user11144: Compose a position where

It's White to move
White can deliver checkmate or stalemate in 1 move
Or if they make a different move, then Black can deliver checkmate or stalemate in one 1 move

I needed 6 white pieces and 11 black pieces. Beat me!
Toned down example for understanding:
[FEN "7k/7P/7K/6BP/8/8/8/8 w - - 0 0"]

After the next White move, Black is either mated or stalemated, depending on if White plays Bf6 or not. (This does not meet the 3rd requirement).

user16650: Just one piece fewer, but how about:
[FEN "5brr/4p1pq/4P1pk/4N1p1/6P1/6PK/6PP/8 w - - 0 1"]

1.Nd7= {Black stalemated} (1.Nf7#) (1.Nxg6 Kxg6#) (1.Nxg6 Qxg6= {White stalemated})


user11144: I knew it! 13 pieces:
[FEN "8/6p1/6P1/6PB/6P1/4p1pQ/4Pp1p/5K1k w - - 0 0"]

1.Qg2#/1.Qxg3=/1.Qh4 g2#/1.Qxh2+ K,gxh2=
Note that 5 pieces just stand around against 1.Qg4, which also ends the game, but in 1.5 moves - still too bad!

11144: <p>Compose a position where</p>
<ol>
<li>It's White to move</li>
<li>White can deliver checkmate or stalemate in 1 move</li>
<li>Or if they make a different move, then Black can deliver checkmate or stalemate in one 1 move</li>
</ol>
<p>I needed 6 white pieces and 11 black pieces. Beat me!</p>
<p>Toned down example for understanding:</p>
<pre><code>[FEN &quot;7k/7P/7K/6BP/8/8/8/8 w - - 0 0&quot;]
</code></pre>
<p>After the next White move, Black is either mated or stalemated, depending on if White plays Bf6 or not. (This does not meet the 3rd requirement).</p>
 16650: <p>Just one piece fewer, but how about:</p>
<pre><code>[FEN &quot;5brr/4p1pq/4P1pk/4N1p1/6P1/6PK/6PP/8 w - - 0 1&quot;]

1.Nd7= {Black stalemated} (1.Nf7#) (1.Nxg6 Kxg6#) (1.Nxg6 Qxg6= {White stalemated})
</code></pre>
 11144: <p>I knew it! 13 pieces:</p>
<pre><code>[FEN &quot;8/6p1/6P1/6PB/6P1/4p1pQ/4Pp1p/5K1k w - - 0 0&quot;]
</code></pre>
<p>1.Qg2#/1.Qxg3=/1.Qh4 g2#/1.Qxh2+ K,gxh2=</p>
<p>Note that 5 pieces just stand around against 1.Qg4, which also ends the game, but in 1.5 moves - <em>still</em> too bad!</p>
