at 17:38

3 Answers 3

15

EDIT 30-Dec-2023: added the R(1,D) position with 32 units. So Challenge 1 is now completely resolved.

Apologies that the first few lines are a bit theoretical. I reach the actual chess as soon as I can! :)

Living & dying moves

In a position, if there are no legal moves, then the game ends immediately: in checkmate (if check) or stalemate (if no check). Otherwise, there is at least one legal move. Then we can describe each legal move as either living (if there's a checkmate still possible after playing it) or dying (if there is no checkmate anywhere ahead in its game tree). Quality of play is not a consideration: all sequences of legal moves are valid.

Living & dead positions

Under FIDE Laws, the game ends immediately in Dead Position (DP) if there's at least one legal move, and all of them are dying. A position with a living move we call living.

So there are four kinds of position: checkmate, stalemate, dead & living. These are exhaustive & mutually exclusive. In particular, stalemate & dead position are disjoint. (This last point wasn't obvious to me at first, but clearly it's the intention of the FIDE Laws.)

Summary
A living move gives a living position or checkmate.
A dying move gives a dead position or stalemate.

Zombies

From a DP, the moves aren't actually played (except over-the-board by accident or from ignorance sometimes: see http://wismuth.com/chess/illegal-moves.html). Such legal but unplayable moves are called zombie moves. Similarly, a legal but unreachable position is called a zombie position.

Formally the game terminates as soon as DP is reached. There is no arbiter involvement needed. I have spoken with the Head International Arbiter at the time (Stewart Reuben) and the Rules Committee member who invented the DP rule (Igor Vereshchagin), who confirmed. When they designed the rule as a replacement for draw by insufficient material, they were aware of the potential for entertainingly complex positions, but reckoned (correctly I think) that almost all practical situations encountered would be very simple. The rule seems to have bedded in well in the last 25 years.

Zombie moves are hypothetical, because in all such positions the game has already finished. But in fact you need hypothetical moves, in order to explore the game tree and determine if a position is living or dead in the first place. So this "walking dead" concept is well-defined and it's also fun.

Avoiding circular definitions

Sometimes, people overload the term "illegal move" to mean "zombie move", as in Francois Labelle's page linked above. A lot of the time such usage doesn't matter, but it can be confusing. Since DP is defined based on the FIDE Laws concept of legality, any redefinition of legality would be circular. So I prefer the perhaps more pedantic terms introduced above. Note: this is just terminology, it doesn't affect the underlying logic.

The three challenges

OP proposed an excellent challenge for each of three increasingly restrictive categories of DP, rephrased as:

  1. any DP,
  2. DP, and some sequence of zombie moves can avoid stalemate indefinitely,
  3. DP, and no sequence of zombie moves can reach stalemate.

A difficulty is that solutions for 3. can have so many units that perhaps nothing new that can be added by loosening the game-end criteria in 2. To avoid overshadowing in this way, I suggest that we modify the categories, to make them mutually exclusive:

  1. DP, and all sequences of zombie moves end in stalemate.
  2. DP, and some, but not all, sequences of zombie moves end in stalemate.
  3. DP, and no sequence of zombie moves ends in stalemate.

(For these purposes, we can ignore Draw by Threefold/Fivefold Repetition & 50/75 move rules. A subject for another day, maybe...)

Types A, B, C & D

To construct a position with certain properties, it's often easier if one player is already in check, or if we're given who has the move. So retroanalysis enthusiasts often sub-divide Task Records into various types:

  • Type B = no check, but we are told who has the move (e.g. White here). (In theory, retro-logic might tell us who is on the move without being told: this is so-called Type A, but this doesn't seem applicable for this question. We would treat it as a special case of Type B.)
  • Type C = one player is in check.
  • Type D = either player might be legitimately be on the move. We require that both sides satisfy the category requirements (1., 2. or 3.)

Therefore we treat each of these Types separately here. Note that if R(n,X) is the maximum number of pieces for Category n with Type X, then R(n,B) >= R(n,D).

Summary of Results

What records have been achieved so far?

Record Modified Challenge 1 Modified Challenge 2 Modified Challenge 3
R(n,B) 32 ID01 (26) (29)
R(n,C) 32 ID02 29 ID04 30 ID06,ID07
R(n,D) 32 ID03 26 ID05 29 ID08

In this table, ID numbers reference the problems below. Any number in brackets is implied by R(n,B) >= R(n,D).

Are these records the best possible? R(1,B) = R(1,C) = R(1,D) = 32 are definitely unbeatable, but all the other records may well be improvable.

At last: some chess!

Here are all the constructions that drive this table:

[title "ID01 R(1,B)=32, White to move, A.Buchanan"]  
[fen "bqn1KN2/rrk1pB2/nb1pPp1p/p1pP1PpP/PpP3P1/1P2N1R1/4Q3/1R4B1 w - - 0 1"]

Note, with White to move there is no e.p. possible since Black's last move could not have been g7-g5. Bf8 would have been captured at home, but we know it's still on the board.

This position would not work as Type D because with BTM it's stalemate/living (depending on e.p.) with BTM. It wouldn't work as Type A because BTM is reachable.

See also two other capture-free positions that work as R(1,B).

[title "ID02 (R1,C)=32, A.Buchanan"]  
[fen "bqn1KN2/rr2pB2/nbkpPp1p/p1pP1PpP/PpP3P1/1P2N1R1/4Q3/1R4B1 w - - 0 1"]

This is just a minor variation, retracting 1 single move from ID01. The same can be done in both of the other two R(1,B) positions linked to.

[title "ID03 R(1,D)=32, A.Buchanan & J.Coakley"]
[fen "qrkn3n/rb1p1p1p/p1pPpPpP/PpP1P1P1/1P6/KRB5/BRN2N2/bQ6 w - - 0 1"]

BTM: 1. ... Bxb2+ 2. KQRBxb2=. WTM can't relieve the looming stalemate. 20+1=21 legal but unplayable moves.

[title "ID04 R(2,C)=29, J.Coakley, The Puzzling Side of Chess 127, 2016"]
[fen "rnb1K1Qk/rb1pN1p1/p1pP2P1/P1P5/5p1p/4pP1P/q3P1BR/5BNR w - - 0 1"]

Mandatory 3 captures yields position in which stalemate is optional.

[title "ID05 R(2,D)=26, J.Coakley, The Puzzling Side of Chess 127, 2016, version"]
[fen "rnb1K1k1/rb1p2p1/p1pP2P1/P1P5/5p1p/4pP1P/4P1BR/5BNR w - - 0 1"]

This is the position 3 mandatory single moves after ID04. It can be reached with either side on the move.

[title "ID06 R(3,C)=30, example 1, A.Buchanan"]  
[fen "qrn1KRRB/brk1pPN1/1p1pPp1p/1b1P1P1P/Pp6/1Pp5/2P5/N5bB w - - 0 1"]

One approach...

[title "ID07 R(3,C)=30, example 2, A.Buchanan"]
[fen "brn1KRRB/brN1pPN1/1bkpPp1p/p1pP1P1P/q1P5/1P6/1Q6/b4B2 w - - 0 1"]  

...and another, achieving the same record.

[title "ID08 R(3,D)=29, A.Buchanan"]  
[fen "qrn1KRRB/brk1pPN1/1p1pPp1p/1P1P1P1P/1p6/1Pp5/2P5/N5bB w - - 0 1"]

One move after ID06, but it can be reached with either player to move.

[title "ID09 Challenge 1, Type B, 31 units, White to move. Last move?, A.Buchanan"]
[fen "brnk1N1B/qnb1pBR1/rbKpP3/p1pP2NQ/P1P2p2/1p3P1p/1P5P/5R2 w - - 0 1"]

Bonus! Last move? Finally, here is a runner-up R(1,B) position in which additionally one can determine the last move. It costs a unit, so is not reflected in the table above, but nevertheless is interesting to illustrate the frequent involvement of en passant in critical DP situations. As with ID01, this can neither be Type A nor Type D.

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: a double pawn move by White is the only way the game could have reached the current position without dying already.

22
  • 1
    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 9, 2017 at 20:13
  • 2
    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, 2017 at 6:37
  • 1
    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, 2017 at 14:35
  • 1
    +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, 2017 at 9:31
  • 1
    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, 2017 at 23:37
12

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"]

*

5
  • Good work. I have experimented a bit and I think you can add Qg1, Rg2 and Rh3 for White (last sequence of legal moves, backwards: e3, Nf1, Qg1, Rg2, Rg1, Bh2, Rh3, Rh2, Bg1, Rg2, Rg1).
    – Annatar
    Sep 25, 2017 at 6:59
  • 2
    @Annatar with Rg2 it's not a dead position as black has fxg2. Sep 25, 2017 at 7:23
  • @RemcoGerlich Oh darn. I focused too much on White's move options, not Black's. Forget about it, then.
    – Annatar
    Sep 25, 2017 at 7:24
  • 4
    This is not a dead position, white can play Bg1, Nh2, Nxg4. Sep 25, 2017 at 7:24
  • 1
    Fixed the cook, with a promotion.
    – Herb
    Sep 25, 2017 at 12:33
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").

10
  • 3
    I like the trick for this 24! Sep 25, 2017 at 10:21
  • 2
    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 :-) Sep 25, 2017 at 10:29
  • 1
    The trick for 24 is fine.
    – supercat
    Sep 25, 2017 at 13:42
  • 1
    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, 2017 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, 2017 at 17:10

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