Retrograde analysis

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This article uses algebraic notation to describe chess moves.

Eric Angelini, Europe Echecs 433, Apr. 1995

Black to move. What has White just played?
Typical example of retrograde analysis problem.

Retrograde analysis is a technique employed by chess problem solvers 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.

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 above. 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 discovered check 2.g2-g4 f4xg3 double check 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.

[edit] Partial retrograde analysis

W. Langstaff, Chess Amateur 1922

Mate in two.
This problem uses partial retrograde analysis method.

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)