|This article uses algebraic notation to describe chess moves.|
In chess, retrograde analysis is a computational method used to solve game positions for optimal play by working backward from known outcomes (e.g. checkmate), such as the construction of endgame tablebases. In game theory at large, this method is called backward induction. For most games, retrograde analysis is only feasible in late game situations of reduced complexity, such as a chess position where few pieces remain in play.
Among chess problem solvers, retrograde analysis is a technique employed to determine which moves were played leading up to a given position. While this technique is rarely needed for solving ordinary chess problems, there is a whole sub-genre of chess problems in which it is an important part; such problems are known as retros. An example with full analysis is available in Thomas Volet's "An Introduction to Retroanalytic Inference" (PDF).
Retros may ask, for example, for a mate in two, but the main puzzle (at least in modern retros) is in explaining the history of the position. This may be important to determine, for example, if castling is disallowed or an en passant pawn capture is possible. Other problems may ask specific questions relating to the history of the position such as "is the bishop on c1 promoted?". This is essentially a matter of logical reasoning, with high appeal for puzzle enthusiasts.
Sometimes it is necessary to determine if a particular position is legal, with "legal" meaning that it could be reached by a series of legal moves, no matter how bad. Another important branch of retrograde analysis problems is proof game problems.
An example of a retrograde analysis problem is shown at right. The solver must deduce White's last move. At first blush, there seems to be no solution: on any square from which the white king could have moved, it would have been under a seemingly impossible double check; however, thinking more we can discover that if white king moved from f5, then the black move before that could be pawn f4xg3, taking the white pawn on g4 en passant. Thus before f4xg3, white must have played pawn g2-g4. But what did Black move before that? The white king on f5 was under check by the bishop on h3 and there was a white pawn on g2. The only possibility is that black moved knight g4-e5 with discovered check. Therefore White's last move was king f5 takes knight on e5. (The entire sequence of moves is thus 1...Ng4-e5+ 2.g2-g4 f4xg3++ 3.Kf5xe5.)
Note that in this example the actual next move is essentially irrelevant; Black has a choice of several relatively trivial ways of delivering instant checkmate. (E.g. Qf3-d5#, Rd6-d5#, etc.).
One might ask: "If the white pawn was on g2 from the start of the game, how could the Black Queen come to be on f3, also a Black Bishop to be on h3? Is this problem valid?" The problem is indeed valid. The initial position has to be legal, but not particularly reasonable. Though the fact that the Black Queen and Bishop were en prise in the initial position may be disappointing to some, it does not invalidate the problem.
Partial retrograde analysis
Some problems use a method called "partial retrograde analysis" (PRA). In these, the history of a position cannot be determined with certainty, but each of the alternative histories demands a different solution. The problem to the left by W. Langstaff (from Chess Amateur 1922) is a relatively simple example; it is a mate in two. It is impossible to determine what move Black played last, but it is clear that he must have either moved the king or rook, or else played g7-g5 (g6-g5 is impossible, since the pawn would have been giving check). Therefore, either Black cannot castle, or White can capture on g6 en passant. It is impossible to determine exactly what Black's last move actually was, so the solution has two lines:
- 1.Ke6 and 2.Rd8# (if Black moved the king or rook)
- 1.hxg6 e.p. (threat: 2.Rd8#) 1...O-O 2.h7# (if Black played g7-g5)
Raymond M. Smullyan wrote two well-received retrograde analysis riddle books:
- The Chess Mysteries of Sherlock Holmes, ISBN 0-8129-2389-8
- The Chess Mysteries of the Arabian Knights, ISBN 0-394-74869-7
- The Retrograde Analysis Corner at Janko.at