World Chess’s columnist explores the retrograde problems of Raymond Smullyan, a polymath, who died last month. A warning to readers: This column is challenging and addictive!
Raymond Smullyan (1919-2017) died last month, a few months short of his 98th birthday. Smullyan was primarily a mathematician and a logician, whose other talents included magic and classical piano. He wasn’t a professional chess player or a chess professional of any sort, but he did contribute to our beloved game. He may not have been the inventor of retrograde analysis, but two of his books, published in 1979 and 1981, did much to popularize the genre.
In retrograde analysis, the aim is not to figure out what to do next, but to figure out what move or moves were made previously to reach the given position. Solving these problems is more a matter of detective work than using traditional chess skills. Indeed, Smullyan’s first book of retrograde puzzles was titled, Chess Mysteries of Sherlock Holmes(Dover, reprinted 2012). Smullyan wrote the book in the style of the original Arthur Conan Doyle books, with Doctor Watson recollecting and narrating Holmes’s intellectual brilliance in solving the puzzles, contrasted with Watson’s less adept – but gradually improving – abilities to draw the proper deductions.
Here are some examples from that book – without the Watson-and-Holmes banter. (Readers looking for that will need to locate a copy of the book.) The first position is, well, elementary.
The assessment of the position is irrelevant; instead, the task is to determine if White has promoted a pawn. The answer is that he must have because White’s bishop can’t have come from c1 – there’s no way that bishop could have entered the game with the pawns still on b2 and d2. The bishop on g3 must therefore be the result of a promotion.
The second example is a bit more difficult. This time the task is to figure out what the last two moves were, given that it’s White to move.
As Black has no material, it’s clear that his last move took the king from a7 to a8. But then what was White’s last move? Black’s king is in check by the bishop on g1, but there’s no way it could have moved there without Black’s king already being in check. Since the bishop couldn’t have moved there, the check must have been a discovery. That piece is no longer on the board, however, so Black must have just taken it on a8, and the only piece that could have moved from the diagonal fragment f2-b6 to a8 is a knight from b6. Ergo, the last moves were 1.Nb6-a8+ Kxa8. (The knight could have taken something on a8, but for simplicity’s sake we’ll assume the square was empty.)
Now for one that’s a bit more complicated, but still quite accessible. The task this time is to determine on which square the White queen was captured.
To solve this, the first thing to do is look for a capture that has only one explanation. Clearly White’s bishops had to be captured on their home squares, but White’s missing queen and knight had to be captured on e6 and h6 – but which was captured where? The key hint comes from the pawn on b3. The only piece it could have captured was Black’s light-squared bishop – obviously it can’t have captured Black’s only other missing piece, which is the dark-squared bishop. But the next question is, how did Black’s light-squared bishop get to b3? Black had to play …dxe6 first. Since White’s queen couldn’t have come out until after axb3 was played, and axb3 couldn’t be played until …dxe6 was played, it follows that a White knight gave itself up on e6, then Black’s bishop was captured on b3, and then White’s queen could get into the game to be captured on h6.
Smullyan added some twists to the retrograde genre, one of which is “monochromatic” chess. In these puzzles, a piece can only move to a square that is of the same color as it started on. That’s no problem for bishops, who fulfill that task by definition, but it’s a severe constraint on all the other pieces. A rook on h1 on an open board, for instance, can never go to the g-, e-, c-, or a-files or to the second, fourth, sixth, or eighth rank. Pawns can only move two spaces on their initial move and then must capture to move again. And knights can never move at all. Here is a (relatively) simple example that illustrates the idea. Given the stipulation that White’s king has made fewer than 14 moves, prove that a pawn promotion has taken place.
At first glance, it seems impossible to solve as none of the remaining pieces were (or could have been) promoted. The key is to think about all the pieces and how they could have been captured. As noted above, knights are completely immobile in monochromatic chess, so that’s a good place to start. White’s knight on b1 could have been captured by Black’s light-squared bishop, a pawn (which would then have promoted), or a promoted light-squared bishop, rook, or queen. The knight on g1 could have perished by Black’s dark-squared bishop, queen, a promoting pawn, or a promoted dark-squared bishop, rook, or queen. Similarly, Black’s g8-knight could have been taken by White’s light-squared bishop, the queen, a promoting pawn, or a promoted light-squared bishop, rook, or queen.
That is no help so far, but what about Black’s b8-knight? It couldn’t be taken by an original rook – none of the knights could be. (The rook on a1 can’t get to the b-file, while the rook on h1 can’t reach the eighth rank.) White’s original queen is stuck on light squares, so she can’t reach the knight, and neither can White’s original bishops. The light squared bishop can’t because b8 is a dark square, while the bishop that started on c1 perished there because the White pawns are still on b2 and d2. White’s knights couldn’t have done it because they can’t move, so that leaves just the king and the pawns.
The king can’t do it because the round trip to b8 and back would take 14 moves, and we were told that the king had made no more than 13 moves in the game. As Black’s knight couldn’t have moved its capture must have been either by a White pawn capturing it on b8 (and thereby promoting), or by a pawn that promoted to a queen, rook, or dark-squared bishop. All four possible ways for the knight to be captured entail White’s promoting a pawn, so Q.E.D.
All of that was just the warmup. The following requires retrograde analysis that can’t be solved with a single flash of insight; several steps are needed to work this one out. And there’s a story behind it.
The players had started a game (a regular one, not monochromatic chess) and then abandoned it for a while to do something else. Some kids were playing around in the room and took one of the pieces from the board – from the h4 square. It is possible to figure out what that piece was, but it won’t be easy! Here’s the position.
Readers who want to try this and are stuck, read only until you get a piece of information that you hadn’t discovered on your own, and then try to solve it from there.
1. The last move had to be cxd8(R)+; otherwise the position would be illegal.
2. The piece captured on d8 could only have been a knight or a bishop, as a Black queen or rook would have had White’s king in check, as there is no possible legal way for Black to have made that move on the previous turn. (Where would the queen or rook have come from?)
3. If the piece on d8 was a bishop, it must have been a promoted bishop, because the original dark-squared bishop that started on f8 couldn’t have reached d8 (or any other square) because of the pawns on e7 and g7.
4. If the piece on d8 was a knight, then at least one of Black’s knights had been promoted.
5. Either way, all the Black pawns are accounted for: seven remain on the board and one has been promoted.
6. Therefore, the piece on h4 couldn’t be a Black pawn.
7. It couldn’t be a White pawn for more or less the same reason: seven White pawns remain on the board, and the eighth just promoted to a rook.
8. It can’t be a Black queen or a Black rook, because both kings would be in check.
9. It can’t be a Black bishop, because if it were it would have been a promoted bishop. But the only promoted Black piece was just captured on d8.
10. It can’t be a Black knight, for the same reason: if it were, then there’s no legal way to account for White’s last move. Black is only missing one pawn, so he can’t have both a knight on h4 and (until White’s very last move) a knight or bishop on d8. The captured piece on d8 was a promoted bishop or knight, so a knight on h4 would require Black to have promoted two pieces.
11. Therefore, it must have been a White piece on h4 – a queen, a rook, a bishop, or a knight. Getting here was the (relatively) easy part; the last part is trickier.
12. It was Black’s h-pawn that promoted (the a6-pawn had to come from b7 and the c4-pawn had to come from f7), and since it couldn’t have jumped the h-pawn it must have made at least one capture.
13. In fact, it could only have made one capture – White has 11 units on the board (remember that there’s a missing White piece on h4).
14. Black’s h-pawn could only have promoted on g1 by taking something on g2 – there’s no other way to get around White’s h- and g-pawns by only making one capture.
15. Therefore, all five Black captures were executed on light squares.
16. Of the six White pieces not visible on the diagram, five are capable of being captured on light squares: the queen, the missing original rook, the light-squared bishop, and both knights.
17. The remaining White piece, the dark-squared bishop, could not have been legally captured by Black’s b7/a6 pawn, the f7/c4 pawn, or the h7/promoted pawn.
18. Therefore, it is a White bishop that is missing from h4!
There are many more puzzles in the book – these only scratch the surface, and except for the last one, are on the easy side. But Smullyan does his best to bring the reader along a step at a time, generally increasing the level of difficulty enough to give the reader a good challenge but not so much that each new puzzle seems like an impossibility. Readers who enjoyed the puzzles in this column will probably enjoy the book as a whole, particularly since his puzzles are sprinkled with entertaining banter.
His second book continues the fictional theme, but this time he offers the puzzles and stories in a new setting – the book is called The Chess Mysteries of the Arabian Knights (Alfred A. Knopf, 1981). Smullyan really is an author for people who like logic puzzles, particularly when they’re mixed with chess.
Dennis Monokroussos is a FIDE master who has written about chess on his blog “The Chess Mind,” since 2005. He has been teaching chess for almost 20 years and for the last 10 years has been making instructional chess videos, which can be found at ChessLecture.com. Between 1995 and 2006, he taught philosophy, including a four-year stint at the University of Notre Dame.