The first thing to notice is that Black is extremely short of moves. 1 . . . Kd4 invites a mate with 2. Nb5+ Ke5 3. d4 mate; 1 . . . Nf7 allows 2. Nxf7+ Kd4 (or 2 . . . Bxf7 3. Qxe4) 3. Nb5 mate; any bishop move is neatly mated by 2. Bf4+] Nxf4 (or 2 . . . Kf6 3. Qg5) 3. Qg7; and any move of the knight from h5 allows Qf4+ followed by Nb5 mate. Since 1 . . . e3 allows immediate mate with 2. d4, this leaves Black with only one non-suicidal move: 1 . . . exd3.
So all White has to do is find a first move which loses none of the above mates and also gives a mate in two after 1 . . . exd3. Sadly, 1. dxe4, eliminating the exd3 possibility, does not work: after 1 . . . Nf7 2. Nxf7+ Bxf7, White no longer has Qxe4.
The only clue - and a vital one - is the laziness of White's rook in all of the above. It must have a role to play, or the composer would not have put it there, but it has so far taken no part. The obvious idea is 1. Bb3 exd3 2. Rc4 but there is no mate after 2 . . . Kd5.
The only other thought that might edge one nearer the answer is to notice that White's queen is not needed to keep the black king hemmed in. If she could be miraculously spirited to the other side of the board then she can mate from a1, b2 or c3.
The solution beautifully ties all these ends together: 1. Rd1]] when 1 . . . exd1=Q+ 2. Qxd1 leaves Black unable to prevent Qa1 mate, while 1 . . . exd3 is met by 2. Qxe2+] dxe2 3. d4 mate, finally revealing the true point of the rook's move to d1.
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