user897: Many years ago, I was shown a problem involving three white pawns promoting to knights and ending up with a ring of five knights encircling the mated king. Does anybody know the starting position?
A few more thoughts:
(1) I think it was mate in seven.  
(2) Unusually, every white move is forced because there is a mate threat from black.  
(3) Unless I have the memory of the board upside down, the black king starts in the middle of the board and ends up on his sixth or even seventh rank surrounded by two pairs of white knights protecting each other and the fifth one executing the mate.  
(4) The white king is somewhere really odd like h7 and the black mate threat comes from a minor piece - a bishop, perhaps?

user852: I don't know about 5 knights, but it has been shown that 3 knights and a king can force mate.
The method is given in Reuben Fine's Basic Chess Endings, ISBN 0-679-14002-6, my copy of which seems to have disappeared, along with the memory of its contents in this instance. I do however, distinctly remember that this endgame situation is covered in the book.

user8860: You must be thinking of this classic Korolkov study (I-II Prize "64" 1937, according to Chernev's Chessboard Magic!; P1288537 in PDB).  It's not a problem, nor in 7 moves, but seems to match the rest closely:
[Title "White to play and win (Korolkov 1937)"]
[Fen "4N3/2PP2P1/6b2/2bN2R1/6pk/p7/K1pn4/8 w - - 0 0"]

1.Rh5+ Kxh5 2.Nf4+ Kh6 3.g8=N+ Kh7 4.Ngf6+ Kh6 5.Nxg4+ Kh7 6.Nef6+ Kg7 7.Ne6+ Kf7 8.d8=N+ Ke7 9.c8=N#

In the starting diagram Black threatens mate with, appropriately, a checking Knight promotion: c1N+ and either Ndb3# or Bd4+ and mate.  If Black diverges from the main line then White usually promotes a Queen with check; exceptions are 1...Kg3 2.Rh1 and 4...Kh8 5.Nxg6#.
It is possible that youur friend showed you the problem without the first two moves, thus making you think that it was a mate in 7.

user757: In one of my books I found this "uncredited" problem, it is probably not the problem you are referring to but I thought you may find it interesting, hope you like it.
2018 Update, thanks to user @Evargalo for adding some light about the author, we can know it was composed by Gijs van Breukelen.
8/3P3k/n2K3p/2p3n1/1b4N1/2p1p1P1/8/3B4 w - - 0 1

1. Nf6+ Kg7 2. Nh5+ Kg6 3. Bc2+ Kxh5 4. d8=Q Nf7+ 5. Ke6 Nxd8+ 6. Kf5 e2 7.Be4 e1=N 8. Bd5 c2 9. Bc4 c1=N 10. Bb5 Nc7 11. Ba4 Ne2 12. Bd1 Nf3 13. Bxe2

White, with material inferiority, must act energically:
1.Nf6+ Kg7 if 1...Kg6 2.Bc2+! Kg7 3.Nh5+ Kf7 4.d8=Q and black has lost the possibility of check on f7.
2.Nh5+ Kg6  The black King cannot go to the last rank, d7 will crown with check.
3.Bc2+ Kxh5 4. d8=Q Nf7+ 5. Ke6 Nxd8+ 6. Kf5 This is the first part of the study: White has accomplished to place the black king in a mate position. In the next set of moves each defensive move by black will be forced.
6...e2 7.Be4! Threat on f3 the only move to prevent checkmate is a sub-promotion.
7...e1=N 8.Bd5!! Threats 9.Bc4 10.Be2# 
8...c2 9.Bc4 c1=N! 10.Bb5! With the idea of checkmate on e8
10...Nc7 11.Ba4! and 4 Knights cannot prevent checkmate
11...Ne2 12.Bd1 Nf3 13.Bxe2 followed by 14.Bxf3#

897: <p>Many years ago, I was shown a problem involving three white pawns promoting to knights and ending up with a ring of five knights encircling the mated king. Does anybody know the starting position?</p>

<p>A few more thoughts:<br>
(1) I think it was mate in seven.  </p>

<p>(2) Unusually, every white move is forced because there is a mate threat from black.  </p>

<p>(3) Unless I have the memory of the board upside down, the black king starts in the middle of the board and ends up on his sixth or even seventh rank surrounded by two pairs of white knights protecting each other and the fifth one executing the mate.  </p>

<p>(4) The white king is somewhere really odd like h7 and the black mate threat comes from a minor piece - a bishop, perhaps?</p>
 852: <p>I don't know about 5 knights, but it has been shown that 3 knights and a king can force mate.</p>

<p>The method is given in Reuben Fine's Basic Chess Endings, ISBN 0-679-14002-6, my copy of which seems to have disappeared, along with the memory of its contents in this instance. I do however, distinctly remember that this endgame situation is covered in the book.</p>
 8860: <p>You must be thinking of this classic Korolkov study (I-II Prize &quot;64&quot; 1937, according to Chernev's <em>Chessboard Magic!</em>; P1288537 in PDB).  It's not a problem, nor in 7 moves, but seems to match the rest closely:</p>
<pre><code>[Title &quot;White to play and win (Korolkov 1937)&quot;]
[Fen &quot;4N3/2PP2P1/6b2/2bN2R1/6pk/p7/K1pn4/8 w - - 0 0&quot;]

1.Rh5+ Kxh5 2.Nf4+ Kh6 3.g8=N+ Kh7 4.Ngf6+ Kh6 5.Nxg4+ Kh7 6.Nef6+ Kg7 7.Ne6+ Kf7 8.d8=N+ Ke7 9.c8=N#
</code></pre>
<p>In the starting diagram Black threatens mate with, appropriately, a checking Knight promotion: c1N+ and either Ndb3# or Bd4+ and mate.  If Black diverges from the main line then White usually promotes a Queen with check; exceptions are 1...Kg3 2.Rh1 and 4...Kh8 5.Nxg6#.</p>
<p>It is possible that youur friend showed you the problem without the first two moves, thus making you think that it was a mate in 7.</p>
 757: <p>In one of my books I found this "uncredited" problem, it is probably not the problem you are referring to but I thought you may find it interesting, hope you like it.</p>

<p>2018 Update, thanks to user @Evargalo for adding some light about the author, we can know it was composed by Gijs van Breukelen.</p>

<pre><code>8/3P3k/n2K3p/2p3n1/1b4N1/2p1p1P1/8/3B4 w - - 0 1

1. Nf6+ Kg7 2. Nh5+ Kg6 3. Bc2+ Kxh5 4. d8=Q Nf7+ 5. Ke6 Nxd8+ 6. Kf5 e2 7.Be4 e1=N 8. Bd5 c2 9. Bc4 c1=N 10. Bb5 Nc7 11. Ba4 Ne2 12. Bd1 Nf3 13. Bxe2
</code></pre>

<p>White, with material inferiority, must act energically:</p>

<p><strong><code>1.Nf6+ Kg7</code></strong> if 1...Kg6 2.Bc2+! Kg7 3.Nh5+ Kf7 4.d8=Q and black has lost the possibility of check on f7.</p>

<p><strong><code>2.Nh5+ Kg6</code></strong>  The black King cannot go to the last rank, d7 will crown with check.</p>

<p><strong><code>3.Bc2+ Kxh5 4. d8=Q Nf7+ 5. Ke6 Nxd8+ 6. Kf5</code></strong> This is the first part of the study: White has accomplished to place the black king in a mate position. In the next set of moves each defensive move by black will be forced.</p>

<p><strong><code>6...e2 7.Be4!</code></strong> Threat on f3 the only move to prevent checkmate is a sub-promotion.</p>

<p><strong><code>7...e1=N 8.Bd5!!</code></strong> Threats 9.Bc4 10.Be2# </p>

<p><strong><code>8...c2 9.Bc4 c1=N! 10.Bb5!</code></strong> With the idea of checkmate on e8</p>

<p><strong><code>10...Nc7 11.Ba4!</code></strong> and 4 Knights cannot prevent checkmate</p>

<p><strong><code>11...Ne2 12.Bd1 Nf3 13.Bxe2</code></strong> followed by 14.Bxf3#</p>
