DEAD RECKONING

{A} A.G.Buchanan
The Problemist July 2001
(1+1) Who moved last?

An off-the-wall idea for chess compositions...

Let me show you a chess problem. (You can click on the diagrams here to enlarge them, if you like.)

Of course, there's not so much excitement left in this game, because neither side can checkmate. But the question I have for you is: who moved last? Yes, it's a genuine problem, yes it's only two pieces. Take a few minutes now to see if you can fathom it.

Given up already? OK, let's compare notes. This chess problem is a surprising consequence of Chess Law 5.2b, [1] which states:

The game is drawn when a position has arisen in which neither player can checkmate the opponent`s king with any series of legal moves. The game is said to end in a 'dead position'. This immediately ends the game, provided that the move producing the position was legal.

Now how does this apply to the problem above? The diagram position itself is obviously dead, but that doesn't tell us who moved last.

Suppose it was Black who just moved. What was the move? Could it have been Kb8-a8 without capture, for example? Aha no, because the prior position with just two kings was already dead. The rule would have triggered then and the game would have ended immediately, without ever giving Black a chance to play Kb8-a8.

OK, so obviously Black must have just captured a piece. In fact it must have been a major piece because with just a bishop or knight, there is no way to checkmate even a co-operating opponent, and again the game would have stopped dead prior to the capture. So what about Kb8Qa8? What's the problem with that?

Here is the sneaky part. In the prior position, with Black in check, what alternatives were there to capturing the queen? Answer: none, because every other square adjacent to the Black king was attacked. So Black was forced to capture. Hence in the prior position it was correct to say that "neither player can possibly checkmate". All possible outcomes are draws. Thus Law 5.2b would have applied, the game would have stopped dead, and Black never could have captured the queen.

The same is true if the checking piece was a rook, or if the Black king was on a7 rather than b8. We've looked at all the possibilities now, and found no legal last move for Black. If Black moved last, the diagram position is not just dead but illegal. So our original supposition was incorrect, and it must be White who moved last.

White could have captured a major piece or a pawn. But whichever unit it was, the White king is not cramped for space, so was not obliged to capture, and could simply have moved somewhere else. The game still had the possibility of ending in a checkmate, and it was only after White chose and played a move which "killed the game" that 5.2b triggered. In fact, there are 73=21 legal last moves for White.

This kind of chess problem trick using Article 5.2b is termed "Dead Reckoning" (DR).

Cool. So are there any other positions to which DR might apply in an interesting way, or is Diagram 1 just a weird anomaly?

Looking around for any other positions with just the two kings on the board, we find another if we shift the White king from c6 to c7 (or equivalently to b6). Once again, the position is legal only if White moved last. [Note: there are only 11 legal prior positions, not 12.]

But apart from rotations, reflections and colour inversions of these two positions, I don't think there are any other DR illegal positions with just two kings on the board. That's nice: it suggests that the property is rare, but not vanishingly so.

And once we started to explore more complex chess positions, we began to find many fascinating varieties of DR position. The goal of this document is to show the power of DR in augmenting retrograde analysis.

So let's look at a slightly more complicated position. Again I would encourage you to try it yourself before reading on.

{B} A.G.Buchanan
Internet Mailing List, Jan 2001
(2+1) Whose move?

Let's clear up one possible issue first. Like {A}, this game in this diagram is over, because no checkmate is possible. Now even though the game is over, it is still meaningful to talk of White or Black having the move. In fact, the definition of a position is not complete until it specifies whose move it is (together with clarifying the status of possible castling rights or en passant). So the question "Whose move?" is meaningful, and of course just gives the opposite answer to the question "Who moved last?". "Whose move?" I find more elegant, and I'll use it from now on.

What happens if White is to move? Then Black's last move was from a7, and must have been a capture, since a knight alone can never deliver checkmate. If Black's last move was to capture a major piece, then a similar argument to the previous problem applies. Black would have been obliged to capture it reaching a draw, so the move could never have been made. But suppose that Black captured a minor piece instead.

With a knight and bishop, White can of course force checkmate. And with two knights, the players can certainly co-operate to engineer a checkmate together, which is all that is needed. The difficulty is that in the prior position with the Black king on a7, where does Black move? The only choices are to capture Nb8 or the minor piece on a8. Since Black is thus obliged to remove one of the mating team, the position is already dead, and the move is never made.

So again, Black has the move.

There are other such illegal positions with just 3 pieces on the board. [How many?] But for now let's move up the complexity curve to the following:

{C} A.G.Buchanan
Internet Mailing List, Jan 2001
(3+1) Whose move?

Hypothesize as before that White is to move. Then although White has certainly got enough material for checkmate, the position is actually dead. The only way that White can avoid stalemating Black is by moving the king or bishop to allow (indeed: compel) Black to capture the bishop, and then of course there is insufficient material remaining for checkmate.

Now we turn to examine what the last move might have been. Black's move was obviously Ka7*Ma8. [By "*" I mean that the move might have been a capture, or it might have been just a regular move without capture. By "M" I mean a Mystery unit, which might even be nothing ("0"). Sometimes we don't care what, if anything, was captured, in which case I'll omit the "M".] So what could the mystery unit be?

Firstly, M cannot be rook or queen. This has nothing to do with 5.2b but is simple retrograde analysis, as the double check is impossible. On the other hand, if M is bishop, knight or nothing at all, then the prior position is legal from a traditional retro standpoint. Note parenthetically that in any case, White's prior move can only have been Pb7-b8B+.

[We determine if candidate last moves are legal from a conventional perspective, because if none of them are legal, then the problem does not use 5.2b at all! For the purposes of this article, we will always try to apply conventional retro reasoning to the diagram first, to make it clearer what the

The DR is in fact easier than the previous two problems: whichever of the three legitimate values M takes, Ka7*Ma8 is forced, and so the move can never be played. So as before, the hypothesis was incorrect, and in fact Black is to move in the diagram. Black is actually in stalemate and the position is dead, but there is no retrospective death, so no contradiction. White with his last move chose stalemate, when he clearly had the option not to.

In {C} we saw a combination of "insufficient material", and stalemate. It's actually unusual for a DR problem to combine different termination methods. Most problems fall into just one of the following three classes:

  1. Class 1: Insufficient material
  2. Class 2: Stalemate
  3. Class 3: Blocked position
Class 1 positions are those where the pieces are free to move around independently, but there just aren't enough of them to ever manage a checkmate, even when the two players are co-operating. As far as Class 2 is concerned. I hope we all know what a stalemate is. And then a Class 3 position is any dead position where both sides are free to move, but there some kind of pawn wall preventing them from capturing one another.

We don't have to consider draw by 50 moves, or draw by repetition any more than we need to consider draw by agreement or resignation. All of these require some choice by at least one of the players. They don't have any impact on whether a position is dead or not.

Problems in the three classes have very different flavours. Class 1 positions are the smallest, and opposing bishops are typically on the same colour squares. Complex Class 3 positions paradoxically involve opposing bishops on opposite colour squares. [2] Class 2 is not fussy about bishops, and contains the greatest variety of themes. Consider {D}.

{D} A.G.Buchanan
Version Internet Mailing List,
Jan 2001
{E} N.D.Elkies
Version Internet Mailing List,
Jan 2001
{F} A.G.Buchanan
Version Internet Mailing List,
Jan 2001
(12+1) White to move and mate in 2 (6+3+1) Place a pawn on a6. What was the last move? (3+4) White to move. Last move? (a) Diagram (b) bK->h8

Hitherto it's been a retro commonplace that without additional information about a legal position, you can only prove that castling is forbidden: you can never prove that it's allowed. This is because any game could be prefixed with a little dance of rooks and knights, to specifically disable some castling right. With Dead Reckoning, this is no longer true!

Almost every move in {D} results in stalemate. The only exception is 1. 0-0 Ke2 which allows 2. Rfe1#. But maybe White moved king or rook earlier in this game, and has lost castling rights?

Black's last move was 0 Kf3Re3, following -1. Re4*e3+. The Black move was forced (wRe3 itself covered g3). So if the current diagram is dead, so was the prior diagram, and the current one would be illegal. Therefore, the diagram is alive, and White must be able to castle. [3]

{E} is a cute example of a "last move" problem, composed by Noam Elkies.

White is in check, so is to move. Regardless of the colour of Pa6, the game must end with Black stalemated on White's coming move (1. Qb7 or perhaps 1. ab7) or the next one (1. Qg2+ Bg2+ 2. Rg2). Black's move may have been one of 11 different candidates: Ba8Mb7+, Bc8Mb7+ or Bc8-b7+. But only in the last case was there a choice, and then only if Pa6 is White. So Pa6 is White, and Black last moved 0. ...Bb7+, instead of 0. ...Ba6, which would have allowed the game to end in checkmate immediately or later.

Now let's solve a Class 3 problem: {F}.

Part (a) first. The kings can move, but the position is dead. If the White king captures the Black bishop, White is stalemated. There are 10 candidate last moves: 0. ...Kh7*Mg8 & 0. ...Kh8*Mg8. Two of them (0. ...Kh7Qg8 & 0. ...Kh7Bg8) imply an impossible retro double check, but the others pass on to the next stage.

Dead reckoning: 0. ...Kh8*Mg8 & 0. ...Kh7*Rg8 are forced, so cannot be played. That leaves two survivors: 0. ...Kh7-g8, 0. ...Kh7Ng8, which both have one alternative forward move: 0. ...Kh7-h8.

The first survivor doesn't yield us any checkmates after playing the alternative, so that one's out. However, the last does give us some options. White can move the knight, and the blocked position can start to break up. We see daylight! So the solution is 0. ...Kh7Nh8, with alternative 0. ...Kh7-h8 1. N~ (1. K~? Kg8 dead) and the knight is captured, opening up the position.

Now part (b). I was tempted to leave it as an exercise, but I can't resist writing the answer here myself. As in (a), there are 10 candidate moves, but now there are three which have promise. 0...Kh7-h8, 0...Kh7Nh8 & 0...Kh7Bh8. All have the alternative 0...Kh7-g8. The first two don't lead anywhere, but the third allows 1 Bg7 ~g7, and the position is wide open.

I call the forward play (after retracting the key) "ghost play". It's somewhat analogous to set play and try play. Often the ghost play is not uniquely determined: the initial position may be tightly constrained, but as soon as the position is released, the position just explodes with choices. But where the forward play is forced, or leads to a unique shortest helpmate or direct mate, that is a bonus. For example, {F}(a) is a helpmate in 3.0 moves after retraction. [4]

I'll end with one more example of each class:

{G} A.G.Buchanan
Internet Mailing List, Jan 2001
{H} A.G.Buchanan, N.D.Elkies
Version Internet Mailing List,
Jan 2001
{I} A.G.Buchanan
Version Internet Mailing List,
Jan 2001
(3+1) White to move. Last move?   (1+11) Last move?   (10+10) Before or after 1 July 1997?

Note that in {I}, there are many promoted units. Class 3 can only make very limited use of queens, rooks, knights, opposite parity bishops, or opposite parity pawns. This doesn't leave much! So legally promoted bishops are not considered a defect at all, since the class would be rather dull without them.

So what was behind the creation of Law 5.2b?

Prior to July 1997, a certain Law 10.4 prevailed:

The game is drawn when one of the following endings arises:
(a) king against king;
(b) king against king with only bishop or knight;
(c) king and bishop against king and bishop, with both bishops
on diagonals of the same colour.
This immediately ends the game.

This itemizes most of the Class 1 cases. This Law prevented pointless races in clock-slapping, but was perceived as over-detailed (citation?). So in 1997 it was replaced with something more generic. This was a nice little present for the world of chess compositions as it opened the door to Class 2 & 3 positions for free. Note that under either version, the game ends immediately. Players don't get to shuffle the pieces around and wait for the flag to fall.

Summary

Dead Reckoning fuses aspects of the known problemist genres of retraction, mutual helpmates and end-game studies, but who could have guessed that these all lay latent within a single rule of chess?

We've been speaking here mainly about DR problems for retrograde analysis. But actually, there are also a whole slew of forward DR problems which have no retrograde content. However that's a somewhat controversial topic for another occasion maybe...

Over 180 DR problems have been composed so far to my knowledge, by about a dozen individuals. Many of these are posted at the website. Thanks for reading.


[1] Dead Reckoning has been part of the Laws of Chess since 1 July 1997. The most recent Laws came into force on 1 July 2005. The information in Article 5.2b is essentially repeated in Article 6.10 (in "The Chess Clock") & Article 9.6 (in "The Drawn Game").

[2] No problem has been devised which combines both Class 1 & Class 3 elements.

[3] Note that the stipulation told us that White had the move. We couldn't have deduced it for ourselves. Actually, a common convention says that White has the move in "mate in n" problems, unless you can prove otherwise. However since this is a tutorial, I spelt it out. There are also standard conventions that say whether castling or en passant are permitted, if the position is ambiguous. In most of the DR problems designed so far, the DR itself resolves any ambiguities, so the conventions are not triggered.

[4] {F}(b) has its own attractions: being the minimal known position exhibiting the Ceriani-Frolkin theme (retro capture of a specific promoted piece). I am trying to get more details on the previous record-holder. DR has allowed a few other long-standing orthodox retro records to be broken. (See here.)

Solutions to Problems {G}-{I}

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