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:
- any DP,
- DP, and some sequence of zombie moves can avoid stalemate indefinitely,
- 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:
- DP, and all sequences of zombie moves end in stalemate.
- DP, and some, but not all, sequences of zombie moves end in stalemate.
- 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.
