Lord Dunsany's Chess Problem

Solution
It is first noted that the black queen is on a white square.

But according to the starting position, the queen starts on her own colour.

That means the black queen and king are not where they started.

The only way for that to happen is if at least one of the black pawns has been moved.

But pawns cannot go backwards.

So the only way the position is legal is if the board is such that black is at the bottom playing up, and white is at the top playing down.

Hence none of the black pawns, being on the $7$th rank, are able to move at this time.

With that in mind, the game progresses as follows.

Note that although the board has been turned round, the ranks and files remain labelled as they are in the diagram, in order to limit confusion.

White moves:
 * $(1): \quad \text N \text g 1 - \text e 2$

Suppose black now moves:
 * $(1): \quad \ldots \qquad \text N \text b 8 - \text a 6$:

White moves:
 * $(2): \quad \text N \text e 2 - \text d 4$:

and whatever black does, white then moves either to $\text N \text d 4 - \text c 6$ or $\text N \text d 4 - \text e 6$ mate.

Now suppose black moves:


 * $(1): \quad \ldots \qquad \text N \text b 8 - \text c 6$:

White moves:
 * $(2): \quad \text N \text e 2 - \text f 4$:

threatening mate next move by $\text N \text f 4 - \text e 6$.

To prevent that, black can delay the inevitable by playing:
 * $(2): \quad \ldots \quad \text N \text c 6 - \text d 4$:

which of course is immediately countered by:
 * $(3): \quad \text Q \times N$:

and there is nothing black can do to prevent:
 * $(4): \quad \text N \text f 4 - \text e 6$ mate

It is worth noting that on move $(2)$, black could play $\text b 7 - \text b 8 (\text Q)$, but by then it is too late to get that new queen out before the hammer blow.

If we explore the possibility of black moving the knight on $\text g 8$, we see that it cannot get itself into position to threaten white's knight in time to prevent the mate in $3$.