6 corrected 1st 2 PGNs
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The Matt Bengtson problem Prof. Elkies mentions is:

[Title "Matt Bengtson, Chess Braintwisters (Burt Hochberg), no. 103. White to move & draw."]
[FEN "4kn2/3p1pPp/4pPpK/6P1/8/2p5/1b6/8 bw - - 0 1"]

The start of the solution-line can't be shown in PGN as White's 1st move is illegal: 1. g8=Black queen Qg7+ (anything else stalemates White) 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 (Kd6 7. Qc5+) 7. Qc6+ Ka7/Ka5 8. Qb7/Qb5, drawing. White's queen is then taken for stalemate. Solution line taken from Chess Braintwisters.


[Title "A Visual Solution Minus The Illegal Move"]
[StartFlipped "0"]
[FEN "4knq1/3p1p1p/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

1... Qg7+ 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 7. Qc6+ Ka5 8. Qb5+ Kxb5

(This is in answer to a comment by Prof. Elkies to Dag Oskar Madsen's answer, but, it seems, the syntax that activates the PGN-viewer in answers doesn't do so in comments.)

A similar oddity from the same book:

[Title "John Beasley, EBUR, 1996. White to move & win."]
[fen "5r1K/2k1P3/3N1Bqp/N2Q1B1P/8/8/8/1R6 w - - 0 0"]

This problem was reprinted as no. 107 in Burt Hochberg's Chess Braintwisters.

White is in check, and his only reasonable move is to capture on f8 (1. Qg8? Qxg8#). But 1. exf8=R allows stalemate after 1. ... Qg7+ because the new rook guards d8 after 2. Bxg7. Similarly, 1. exf8=N guards d7 allowing 1. ... Qh7+ and stalemate, and 1. exf8=Q/B guards d6 allowing 1. ... Qg8+ and stalemate. The answer is 1. exf8! (no promotion). Now 1. ... Qg7+ 2. Bxg7 releases d8; 1. ... Qh7+ 2. Bxh7 releases d7; 1. ... Qg8 2. Qxg8 releases d6. 1. ... Qxf6+ destroys the stalemate at once, and White wins. Solution line and discussion taken from Chess Braintwisters.

The Matt Bengtson problem Prof. Elkies mentions is:

[Title "Matt Bengtson, Chess Braintwisters (Burt Hochberg), no. 103. White to move & draw."]
[FEN "4kn2/3p1pPp/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

The start of the solution-line can't be shown in PGN as White's 1st move is illegal: 1. g8=Black queen Qg7+ (anything else stalemates White) 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 (Kd6 7. Qc5+) 7. Qc6+ Ka7/Ka5 8. Qb7/Qb5, drawing. White's queen is then taken for stalemate. Solution line taken from Chess Braintwisters.


[Title "A Visual Solution Minus The Illegal Move"]
[FEN "4knq1/3p1p1p/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

1... Qg7+ 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 7. Qc6+ Ka5 8. Qb5+ Kxb5

(This is in answer to a comment by Prof. Elkies to Dag Oskar Madsen's answer, but, it seems, the syntax that activates the PGN-viewer in answers doesn't do so in comments.)

A similar oddity from the same book:

[Title "John Beasley, EBUR, 1996. White to move & win."]
[fen "5r1K/2k1P3/3N1Bqp/N2Q1B1P/8/8/8/1R6 w - - 0 0"]

This problem was reprinted as no. 107 in Burt Hochberg's Chess Braintwisters.

White is in check, and his only reasonable move is to capture on f8 (1. Qg8? Qxg8#). But 1. exf8=R allows stalemate after 1. ... Qg7+ because the new rook guards d8 after 2. Bxg7. Similarly, 1. exf8=N guards d7 allowing 1. ... Qh7+ and stalemate, and 1. exf8=Q/B guards d6 allowing 1. ... Qg8+ and stalemate. The answer is 1. exf8! (no promotion). Now 1. ... Qg7+ 2. Bxg7 releases d8; 1. ... Qh7+ 2. Bxh7 releases d7; 1. ... Qg8 2. Qxg8 releases d6. 1. ... Qxf6+ destroys the stalemate at once, and White wins. Solution line and discussion taken from Chess Braintwisters.

The Matt Bengtson problem Prof. Elkies mentions is:

[Title "Matt Bengtson, Chess Braintwisters (Burt Hochberg), no. 103. White to move & draw."]
[FEN "4kn2/3p1pPp/4pPpK/6P1/8/2p5/1b6/8 w - - 0 1"]

The start of the solution-line can't be shown in PGN as White's 1st move is illegal: 1. g8=Black queen Qg7+ (anything else stalemates White) 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 (Kd6 7. Qc5+) 7. Qc6+ Ka7/Ka5 8. Qb7/Qb5, drawing. White's queen is then taken for stalemate. Solution line taken from Chess Braintwisters.


[Title "A Visual Solution Minus The Illegal Move"]
[StartFlipped "0"]
[FEN "4knq1/3p1p1p/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

1... Qg7+ 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 7. Qc6+ Ka5 8. Qb5+ Kxb5

(This is in answer to a comment by Prof. Elkies to Dag Oskar Madsen's answer, but, it seems, the syntax that activates the PGN-viewer in answers doesn't do so in comments.)

A similar oddity from the same book:

[Title "John Beasley, EBUR, 1996. White to move & win."]
[fen "5r1K/2k1P3/3N1Bqp/N2Q1B1P/8/8/8/1R6 w - - 0 0"]

This problem was reprinted as no. 107 in Burt Hochberg's Chess Braintwisters.

White is in check, and his only reasonable move is to capture on f8 (1. Qg8? Qxg8#). But 1. exf8=R allows stalemate after 1. ... Qg7+ because the new rook guards d8 after 2. Bxg7. Similarly, 1. exf8=N guards d7 allowing 1. ... Qh7+ and stalemate, and 1. exf8=Q/B guards d6 allowing 1. ... Qg8+ and stalemate. The answer is 1. exf8! (no promotion). Now 1. ... Qg7+ 2. Bxg7 releases d8; 1. ... Qh7+ 2. Bxh7 releases d7; 1. ... Qg8 2. Qxg8 releases d6. 1. ... Qxf6+ destroys the stalemate at once, and White wins. Solution line and discussion taken from Chess Braintwisters.

5 A PGN for visual learners-it hurts my brain trying to do it in my head.
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The Matt Bengtson problem Prof. Elkies mentions is:

[Title "Matt Bengtson, Chess Braintwisters (Burt Hochberg), no. 103. White to move & draw."]
[FEN "4kn2/3p1pPp/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

[Title "The Game After The Illegal Move"]
[FEN "4knq1/3p1p1p/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

The start of the solution-line can't be shown in PGN as White's 1st move is illegal: 1. g8=Black queen Qg7+ (anything else stalemates White) 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 (Kd6 7. Qc5+) 7. Qc6+ Ka7/Ka5 8. Qb7/Qb5, drawing. White's queen is then taken for stalemate. Solution line taken from Chess Braintwisters.


[Title "A Visual Solution Minus The Illegal Move"]
[FEN "4knq1/3p1p1p/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

1... Qg7+ 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 7. Qc6+ Ka5 8. Qb5+ Kxb5

(This is in answer to a comment by Prof. Elkies to Dag Oskar Madsen's answer, but, it seems, the syntax that activates the PGN-viewer in answers doesn't do so in comments.)

A similar oddity from the same book:

[Title "John Beasley, EBUR, 1996. White to move & win."]
[fen "5r1K/2k1P3/3N1Bqp/N2Q1B1P/8/8/8/1R6 w - - 0 0"]

This problem was reprinted as no. 107 in Burt Hochberg's Chess Braintwisters.

White is in check, and his only reasonable move is to capture on f8 (1. Qg8? Qxg8#). But 1. exf8=R allows stalemate after 1. ... Qg7+ because the new rook guards d8 after 2. Bxg7. Similarly, 1. exf8=N guards d7 allowing 1. ... Qh7+ and stalemate, and 1. exf8=Q/B guards d6 allowing 1. ... Qg8+ and stalemate. The answer is 1. exf8! (no promotion). Now 1. ... Qg7+ 2. Bxg7 releases d8; 1. ... Qh7+ 2. Bxh7 releases d7; 1. ... Qg8 2. Qxg8 releases d6. 1. ... Qxf6+ destroys the stalemate at once, and White wins. Solution line and discussion taken from Chess Braintwisters.

The Matt Bengtson problem Prof. Elkies mentions is:

[Title "Matt Bengtson, Chess Braintwisters (Burt Hochberg), no. 103. White to move & draw."]
[FEN "4kn2/3p1pPp/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

[Title "The Game After The Illegal Move"]
[FEN "4knq1/3p1p1p/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

The start of the solution-line can't be shown in PGN as White's 1st move is illegal: 1. g8=Black queen Qg7+ (anything else stalemates White) 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 (Kd6 7. Qc5+) 7. Qc6+ Ka7/Ka5 8. Qb7/Qb5, drawing. White's queen is then taken for stalemate. Solution line taken from Chess Braintwisters.

(This is in answer to a comment by Prof. Elkies to Dag Oskar Madsen's answer, but, it seems, the syntax that activates the PGN-viewer in answers doesn't do so in comments.)

A similar oddity from the same book:

[Title "John Beasley, EBUR, 1996. White to move & win."]
[fen "5r1K/2k1P3/3N1Bqp/N2Q1B1P/8/8/8/1R6 w - - 0 0"]

This problem was reprinted as no. 107 in Burt Hochberg's Chess Braintwisters.

White is in check, and his only reasonable move is to capture on f8 (1. Qg8? Qxg8#). But 1. exf8=R allows stalemate after 1. ... Qg7+ because the new rook guards d8 after 2. Bxg7. Similarly, 1. exf8=N guards d7 allowing 1. ... Qh7+ and stalemate, and 1. exf8=Q/B guards d6 allowing 1. ... Qg8+ and stalemate. The answer is 1. exf8! (no promotion). Now 1. ... Qg7+ 2. Bxg7 releases d8; 1. ... Qh7+ 2. Bxh7 releases d7; 1. ... Qg8 2. Qxg8 releases d6. 1. ... Qxf6+ destroys the stalemate at once, and White wins. Solution line and discussion taken from Chess Braintwisters.

The Matt Bengtson problem Prof. Elkies mentions is:

[Title "Matt Bengtson, Chess Braintwisters (Burt Hochberg), no. 103. White to move & draw."]
[FEN "4kn2/3p1pPp/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

The start of the solution-line can't be shown in PGN as White's 1st move is illegal: 1. g8=Black queen Qg7+ (anything else stalemates White) 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 (Kd6 7. Qc5+) 7. Qc6+ Ka7/Ka5 8. Qb7/Qb5, drawing. White's queen is then taken for stalemate. Solution line taken from Chess Braintwisters.


[Title "A Visual Solution Minus The Illegal Move"]
[FEN "4knq1/3p1p1p/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

1... Qg7+ 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 7. Qc6+ Ka5 8. Qb5+ Kxb5

(This is in answer to a comment by Prof. Elkies to Dag Oskar Madsen's answer, but, it seems, the syntax that activates the PGN-viewer in answers doesn't do so in comments.)

A similar oddity from the same book:

[Title "John Beasley, EBUR, 1996. White to move & win."]
[fen "5r1K/2k1P3/3N1Bqp/N2Q1B1P/8/8/8/1R6 w - - 0 0"]

This problem was reprinted as no. 107 in Burt Hochberg's Chess Braintwisters.

White is in check, and his only reasonable move is to capture on f8 (1. Qg8? Qxg8#). But 1. exf8=R allows stalemate after 1. ... Qg7+ because the new rook guards d8 after 2. Bxg7. Similarly, 1. exf8=N guards d7 allowing 1. ... Qh7+ and stalemate, and 1. exf8=Q/B guards d6 allowing 1. ... Qg8+ and stalemate. The answer is 1. exf8! (no promotion). Now 1. ... Qg7+ 2. Bxg7 releases d8; 1. ... Qh7+ 2. Bxh7 releases d7; 1. ... Qg8 2. Qxg8 releases d6. 1. ... Qxf6+ destroys the stalemate at once, and White wins. Solution line and discussion taken from Chess Braintwisters.

4 A PGN for visual learners-it hurts my brain trying to do it in my head.
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The Matt Bengtson problem Prof. Elkies mentions is:

[Title "Matt Bengtson, Chess Braintwisters (Burt Hochberg), no. 103. White to move & draw."]
[fen[FEN "4kn2/3p1pPp/4pPpK/6P1/8/2p5/1b6/8 wb - - 0 0"]1"]

[Title "The Game After The Illegal Move"]
1.[FEN "4knq1/3p1p1p/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]
 

The start of the solution-line can't be shown in PGN as White's 1st move is illegal: 1. g8=Black queen Qg7+ (anything else stalemates White) 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 (Kd6 7. Qc5+) 7. Qc6+ Ka7/Ka5 8. Qb7/Qb5, drawing. White's queen is then taken for stalemate. Solution line taken from Chess Braintwisters.

(This is in answer to a comment by Prof. Elkies to Dag Oskar Madsen's answer, but, it seems, the syntax that activates the PGN-viewer in answers doesn't do so in comments.)

A similar oddity from the same book:

[Title "John Beasley, EBUR, 1996. White to move & win."]
[fen "5r1K/2k1P3/3N1Bqp/N2Q1B1P/8/8/8/1R6 w - - 0 0"]

This problem was reprinted as no. 107 in Burt Hochberg's Chess Braintwisters.

White is in check, and his only reasonable move is to capture on f8 (1. Qg8? Qxg8#). But 1. exf8=R allows stalemate after 1. ... Qg7+ because the new rook guards d8 after 2. Bxg7. Similarly, 1. exf8=N guards d7 allowing 1. ... Qh7+ and stalemate, and 1. exf8=Q/B guards d6 allowing 1. ... Qg8+ and stalemate. The answer is 1. exf8! (no promotion). Now 1. ... Qg7+ 2. Bxg7 releases d8; 1. ... Qh7+ 2. Bxh7 releases d7; 1. ... Qg8 2. Qxg8 releases d6. 1. ... Qxf6+ destroys the stalemate at once, and White wins. Solution line and discussion taken from Chess Braintwisters.

The Matt Bengtson problem Prof. Elkies mentions is:

[Title "Matt Bengtson, Chess Braintwisters (Burt Hochberg), no. 103. White to move & draw."]
[fen "4kn2/3p1pPp/4pPpK/6P1/8/2p5/1b6/8 w - - 0 0"]

1.

The start of the solution-line can't be shown in PGN as White's 1st move is illegal: 1. g8=Black queen Qg7+ (anything else stalemates White) 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 (Kd6 7. Qc5+) 7. Qc6+ Ka7/Ka5 8. Qb7/Qb5, drawing. White's queen is then taken for stalemate. Solution line taken from Chess Braintwisters.

(This is in answer to a comment by Prof. Elkies to Dag Oskar Madsen's answer, but, it seems, the syntax that activates the PGN-viewer in answers doesn't do so in comments.)

A similar oddity from the same book:

[Title "John Beasley, EBUR, 1996. White to move & win."]
[fen "5r1K/2k1P3/3N1Bqp/N2Q1B1P/8/8/8/1R6 w - - 0 0"]

This problem was reprinted as no. 107 in Burt Hochberg's Chess Braintwisters.

White is in check, and his only reasonable move is to capture on f8 (1. Qg8? Qxg8#). But 1. exf8=R allows stalemate after 1. ... Qg7+ because the new rook guards d8 after 2. Bxg7. Similarly, 1. exf8=N guards d7 allowing 1. ... Qh7+ and stalemate, and 1. exf8=Q/B guards d6 allowing 1. ... Qg8+ and stalemate. The answer is 1. exf8! (no promotion). Now 1. ... Qg7+ 2. Bxg7 releases d8; 1. ... Qh7+ 2. Bxh7 releases d7; 1. ... Qg8 2. Qxg8 releases d6. 1. ... Qxf6+ destroys the stalemate at once, and White wins. Solution line and discussion taken from Chess Braintwisters.

The Matt Bengtson problem Prof. Elkies mentions is:

[Title "Matt Bengtson, Chess Braintwisters (Burt Hochberg), no. 103. White to move & draw."]
[FEN "4kn2/3p1pPp/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]

[Title "The Game After The Illegal Move"]
[FEN "4knq1/3p1p1p/4pPpK/6P1/8/2p5/1b6/8 b - - 0 1"]
 

The start of the solution-line can't be shown in PGN as White's 1st move is illegal: 1. g8=Black queen Qg7+ (anything else stalemates White) 2. fxg7 c2 3. g8=Q c1=Q 4. Qxf7+ Kd8 5. Qe8+ Kc7 6. Qc8+ Kb6 (Kd6 7. Qc5+) 7. Qc6+ Ka7/Ka5 8. Qb7/Qb5, drawing. White's queen is then taken for stalemate. Solution line taken from Chess Braintwisters.

(This is in answer to a comment by Prof. Elkies to Dag Oskar Madsen's answer, but, it seems, the syntax that activates the PGN-viewer in answers doesn't do so in comments.)

A similar oddity from the same book:

[Title "John Beasley, EBUR, 1996. White to move & win."]
[fen "5r1K/2k1P3/3N1Bqp/N2Q1B1P/8/8/8/1R6 w - - 0 0"]

This problem was reprinted as no. 107 in Burt Hochberg's Chess Braintwisters.

White is in check, and his only reasonable move is to capture on f8 (1. Qg8? Qxg8#). But 1. exf8=R allows stalemate after 1. ... Qg7+ because the new rook guards d8 after 2. Bxg7. Similarly, 1. exf8=N guards d7 allowing 1. ... Qh7+ and stalemate, and 1. exf8=Q/B guards d6 allowing 1. ... Qg8+ and stalemate. The answer is 1. exf8! (no promotion). Now 1. ... Qg7+ 2. Bxg7 releases d8; 1. ... Qh7+ 2. Bxh7 releases d7; 1. ... Qg8 2. Qxg8 releases d6. 1. ... Qxf6+ destroys the stalemate at once, and White wins. Solution line and discussion taken from Chess Braintwisters.

3 correct a typo in my copying from Hochberg
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2 Provided a further example where wP reaches rank 8 & doesn't promote, as per comment.
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1
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