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Known as Fool's Mate because of the extreme unlikelihood of contriving to be checkmated in such a manner, the shortest possible game of chess is as follows:

1.f4 (or 1.f3, but 1.f4 is actually a legitimate move, known as Bird's Opening, and therefore slightly more plausible)
1...e5 (or 1...e6, but again, 1...e5 is a known opening move in response to Bird's Opening, called the From Gambit)
2.g4?? (There's no excuse for this move, however)
2...Qh4++

+----+----+----+----+----+----+----+----+
| r  | n  | b  |    | k  | b  | n  | r  |
+----+----+----+----+----+----+----+----+ 
| p  | p  | p  | p  |    | p  | p  | p  |
+----+----+----+----+----+----+----+----+
|    |    |    |    |    |    |    |    | 
+----+----+----+----+----+----+----+----+
|    |    |    |    | p  |    |    |    | 
+----+----+----+----+----+----+----+----+
|    |    |    |    |    | P  | P  | q  | 
+----+----+----+----+----+----+----+----+
|    |    |    |    |    |    |    |    | 
+----+----+----+----+----+----+----+----+
| P  | P  | P  | P  | P  |    |    | P  | 
+----+----+----+----+----+----+----+----+ 
| R  | N  | B  | Q  | K  | B  | N  | R  |
+----+----+----+----+----+----+----+----+

White is checkmated along the vulnerable e1-h4 diagonal, unable either to move his king or interpose another piece. At this point, when demonstrating Fool's Mate to a beginner, they will typically stare at the board with a perplexed expression for a number of minutes, unable to believe that this is possible.

Though it might be thought that no serious or even semi-serious chess player could possibly make a mistake of this magnitude, I have in fact seen this position with colours reversed in an Irish chess tournament, with a friend of mine playing the winning side. The game went:

1.e4 g5 (This is a known, if slightly dubious, opening variation known as The Grob, after Henry Grob who invented it.)
2.d4 f5?? Incomprehensible on so many levels.
3.Qh5++

+----+----+----+----+----+----+----+----+
| r  | n  | b  | q  | k  | b  | n  | r  |
+----+----+----+----+----+----+----+----+ 
| p  | b  | p  | p  | p  |    |    | p  |
+----+----+----+----+----+----+----+----+
|    |    |    |    |    |    |    |    | 
+----+----+----+----+----+----+----+----+
|    |    |    |    |    | p  | p  | Q  | 
+----+----+----+----+----+----+----+----+
|    |    |    | P  | P  |    |    |    | 
+----+----+----+----+----+----+----+----+
|    |    |    |    |    |    |    |    | 
+----+----+----+----+----+----+----+----+
| P  | P  | P  |    |    | P  | P  | P  | 
+----+----+----+----+----+----+----+----+ 
| R  | N  | B  |    | K  | B  | N  | R  |
+----+----+----+----+----+----+----+----+

My friend jumped up from the board with a grin and immediately motioned me to come over and see, and his much younger opponent burst into tears, after spending the required 30 seconds trying to figure out why the position was actually checkmate. We wondered whether or not to tell him that, having fallen into a version of Fool's Mate, he was now a fool, but reconsidered. After all, his feet didn't touch the floor when sitting down at the board, and besides, falling into Scholar's Mate doesn't make you a scholar.