It is possible, and three moves is as fast as White can ever manage. One version runs 1.e4 f6 2.d4 g5 3.Qh5 mate.
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h 1 Here is the finish, so you can see exactly which squares matter.
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h 1 Why each move is needed
1…f6 is the move that opens the road. Without it the pawn on f7 blocks the diagonal and nothing is possible.
2.d4 looks like a spare move, and in a sense it is. White has no way to punish 1…f6 immediately: 2.Qh5+ is met by 2…g6, the pawn blocks the check, and the queen has to run. So White plays a useful developing move and waits.
2…g5 is the second gift. Now the g6 square can never be filled by that pawn, because the pawn has walked past it. The picture is the same one that produces the two-move mate with colours reversed: an open diagonal to the king plus no blocker.
3.Qh5 arrives and Black is lost. Check the king’s options in the diagram: f7 is attacked by the queen, e7 and d7 hold his own pawns, d8 his queen, f8 his bishop. The knight on g8 can reach f6 and h6 but neither square is on the diagonal, so there is no interposition, and nothing on the board attacks h5.
Nothing here is forced
That word matters. Black chooses 1…f6, and Black chooses 2…g5. Play 2…e6 or 2…Nf6 or almost anything else and there is no mate, only a white queen that will need a second move soon.
So the answer to “can you checkmate in three moves” is yes in the same way that you can win a race if the other runner sits down. I have had it happen to me once, in a five-minute game, against an opponent who was clearly playing with one hand and eating with the other.
Mate in 3 in a puzzle
A puzzle that says mate in 3 is asking for something much stronger: three of your moves that force mate against every reply. That is six half-moves of calculation, and the branching is where the difficulty lives. Your first move usually removes a defender or shuts a flight square, your second forces the king to a particular square, and only the third gives the mate.
The practical method is to picture the final position first and work back to it. Then check the defences you have not looked at, which is always the move you have unconsciously decided is impossible. Sets of three-move mates are the right difficulty once one- and two-movers stop taking you any time at all, and if you want the fastest mates in general, the record for both colours is a short read.