Puzzle 16 soluton

Here's Black's first move, Gf3+:

 

White has one legal move, Kf1. Black then mates with either Gh1 or Gh2. Here's the final position in the latter case:

Puzzle 15 solution

Here's White's first move, Gh5+.

Black has three legal moves: g6, Ng6, and Gf7. If Black plays g6, White mates with Gg7. If Black plays Ng6, White mates with Gxg7. If Black plays Gf7, White mates with Gxf7. Since checkmates with more of the checkmated player's pieces left on the board are more impressive, here's the first checkmate mentioned above:

Puzzle 14 solution

Here's Black's first move, Ga1+:

White has five legal moves: Gb1, Gc1, Gd1, Ge1, and Kf2. Only the first two prevent Black from checkmating White on Black's second move. If White plays Gd1, Black mates with Gxd1. If White plays Ge1, Black mates with Gxe1. If White plays Kf2, Black mates with Gh1. So we'll assume White plays Gb1. Here's Black's second move, Gxb1+:

White has only one legal move here, Kf2. Here's Black's final move, Gxe1 mate:

Puzzle 13 solution

Here's White's first move, Ng4+:

Black is in double check, and has four possible replies: Kh7, Kg7, Kg6, and Kh5. Regardless of which reply Black chooses, White will checkmate Black with the move Gh6 (Gargantua to h6). Here's what the checkmate looks like when Black chooses Kh7:

Puzzle 12 solution

Here's Black's helping move, Bg7:

 Here's White's reply, Gd6 mate:

Puzzle 11 solution

Here's Black's helping move, Rc6:

 Here's White's reply, Gf7 mate:

Puzzle 10 solution

Here's Black's helping move, Gh4:

 Here's White's reply, Gg7 mate:

Puzzle 9 solution

Here's Black's helping move, Ne7:

Here's White's reply, Qf6 mate:

 Note that the first move could also have been Nd6, Nd4, Ne3, Ng3, Nxh4, or Nh6.
 

Puzzle 8 solution

Here's Black's helping move, Ng8:

 Here's White's reply, Qg6 mate:

 Note that the first move could also have been Rg8.