SOLUTIONS TO DR PROBLEMS

Solutions to other DR problems will be included here in time.

001 002 004 009 011 014 015 020 023 040 041 054 059 061 062 063 065 070 072 082 085 087 093 102 103 106 110 132 134 135 140 144 148 160 161 162 163

001
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.]

002
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 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 Black has the move.

004
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.

009
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.

011
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..

014
(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).

015
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!

020
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.

023
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.

040
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.

041
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 castle.

054
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#.

059
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!)

061
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 legal.

062
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.

063
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.

065
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 once. RT

070
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#.

072
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#.

082
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.

085
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 alive.

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

087
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

093
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.

102
(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=.

103
Dead position. Alternatives to 0…Pf7-f5 are dead so e.p. illegal.

106
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

110
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

132
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+.

134
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.

135
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

140
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.

144
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.

148
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.

160
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.

161
Dead position. All candidate last moves are forced except for 0…Pb7-b5, which has living alternative 0 …Pb7-b6+.

162
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#.

163
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#.

Home