A quick chess

Proof that the number of half-moves, say, n, is greater or equal than 15.

We will show that the black queen needs at least 7 half-moves, and that then at least one more white half move is needed.

1. d6 and Qd7 are obviously necessary. - 2 half-moves.

2. The queen has to capture a knight, two bishops and the other queen. - 4 half-moves.

3. We state that, if the queen captures at every half-move (after the second) the fourth piece to be captured has to be the queen's bishop (so that she has to go to e1 in order to be captured by the king, adding the last half-move).

Assume this statement is not true. Then, the bishop has to be captured at the third move (it is easy to see that it is impossible to get to c1 in less than 3 moves). There is only one way to get to c1 in three moves from Qd7: Qg4 Qd1 Qc1, implying that there were pieces to be captured in those squares. This leads to only three possible situations:

and the King has to capture the Queen.

and the King has to capture the Queen, or

obviously not the case.

Hence, as we stated, the black needs to do at least 7 half-moves, plus the last half-move by the white King, $7\cdot2+1=15\implies\boxed{n\geq15}$

Note: I assumed it is the King who captures the Queen, since there is no way that the Queen ends in h1 or b1. I also assumed black doesn't use the knights.

The fastest game leading to it has 15 half-moves; moreover, it is unique:

1. e3 d6
2. Bb5 Qd7
3. Ne2 Qxb5
4. 0-0 Qxe2
5. f3 Qxd1
6. Kf2 Qxc1
7. Rh1 Qe1+
8. Kxe1

This also has a theme: White castled, but then returned the king and rook back, for an uncastling theme .

This kind of puzzles, where you have to construct a game leading to a given position, is called a proof game . (Normally, proof games give you the number of half-moves you have, and you need to construct the game leading to it; for obvious reasons it's difficult to duplicate here, so you get "find the minimum number of moves" instead.) This composition is made by Andrew Buchanan, 2005.