A shortest proof game (or simply proof game) is a type of chess problem in which the solver must construct a game, starting from the initial position in chess, which ends with a given position after a given number of moves (sometimes no number of moves is given and the task is simply to construct the shortest possible game ending with the given position).
When published, shortest proof games will normally present the solver with a diagram - which is the final position to be reached - and a caption such as "SPG in 9.0". "SPG" here is short for "shortest proof game" and the "9.0" indicates how many moves must be played to reach the position; 9.0 means the position is reached after black's ninth move, 7.5 would mean the position is reached after seven and a half moves (that is, after white's seventh move) and so on. Sometimes the caption may be more verbose, for example "Position after white's seventh move. How did the game go?".
All published SPGs will have only one solution: not only must the moves in the solution be unique, the order of them must also be unique. They can present quite a stern challenge to the solver, especially as assumptions which might be made from a glance at the initial position often turn out to be incorrect (a piece apparently standing on its initial square may turn out to actually be a promoted pawn, for instance).
A (relatively) simple example is given to the right. It is a version by Andrei Frolkin of a problem by Ernest Clement Mortimer, and was published in Shortest Proof Games (1991). It is an SPG in 4.0. It is natural to assume that the solution will involve the white knight leaving g1, capturing the d7 and e7 pawns and the g8 knight (see algebraic notation), and then being captured itself, but in fact the solution carries an element of paradox quite common in SPGs: it is the knight that started on b8 that has been captured and the knight now on that square has come from g8. The solution (the only possible way to reach the position after four moves) is 1. Nf3 e5 2. Nxe5 Ne7 3. Nxd7 Nec6 4. Nxb8 Nxb8.
Most SPGs have a solution from about six to about thirty moves, although examples with unique solutions more than fifty moves long have been devised.
There are a number of variations on SPGs. The problem may carry a stipulation similar to "Find a game with 8.b7-b8=N mate", which simply means a game must be constructed starting from the initial position and ending on the given move number with the given move. Or it may be a one-sided proof game, in which only white makes moves (this is the SPG analogue to the series-mover in other types of chess problems). An alternative rule-set may also be specified (such as circe chess or losing chess), or a fairy piece may be substituted for an orthodox piece.
A number of chess problem composers have specialised in SPGs, with one of the most notable examples being Michel Caillaud who did much to popularise the genre in the 1970s and 1980s.
Gerd Willts and Andrei Frolkin, Shortest Proof Games (1991) - published in German but written in English. Includes 170 examples.