12+10. Man füge einen weißen Turm hinzu; dann Matt in einem Zug
The only way to give mate in 1 is to add the wh. Rook on b1 or c3
and play 1. b3#. Now, what is the right square for the Rook ?
Counting missing White pieces: including the Rook which is to be placed,
and the Wh. KB which fell on its home square (f1), there are two other
missing pieces: a wR and a wN. One of these fell on f6.
Now, White's h-pawn made one capture, on a black square, to reach
g3. White's c-pawn captured the remaining Bl. pieces via c2xd3xe4xf5xg6xh7,
all on white squares. So White's h-pawn must have captured the Black
KB, and White's c-pawn captured a bP (or its promoted replacement),
two N's, a Q and a R. For the Bl. b-pawn to be captured by W's c-pawn,
it would have to have made at least two captures (or else it made one
capture to get to the c-file and the c pawn made one capture to get
to the d-file). In either case, there is only one remaining White piece
to be captured (whichever one was not captured on f6). So the Black
b-pawn must have promoted and made at most one capture in doing so.
It must therefore have captured a piece on a2 and promoted on a1.
So, the wN and wR were captured on a2 and f6 (though not necessarily
in that order). However, Bl.'s KB could not have gotten out to be captured
on g3 until after the capture on f6. As the Wh. QR was permanently confined
to a1-a2-b1, neither wR could have gotten out to be captured until after
the capture on f6. Thus, the wN was captured on f6, so a wR (necessarily
the QR) was captured on a2. So it must be the Wh. KR that is supporting
the pawn delivering mate. Since the Wh. KR cannot get to b1, it must
be on c3.
Solution by Philippe Schnoebelen