Can you explain the difference in last week's puzzle between a8=B!! and a8=Q? ? I am a bit puzzled...

The problem is there is a wrong conclusion in the solution. The position after 12... Kxf1 is a draw, not a win. After 4 a8B Kf1 5 Bxh1 black has Bh2, so there is no stalemate and white wins.

Let's see...

1. Ne2 Bxe2 (1...Rd3 2.Nxd3 +-) 2.d8=Q Rg3+ (2...Bc4+ 3.Kh7 Rc1 (3...Bd3+ 4.Kh8 +-) 4.Qa5+ K[g,h]4 5.Qd2 g1=Q 6.Bxg1 Rxg1 7.Q[d4,h2]+ +-) 3.Kh7 g1=Q (3... Bd3+ 4.Kh8 +-) 4.Bxg1 Rxg1 5.Qe8+ +-

I think I got most of it.

1.Ne2 g1Q+ and 2.Bxg1 Bh3 or 2.Nxg1 Ba6

Yes, but: 1.Ne2 g1=Q+ 2.Bxg1 Bh3 3.Nf4+ Kh6 (3... K[g5,h4] 4.d8=Q+ or 3... Kg[4,6] 4.Nxh3 Rd3 5.Nf4+) 4.Nxh3 Rd3 5.Be3+ +-

Oh, and I mistyped the first side variation in my last post. It should obviously read 1... Rd3 2.Nf4+ K~ 3.Nxd3.

It's easy to make mistakes when you're solving without a board.

And again I noticed a typo. White should obviously play Nf2+ to pick up black's rook after 3...Kg4 and not the impossible "4.Nf4+". It would be nice if you could edit your posts.

after 5.Be3 Rxe3 6.Nf4 fails to 6....Re7 7.d8Q Rg7+ 8.Kf8 Rg8 9.Kxg8 stalemate.
So 6.Nf2 Rg3+ 7. Kh8 wins (7.Kf8 Rf3 8.Ke7 Re3+ etc.)

Cute. I had completely missed the stalemate after 6.d8=Q?? 6...Re8+.

Question about last week:

Why can't white play 1. exf3, and avoid all stalemates?

1.exf3 Bf2+ 2.Ke2 Kg2 3.f8Q h1Q
Black will have sufficient counterplay