by Andrew Buchanan

Compositions referenced in the text are to be found in the following files:
{A} to {I}
{J} to {R}
{S} to {W}

I thank Gianni Donati, Noam Elkies, Joost de Heer, François Perruchaud, Ronald Turnbull & Lynn Yarbrough for their time and encouragement.


Dead Reckoning (DR) compositions exploit Article 1.3 of the FIDE Laws of Chess (A1.3), which states: "If the position is such that neither player can possibly checkmate, the game is drawn." This is known as a dead position. Surprisingly, a fertile new genre of chess composition derives from this apparently dry rule. [1]

The paradigm example of DR is {A}, an obviously dead position. Now suppose Black just moved. It wasn't 0…Kb8-a8 because the prior position would already have been dead, and the game over, before the current diagram could be reached.

So Black just captured a piece. Moreover, the piece was a major piece, because king + a minor piece cannot mate even a co-operating opponent. What about 0…Kb8xRa8 for example? This is the nub of it. In the prior position, the rook capture was forced, leading inevitably to the bare kings position. Hence it was correct to say in the prior position that neither player can possibly checkmate. Thus the game would have stopped dead prior to the rook capture.

Likewise Black did not just capture a queen, nor move from a7. That exhausts the possibilities, so in fact White not Black moved last, and the White king captured a pawn or major piece on c6. There are 7 squares that the White king might have occupied prior to the capture, and from any of them, instead of making that capture, the White king could simply have moved away somewhere else. The prior position was not dead, and the game could have ultimately ended (if the players so chose) in a Black victory. So White it was that moved last. End of Solution.

This position can equally be captioned: "Whose move?" In a position which somehow terminates the game, it is still meaningful to specify who has the move, even though no more moves can actually be played. In fact it's mandatory to specify this. Whether the game has terminated (by checkmate, stalemate or death) is determined by examining the legal moves. [2] One can only say which moves are legal if one knows whose has the move. By the Laws, a player is said to "have the move", when his opponent's move has been made. [3]

To date, half a dozen individuals have designed over 150 DR compositions. Most are Retrograde Analysis, although Ronald Turnbull discovered how to apply the concept to orthodox non-retrograde problems. [4] The current article focuses on the application of DR to those retro perennials: castling and en passant.

How does DR interact with the chess composition conventions for castling and en passant? For most of the problems in this article, the answer is simple. The conventions are designed to apply in situations where there is irreducible uncertainty as to whether the move is legal. But all the problems here are specifically designed for DR itself to remove the uncertainty. So these conventions are never triggered. It may be possible to devise compositions in which the interaction between Article 1.3 and the conventions are much more subtle, but those will be saved for another occasion. [5]


In orthodox retro compositions, it is impossible to prove that castling is legal. Any proof game that preserves some castling right may be prefixed by a sequence of rook and knight moves to specifically disable that right. However in Dead Reckoning, that limitation does not apply.

Consider {B}. Unless White can castle, stalemate looms, and the position is dead. For suppose White cannot castle. Black's last move was either 0…Kh2xNg1 (following -1 Nh3-(x)g1+) or 0…Kh2-(x)Mg1 (following -1 h3xg4+), where M=R/N/0 to avoid an impossible double check. In any case, Black's move was forced, and hence illegal under DR. So White definitely can castle and 1 0-0-0# avoids stalemate. However, since the White king never moved, the White queen on h5 is not original. This accounts for all the White units, so Black's last was 0…Kh2-g1 without capture. Also check that the position can unwind legally, with Black having played ...Kh3-g2 earlier, prior to White's h2-h3. End of solution. [6]

We searched for other positions where DR determines castling or e.p. availability. Twelve tasks were defined, assuming White to move:

Code Position Alive/Dead? Type of Conditional Move? Is the move permitted? Examples
DWY Dead White Castling Yes {C}
DWN Dead White Castling No {D}
AWY Alive White Castling Yes {B}, {E}, {V}, {W}
AWN Alive White Castling No {E} again
DBY Dead Black Castling Yes {F}
DBN Dead Black Castling No {G}
ABY Alive Black Castling Yes {H}
ABN Alive Black Castling No {I}
DEY Dead En Passant Yes {J}
DEN Dead En Passant No {K}
AEY Alive En Passant Yes {L}, {I} again
AEN Alive En Passant No {L} again

Alive and dead positions present different challenges to DR design, so are distinguished. As with the "whose move" question, it is meaningful to specify castling and en passant availability at game-end. At each move, it is necessary to determine whether the game has terminated by checkmate, stalemate or death, and this can only be done by examining the legal moves.

When considering castling, we ask which castling possibilities have been permanently disrupted by prior king or rook move. This is the notion of castling rights which underpins the notion of position in the Laws of Chess. [11] It is meaningful to ask this even if (as in many of the diagrams) an intermediate piece currently blocks the castling, or even if there is no way that castling could ever take place. [12]

On the other hand, the definition of en passant just depends on whether the capture can actually be executed.

Each of these compositions has a unique last move. Relaxing this requirement yields a separate set of tasks. For example, in {G}, remove bBf8, bPf7, bPg7 & bPh7. The minimal known position where DR allows a definite statement to be made about a conditional move is {M}. [11]

Some compositions appear twice in the table. An unavailable conditional move can almost never imply that the position is alive; to do that a second, available conditional move is required, as in {E}, {I} & {L}. An interesting exception is {N}.

It is also possible to construct paradoxical positions where DR can resolve the availability of a conditional move, but we (unless we have recourse to the conventions) we cannot know whether the position is alive or dead. See {O}, which has the code ?BN.


{P}& {Q} are of a different species.

The previous section considered conditional moves taking place forward in play from the diagram. Note however that in {C}, castling is legal in forward play only as a side effect of being legal in the "ghost play" following the retraction of the last move. Ghostly doings were also important in {D} & {K}. In {P} & arguably {Q} the play is "all ghost", as there is no question of a forward conditional move at all.

Ghost play in DR is somewhat analogous to set play or try play in orthodox problems: it is an alternative avenue to exhibit interesting functionality. If the diagram is dead, the composer may tacitly attempt to render the ghost play as coherent as the setting permits, without adding any extra material. In the simplest form, a unique ghostly line avoids the draw (e.g. {C}, {F}, {P} & {U}). But other compositions may ghostily contain a help mate or even direct mate (e.g. h#1 in {D} or h#1 in each twin of {R}, or #2 in {S}). [12]

Leaving ghosts aside, there are other ways that conditional moves can appear, e.g. in one of the solutions to {R}. It seems difficult to create problems of this "backwards" form that involve castling: {S} is a rare exception.

A historical factoid: DR has been part of the Laws of Chess since 1 July 1997. {T} is my response to a composite orthodox/DR task proposed by Ronald Turnbull. The point is that the en passant convention applies to the orthodox half of the problem only. Ronald's fine idea surely deserves a deeper implementation than my somewhat light proof-of-concept. Perhaps some reader can supply one.

And finally, how might one view the putative en passant on g3 in {U}?


An early strategic decision in DR design was to avoid too much traditional retro logic, so that the DR content was not drowned. However it is non-trivial to extend DR techniques to include state-of-the-art retro concepts such as retro-knots, retro-sequences and tempo.

To illustrate the challenges that come up, consider a retro-knot, i.e. a configuration of self-retro-blocking units. In an orthodox retro composition, opposing units in a retro-knot may block or pin one another, and units may be retro-immobilized by the threat of an impossible check (e.g. bNa1 retro-immobilized by wKd4). But in a DR position, forward play must lead to a draw, and the tricks mentioned usually allow some checkmate. So DR retro-knots are more limited than in orthodox problems, generally relying on friend blocking friend, fixed trellises of pawns, and knowing who moved last. The kingsides of {F} & {O} are typical examples.

Some nits on the "twelve tasks" presented in the table above. It would be nice to find check-free alternatives to {H} & {I}, and also to avoid the obtrusive bishop found in {F} and {K}. I would welcome suggestions here.

We can also ask which logical combinations of multiple conditional moves can be realized under DR. We can explore the interaction with the conventions covering castling & en passant. The neuron-knotting interaction of DR with AP also remains to be investigated.

The exploration ahead will show how far DR can lead to novel, lively and amusing chess problems.


{C} Dead position. All 16 White pieces remain, so bPh5 has not just come from g6. To avoid retro-death, last move was 0 …Ph7-h5 with alternative 0 …Ph7-h6 1 0-0 h5 2 Ra1#. So castling is legal.
{D} Dead position. All White pawns remain, so both White rooks are original. White's only missing piece is a black-squared bishop; hence bPg6 has not captured to reach this position. 0...g7-g6 has no alternatives, so is not a possible last move. The last move can only be 0…f4-f3 with alternatives 0…fxg3ep. or 0…fxe3ep. Either way, White's kingside castling rights are disrupted by necessity for wQR to reach f1 or h1. Note: a & b pawns never cross-captured, because of wBb1.
{E} Dead unless 1 0-0 is legal (after which 1 …Kxe2 2 Bc4+ Ke3 3 Ra3#). Last move was 0 …Ke4-e3 (forced), so kingside castling is legal. A White pawn captured a Black rook's pawn, which must have promoted (Black has made no captures in this game). So a White rook moved: to allow the promotion. By DR, the king's rook did not move, so by elimination it was the queen's rook that did move.
{F} Dead position. The White f-pawns come from files c through f, making 6 captures, including Black's e-pawn. So White b-pawn promoted without capturing. bPa3 or bPa5 captured onto the a-file prior to this promotion. bPf3 is original g-pawn. So Black didn't just capture. Stalemate was inevitable unless last move was 0 …Pa4-a3 and castling legal. Then 0 …0-0-0 1 a3 Rg8#.
{G} Dead position. Black's last move was 0…Pa6-a5 (alternatives dead), 0…Kd8-e8 (forced), 0…Kd8xNe8 (forced), 0…Rg8-h8 (alternatives dead) or 0…Rg8xNh8 (alternative living, so by elimination this was the last move, and castling is illegal.) Note: 0…Pb6xa5 retro-blocks wK.
{H} Dead unless 2 0-0 legal after 1...Kxe6. Black just played 1 d7-d6+ (alternatives dead). So castling is legal. The full solution is 1...Kxe6 2 0-0 Kxe7 3 Kh7 Ke6 4 Rg8 Kf7 5 Rg5 fxg5 6 Rh8 g6#.
{I} All missing Black units were captured by White pawns. If Black's last move was the single step, then with no DR, Black could castle, since the g & h pawns can cross-capture to release the king's rook. However the prior position is dead. So Black's last move was the double step, and e.p. is legal. bK must have moved earlier to release bKR.
{J} Dead position. All candidate last moves are forced except for 0…Pb7-b5, which has living alternative 0 …Pb7-b6+.
{K} Dead position. bB is promoted f-pawn. Original Black e-pawn is now bPb4, so 0…a5xb4 didn't happen. By pawn capture parity, Black f, g & h pawns never captured. So candidate last moves are 0…Ph7-h5 (dead), 0…Ph6-h5 (dead) and 0…Pg4-g3 (alternative 0…Pg4xPh3ep!)
{L} If no e.p., White's next move is stalemate. 1 a5xb6ep cxb6+ 2 Kxb6= but 1 e5xd6ep exd6#. (1...cxd6+ also is not dead.) Candidates for Black's last move are 0...axb5 or 0...b7-b5 or 0...d7-d5 (not 0...d7xe6 because this implies 5 Black pawn captures). The alternative 0...c4xd3ep is illegal (0 d2-d4 is impossible because White pawns could never have reached the current configuration). 0...axb5 is singular (= "has no alternatives"), while 0...b7-b5 has alternatives only 0...b7-b6+ 1 a5xb6 c7xb6+ 2 Kxb6= or 0...b7xc6 1 ~=. So Black's last move was 0...d7-d5.
{M} Dead position. Alternatives to 0…Pf7-f5 are dead so e.p. illegal.
{N} If e.p., is legal: dead draw. But then last move was 0 Pb2-b4, and the only alternative, 0 Pb2-b3, is also dead. So e.p. is illegal, and the position is mate.
{O} Note that bPg6 is the original g-pawn (to allow bBg8 to leave) so bPg5 is original e-pawn (to allow bBh6 to escape), and bPf3 is original h-pawn. With bPd6 as original c-pawn, all Black captures are accounted for. The retro-knot releases by cxd6 allowing the wK to escape, then bPg5 comes from f6. So last move was 0…f4-f3 or 0…a7-a5 or 0…a6-a5. The first of these has no living alternative (even 0…fxg3ep=). But the second is living since 1 b5xa6ep f2 2 a7 f1=Q/R#, while the third (where the position is dead) has the alternative 0…f2 1 b5xa6 then e.g. 1…f1=Q/R#. In either of these two cases, Black cannot castle because the bK must have moved to release the bQR.
{P} Dead position. White made 5 pawn captures. So Black has not just played 0…Pe7-e5. Black made at least 5 pawn captures, but not White g & h pawns which never captured. Not 0…gxf5 nor 0…fxe4 (too many captures). Not 0…Pe6-e5 (Pe6xf5 earlier captured White light-squared bishop). Not 0…Ke7-f8 (impossible check). Candidates are 0 …Pf5-f4, 0…Rg8-h8, 0 …Kg8-f8, (all alternatives dead) or 0 …Ke8 (with alternative 0 …0-0 1 c4 Rd8 2 cxd8 etc). So the Black king never moved until 0…Ke8-f8.
{Q} The position is dead, since White cannot avoid stalemating Black, even after 1 Ng2 hxg2. Black's last move was 0…h4-h3; by DR this cannot have been forced, so 0…hxg3ep+ was legal. This leads to 1 Kf1 gxh2 2 Nd3 hxg1N 3 Nf2#.
{R} 1 K/Qxc3= looms. Black's last was 0...c4-c3+ or 0...b4xMc3+. The former is only non-singular following 0 d2-d4, when 0...c4xd3ep allows 1 Bxb3#. The latter is only non-singular following 0 c2-c4, when 0...d5xc4 allows 1 Qe8#.
{S} Black to move, else retrostalemate soon. Position dead. All 15 Black units were captured by White pawns. White's last move was not a capture (0 a2xb3 would retro-block the Black a-pawn, 0 g6xh7 would uncapture the unmoved h-pawn, leading to retrostalemate). Without DR, 0 Be8(f7,g6)-h5 Kh4-h3 -1 Pg2-g3+ K?-h4 unwinds the position. But 0 Bh5 has no alternatives which lead to life. 0 0-0 unwinds with 0…Kg2-h3, and has living alternatives.
{T} (a) White's last move could have been 0…Pa2/3-a4, so e.p. is not necessarily legal. By convention, e.p. is not permitted. The chase to the draw is 1 f5 a5 2 f4 a6 3 f3 a7=. (b) Time passes, and DR now applies. The current position is dead (even after 1 f6) unless en passant is legal, so the earlier solution will not work. 0…Pa3-a4 has no living alternative. So White did play 0…Pa2-a4 and the en passant capture is on. 1 bxa3ep bxa3 2 b2 Bxb2 3 f6 and the game is still alive, but now White can play 3…Bxf6=.
{U} The g3e.p. question is a red herring, I'm afraid. Naughty of me to have hinted otherwise. If Black's last move was 0…Rd5-(x)Mb5++, what is M? White is missing Q, P. The White c-pawn never captured or promoted, and if M=Q, then the wQ moved in without capture, so Black was already in check from wB. So M=0, and the move was forced. Black must have played 0…b4xc3ep++, and had the alternative 0…b4xNa3+. White can avoid stalemate only by 1 BxRb5#, defended by the pinned wPc4. Black could not have played 0…hxg3ep, because White's last move was known to be 0 c2-c4, prior to which -1 Rd5-b5+ Nb5-a3+.
{V} The position if legal is dead unless castling is on. Black's last move was 0…Kh1-(x)g1, and forced, so by DR castling is legal. If White's last was -1 Pg2-g3, the Black king could only have reached its position by dislodging the White king. So White's last was -1 Rg2-(x)g1. If the White rook captured any piece here, then Black has no prior move. Note that earlier, the White rook must have captured on g2 or f2 to avoid perpetual retrogression.
{W} By DR, castling is legal, and 1 Ndxf2 d6 2 0-0-0#. If Black played 0…g3xBf2+, then there were 12 White pawn captures, but also the White bishop captured on f2. However the Black a-pawn did not capture, or promote. Contradiction. So Black's last was 0…e3xBf2+. Now the Black g & h pawns were not captured by pawns either, so before came -1 Bg3-f2+ e4-e3.

Version 1.0 Originally published in StrateGems 16, Oct-Dec 2001.
Version 1.1 Published here 4 Dec 2001, correcting a very few typos [11], & with minor formatting changes.
Version 1.2 Published here 19 Oct 2003, with substantial text revision, including the stipulations, but the diagrams & problem content (except in one marked case) have not been tinkered with.[12]
Version 1.3 Published here 24 Oct 2009, changing to point to latest version of the Laws.
Version 1.4 Published here 2 Feb 2013, with many text changes, as an exercise to get back into the swing of writing in HTML and about chess.

[1] This has been a rule of chess since 1997 and appears in Articles 1.3, 5.2.b, 6.9, 7.4.b & 9.6 of the 2009 Laws. I don't think any rule appears more often, in what is otherwise a relatively concise document!
[2] See articles 1.2, 5.1.a, 5.2.a. Note that every stalemate is a dead position. One might also try to argue that every dead position is a stalemate, asserting that the termination of the game means there are no legal moves. I don't agree, because this would imply that there are two points in a move at which the legality of moves are determined: first at the beginning of analysis and then after the result of the game is decided, when all moves are impossible because the game has been terminated. But that doesn't make these moves illegal. So there are three mutually exclusive states of a game: alive, checkmate & dead. 
[3] Article 1.2. One might argue that White does not have the move at the start of the game! This tiny oversight is irrelevant to DR, but it would be nice if it were fixed some time.
[4] Ronald Turnbull "Dead Reckoning" The Problemist, 7/2001. See also {T} in this article. His other retro DR problems include the highly efficient {V}.
[5] When I wrote this originally, I was confused about the conventions, largely because they are so badly explained in the Codex. So earlier versions of this document took a different stance, stating that: "For simplicity, when using DR, the orthodox castling and en passant conventions are suspended." It was almost as if DR was a fairy convention (which it isn't - it's the way the Laws of Chess work). I now think that the conventions will only trigger once all other reasoning is exhausted. And for all the problems here, they never do trigger, because the DR logic answers the questions. Equally, I no longer consider it's necessary to label DR compositions as such in the stipulation. These are mainstream, not fairy, compositions.
[6] The first publication of a position where castling can be proved to be legal was in the first version of the tutorial. It was sent to the retro mailing list on 24 Jan 2001. The composition was: "Qh7, Bg6, Be5, Pc4, Pd4, Pf4, Nd3, Nf3, Pa2, Pb2, Pe2, Pg2, Ke1, Rh1/ Ke3. 14+1 White to move; what was the last move, and can White mate in two?"
[7] The notion of position only matters in the Laws when it comes to determining whether it has been repeated. It's not surprising then that the definition is tucked away in Law 9.2.
[8] For those problems shown in the table, it happens that DR permanently prevents castling (i.e. proves a rook or king moved) just in those diagrams in which castling is temporarily prevented (e.g. by an intervening piece). This correlation between temporary and permanent prevention is not a rule. One counter-example is the simplified version of {G}, another in the opposite sense is {W}.
[9] Ideally, the uniqueness of the last move would also be due to DR. But {E} & {H} are both interesting enough without this additional property.
[10] The #2 in {S} is also a h#1.5, of course.
[11] Errata to version in StrateGems: François Perruchaud's name changed in stipulations for {B} & {J} to his preferred form. Stipulation of {H} changed to highlight castling question. Precise stipulation of {P} adjusted to match original in The Problemist. Solution to {H}: helpmate spelled out and "black-first" notation is used. Solution to {S}: "uncapture the a-pawn" replaced by "uncapture the unmoved h-pawn". Solution to {U}: obviously incorrect rank or file corrected in several places. Solution to {V}: "retrostalemate by repetition" replaced by the more standard term "perpetual retrogression".
[12] If a problem's stipulation changes, essentially as a matter of clarification, this to my mind implies neither a correction nor a new version.

Back to home page