SOLUTIONS TO DR PROBLEMS
Solutions to other DR problems will be included here in time.
001 002 004 009
011 014 015 020
023 040 041 054
059 061 062 063
065 070 072 082
085 087 093 102
103 106 110 132
134 135 140 144
148 160 161 162
The relevant Rule of [the 1997 Laws of] Chess is Article 1.3 (Dead Game): "If
the position is such that neither player can possibly checkmate, the game
is drawn." Article 9.6 adds: "This immediately ends the
game." The diagram shows a game thus drawn (Dead Game). The last
move to the diagram must have been a capture of Q or R, else the game was
dead before that move. Now suppose Black has just captured Q or R, from a7
say. Black's only move to escape check is Kxa8, with a drawn game to
follow. Accordingly, no checkmate is possible from this position: Article
1.3 applies and the move Kxa8 does not take place. So Black didn't just
move, and White did. RT
[Note that the in the current 2005 version of the Laws of Chess, 5.2b
is the relevant article.]
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 is obliged to
capture it, so DR still triggers, the prior position is dead and the
move could never have been made. But suppose that Black captured a minor
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 Black has the move.
If White has the 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. So what could the mystery unit be? Firstly, M
cannot be rook or queen. This has nothing to do with DR but is simple
retrograde analysis. 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+.
Now DR: 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.
After White moves, the game is pat, or Black is forced to take the Pawn.
So the game is dead. Black's last move was Kb8*Ma8. If M=Q/R the position
is illegal. If M=B/N/0, Black has sole alternative Kxa7, but if M=B/0 then
insufficient mating material remains. So M=N. This composition again mixes
Class 1 & Class 2 elements.
White is in check, so is to move. Regardless of the color of Pa6, the game
must end with Black stalemated on White's coming move (1 Qxb7 or 1 axb7)
or the next one (1 Qg2+ Bxg2+ 2 Rxg2). Black's move may have been one of
11 different candidates: Ba8xMb7+, Bc8xMb7+ 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 ...Bxa6, which can
allow the game to end in checkmate immediately or later..
(a) 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: Kh7*Mg8, Kh8*Mg8. Two of them (Kh7xQg8 and Kh7xBg8) imply an
impossible retro double check, but the others pass on to the next stage.
Dead reckoning: Kh8*Mg8 & Kh7*Rg8 are forced, so cannot be played.
That leaves two survivors: Kh7-g8, Kh7xNg8, which both have one
alternative forward move: 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...Kh7xNh8, with sub line
0...Kh7-h8 1 N~ (1 K~? Kxg8 dead).
(b) As in (a), there are 10 candidate moves, but now there are three
which have promise. Kh7-h8, Kh7xNh8 & Kh7xBh8. All have the
alternative Kh7-g8. The first two don't lead anywhere, but the third
allows 1 Bxg7 ~xg7, and the position is wide open.
This is a minimal known position exhibiting the Ceriani-Frolkin
theme (retro capture of a specific promoted piece).
If Black moved last from b1/2, then White has no prior move (a condition
called "retropat"). So White moved last. So Black to move, but
in pat. If White's last move was with a Pawn, then this blocks in the
black King or the white Bishop. The only candidates to avoid an impossible
check are Bh2*Mg1 or Kf1*M'e1. M=N/B/0 or M'=B/0 does not relieve the
retropat. M=Q/R or M'=Q/R means the capture was forced, so DR prevents it.
So the last move was Kf1*Ne1. Check that M/M'=Q/R are legal retractions
unless DR applied!
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.
White cannot avert the looming stalemate. Each square adjacent to the
king is covered by at least two units or a pawn which cannot advance.
Moreover, White cannot check Black. So the position is dead.
Retro: the Black king may have come from b7 or d8. If from b7, he may
have captured rook, knight or nothing. If from d8, he may have captured
bishop, knight or nothing. Other options result in impossible double
check. The position unwinds in any of these six cases.
DR: 0…Kb7*Mc8 has the alternative Kb7-a8. However, once in the
corner, White again cannot relieve the stalemate, even if there is a rook
or knight on c8. 0…Kd8*Mc8 has the alternative 0…Kd8-e7 (unless there
was a knight on c8, in which case there is no sub). However, all the
squares around e7 remain covered whatever White does. The only way to
avoid the stalemate is if there was no unit on c8, in which case White can
play 1 c8N#. Note that Bb6 covers d8. So Black’s last move was 0…Kd8-c8.
Whoever has the move, no mate can ever occur, so dead. Every missing Pawn
promoted to a Bishop. This costs 2 captures/file, from e to h, which
together with the b-Pawn captures makes 12 minimum. The d-Pawns never
captured, and the Bishops on b1/8 did not just promote. Thus the last move
could only have been a banal Bishop move without capture, and the diagram
is illegal by DR.
Almost every move 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? Black's last move was 0 …Kf3xRe3, following -1 Re4*e3+. The Black
move was forced (wRe3 itself covered g3). So if the diagram is dead it is
illegal. Therefore, the diagram is alive, and White must be able to
The kings can't pass, so the wK can never reach bBf8, which is the only
vulnerable part of the wall. The wB can pass the bK, if the parity is
right (note there are a couple of resources for changing that parity), but
there is nothing for the wB to see if it does pass bK. The only way that
the game might avoid a draw is if Black is checkmated. This can only take
place on e1, if the wK is on c1, and the wB moves from g8 to f7 check.
However, prior to this, Black must make a waiting move, and there are no
spare Black moves. Conclusion: the position is dead.
Black might have played 0...Ke8*d8 (but singular) or 0…Pa5-a4. At
first 0…Pa5-a4 looks to be unhelpful, but there is the alternative
0...Ke8. Now, White must try first to get the wB past the bK. He appears
to have two pawns moves to change parity, but if he uses the a-pawn, then
that will take Black's only waiting move prior to the check.
The following moves show the first attempt: 1 Ka6 Kd8 2 Bb7 Ke8 3 Bc8
Kd8 4 Kb7 (not 4 Bd7= or 4 Bb7 going back to an earlier position) 4...Ke8
5 Ka6/8 (5 Bd7+ Kd8 6 Bc8 (to avoid checkmate, but been in this position
before)) 5...Kd8 6 g6 Ke8 7 Bd7+ Kd8 8 Be8 Kc8 9 Bf7 Kd8 10 Bg8 Ke8. The
bishop is there, now try the wK: 11 Kb7 Kd8 and now there is no way to
advance or to lose parity. So something is missing.
The key idea is that White must use the diagonal a6-c8 to change the
parity before the bishop goes past the bK. This saves the Pg4 move for use
near the end. The solution can follow the first attempt as far as 4...Ke8
then 5 Ka8 Kd8 6 Ba6 Ke8 7 Bb7 Kd8 8 Bc8 Ke8 9 Bd7+ Kd8 10 Be8 Kc8 11 Bf7
Kd8 12 Bg8 Ke8. Now for the wK: 12 Kb7 Kd8 13 g5 Ke8 14 Kc8 a4 15 Bf7#.
Dead position. bB is promoted, on light square. 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!)
Dead position. Black never captured. To avoid retro-death, last move was 0
…Ph7-h5 with alternative 0 …Ph7-h6 1 0-0 h5 2 Ra1#. So castling is
If no e.p., White's next move is stalemate. 1 a5xb6ep cxb6+ 2 Kxb6= but 1
e5xd6ep exd6#. Black's last move was 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, 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.
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.
All missing black units have been captured by white pawns, and White's g
and h pawns have died on file without capturing. Apart from those two
pawns, all missing white units have been captured by black pawns. Black's
last move was not a capture. It wasn't Pc6-c5 as this pawn came from a7.
Light-square wB died by Pe6xf5, so the pawn now on f4 came from e7, and
the pawn now on f4 came from c7. So Black didn't just move this e-pawn,
nor did he move K from e7 (improper check).
The diagram itself is a dead position. Black's last move wasn't with
f-pawn or Rook as these moves were from dead positions. The same goes for
any move to the diagram by black King - except for Ke8-f8. If Black just
made that move, and if he can still castle, the position before that move
was alive: he could have played 0-0 followed by wPc3-c4; bRd8; Pc7xd8 with
a live game after the promotion. As this is the only way there could have
been a legal move to the diagram, we know that Black did play Ke8-f8 from
a position where he could have castled. So the Black King has moved just
Dead unless 1…0-0 legal after 1 Kxe6. Black just played 0...d7-d6+
(alternatives dead). So castling is legal, and after 2 Kxe7, White can
helpmate later by 6 g6#.
Dead position. The White f-pawns come from files c through f, and 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#.
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 never
captured). So a White rook moved: by DR, it was the queen's rook.
Here it appears that White cannot avoid giving stalemate. However, Black's
last move to the diagram was Kh2-g1 - possibly with capture of some white
unit. Black King was here escaping check from the white Rook, which had
just made some capture on h3. That move was Black's only legal one.
Accordingly, if Position B is dead then so was the position before
that black move. That black move took place, so position B is still
We now notice that there is just one way that White can avoid giving
stalemate - and that is by giving checkmate! This he can do in just one
way: by castling. The position before Black's last move was still alive
only if this castling option is available to White. And so we have a
position, never before seen in orthodox chessplay, where we can say with
certainty that White has not lost the right to castle. The history of the
Queen's Rook is that it has no history: it has remained where it started,
on a1. RT
Far from anyone achieving mate in 3, Black is about to get stalemated. So
what was Black's last move to the diagram? If it was Kg8-h8 (or with
capture) that move was forced and led to a dead position: so was itself a
dead move. If it was Pa6-a5, then Black did have a choice. He could
instead have played Pa6xb5+. However, this is followed by wKa5 or wKxb5,
both stalemate. And so the position with bP on a6 is also dead.
Accordingly, Black just played Pa7-a5, and White may play 1.bxa6 e.p. bxa6
2.b7 a5 3.b8Q#. RT
If e.p., is illegal: mate, otherwise dead draw. If last move was 0 Pb2-b4,
then the only alternative, 0 Pb2-b3, is also dead.
(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) The current position is dead (even
after 1 f6) unless en passant is legal, so by DR 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=.
Dead position. Alternatives to 0…Pf7-f5 are dead so e.p. illegal.
White starts, and with Black's help stalemates him on the 7th
move. Provided Black plays b7-b5, stalemate of Black in 7 moves seems
inevitable. Indeed, it is inevitable... and 1.b5 is Dead Game with no
further play. So Black must advance more slowly, and White must play so as
to allow still-alive game ... by checkmate to White! 1... Kg7 2.b6 Kf6
3.b5 Ke5 4.b4 Kd4 5.b3 Kc3 6.a4 Kb2 7.a3+. Now the game is still alive as
7...Kxa1 8.b2#. Accordingly, White may play 7...Kxb2=. Note that 2.a4?
turns out not to work - wK can't get through to b2. RT
Not 1.Kb1 Sxd2+? 2.Ka1 Kb3 - the position is dead after 1...Sxd2, but
1.Kb1 Kb3 2.Ka1 Sxd2=. Not 1.d1Q? Sd2 2.Qb3+ Kxb3 as after 2.Qd3+ White
must capture bQ with K or S, leading to stalemate or draw, so that
position is dead, but 1.d1B Sd2 2.Bb3 Kxb3. Finally 1.d1R Kb3 2.Rd2 Sxd2. RT
The g3e.p. question is a red herring. 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+.
The retro-knot releases by cxd6 allowing the wK to escape, then bPg5 comes
from f6. Note that bPg6 is the original g-pawn (to allow bBg8 to leave) so
bPg5 is e-pawn, and bPf3 is h-pawn. With bPd6 (c-pawn) all Black captures
are accounted for. 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 allows 0…a7-a6 1 b5xa6ep f2 2 a7 f1=Q/R# (so the position is
living) 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.
All missing Black units were captured by White pawns. If Black's last move
was the single step, then from an orthodox perspective, Black can 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
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.
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. Unless White
can castle, stalemate looms, and the position is dead. 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.
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.
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
Dead position. All candidate last moves are forced except for 0…Pb7-b5,
which has living alternative 0 …Pb7-b6+.
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#.
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