Puzzle 32 solution

Here's Black's first move, Bxg2+:

White has one legal move, Kg1. Black then mates with Gh3:

Puzzle 31 solution

Here's Black's helping move, Kf7:

Here's White's reply, Gh8 mate:

Puzzle 30 solution

Here's Black's first move, Gh2+:

White has one legal move, Kf2. Black then mates with Bg3:

Puzzle 29 solution

Here's White's first move, Ge8+.

Black has one legal move, Kh7. White then mates with Gxg7:

Puzzle 28 solution

Here's Black's first move, Nxf3+:

White has two legal moves, Kh1 and Kf1. In either case, Black mates with Gh2. Here's the final position when White plays Kf1:

Puzzle 27 solution

Here's White's first move, Rxh6+.

 Black has one legal move, gxh6. White then mates with Gxh6:

Puzzle 26 solution

Here's Black's first move, Ge3+:

White has two legal moves, Kh1 and Kh2. In either case, Black mates with  Gf1. Here's the final position when White plays Kh2:

Puzzle 25 solution

Here's White's first move, Re7+.

Black has two legal moves, Kf8 and Kg8. In either case, White mates with Gxg6. Here's the final position when Black plays Kf8:

Puzzle 24 solution

Here's Black's first move, Gc3+:

White has one legal move, Kf2. Black then mates with Gd1:

Puzzle 23 solution

Here's White's first move, Gxd6+.

Black has one legal move, Kf3. White then mates with Gh2:

Puzzle 22 solution

Here's Black's first move, Gd2+:

White has two legal moves, Ka1 and Ka3. In either case, Black mates with Gc2. Here's the final position when White plays Ka3:

Puzzle 21 solution

Here's White's first move, Rxb8+.

Black has one legal move, Kxb8. White then mates with Ga7:

Puzzle 20 solution

Here's Black's first move, Ge1+:

White has one legal move, Kh2. Black then mates with Gf1:

Puzzle 19 solution

Here's White's first move, Gc8+.

Black has one legal move, Kf6. White then mates with Gf5:

Puzzle 18 solution

Here's Black's first move, Gxf3+:

 

White has one legal move, Gg2. Black then mates with Rxh2:

 

Puzzle 17 solution

Here's White's first move, Gd5+.

 

Black has two legal moves, Kc8 and Kb8. If White plays Kc8, Black mates with Gxa8. If White plays Kb8, Black mates with Gc6. Here's the final position in the latter case:

 

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: