Answers.Body: <p>It is true that for larger numbers of moves it becomes easier. From the following solutions, the ones from n=2 to n=8 are clearly optimal. The solution for n=1 looks good, but it may not be optimal.</p>
  <h3>All pieces have 1 move</h3>
  <pre><code>[FEN &quot;8/8/8/1pp5/8/K7/8/k7 w - - 0 1&quot;]
  </code></pre>
  <h3>2 moves (found by ghilesZ)</h3>
  <pre><code>[FEN &quot;2k5/K1p5/8/8/8/8/8/8 w - - 0 1&quot;]
  </code></pre>
  <h3>3 moves</h3>
  <pre><code>[FEN &quot;8/8/8/8/8/8/K7/2k5 w KQkq - 0 1&quot;]
  </code></pre>
  <h3>4 moves</h3>
  <pre><code>[FEN &quot;8/8/8/8/8/K7/8/2k5 w KQkq - 0 1&quot;]
  </code></pre>
  <h3>5 moves</h3>
  <pre><code>[FEN &quot;8/8/8/8/8/K7/8/3k4 w KQkq - 0 1&quot;]
  </code></pre>
  <h3>6 moves</h3>
  <pre><code>[FEN &quot;8/8/8/8/8/1K6/3k4/8 w KQkq - 0 1&quot;]
  </code></pre>
  <h3>7 moves</h3>
  <pre><code>[FEN &quot;8/8/8/8/1K6/8/3k4/8 w KQkq - 0 1&quot;]
  </code></pre>
  <h3>8 moves</h3>
  <pre><code>[FEN &quot;8/8/8/8/1K6/8/4k3/8 w KQkq - 0 1&quot;]
  </code></pre>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-04-17T08:30:22.103Z
Answers.Text: I totally overlooked the 2 move with 3 pieces solution. Perfect! 1 move can also be done with 2 black pawns, but that is hardly a real improvement.
Answers.UserDisplayName: Hauke Reddmann
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-04-17T11:41:22.963Z
Answers.Text: I think I've found a 3 unit solution for 2 moves: 8/8/8/8/8/8/k1P5/2K5 b - - 0 1. It is optimal as it can not be achieved with only 2 kings.
Answers.UserDisplayName: ghilesZ
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-04-17T16:33:55.513Z
Answers.Text: @HaukeReddmann I found the solution with 2 black pawns, which is better according to the given definition of "lightness", so I updated the answer.
Answers.UserDisplayName: wimi
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-04-17T16:34:54.133Z
Answers.Text: @ghilesZ good find! I added it to the answer (or a symmetric one, because "rather black than white" according to the question).
Answers.UserDisplayName: wimi
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-04-17T20:51:09.773Z
Answers.Text: @ghilesZ: And I overlooked that too...BTW, "rather black than white material" is sort of problem chess standard.
Answers.UserDisplayName: Hauke Reddmann
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-04-16T21:01:48.760Z
Answers.LastActivityDate: 2021-04-17T16:32:24.007Z
Answers.LastEditDate: 2021-04-17T16:32:24.007Z
Answers.UserDisplayName: wimi
Body: <pre><code>[FEN &quot;8/8/8/8/pp6/kp6/1p6/1K6 w KQkq - 0 1&quot;]
  </code></pre>
  <p>This is a well-known position: both sides are stalemated (if it's their move). It's the lightest position with this property.</p>
  <p>Devise the lightest positions where <em>each</em> piece of each side has exactly n moves. (So this would be n=0, you may go up to n=8, which is of course solved by putting only the kings into the free field.)</p>
  <p>&quot;Lightest&quot; is defined in the standard way: first piece count, rather pawns than figures, rather black than white.</p>
  <p>Experts, please hold you back, it's a very easy puzzle. :-)</p>
  
Tags: <problems><puzzles><records><mathematics>
Title: A mobility puzzle
ThreadText: # A mobility puzzle
  
  (Asked by: username_0 on 2021-04-16)
  
  [FEN "8/8/8/8/pp6/kp6/1p6/1K6 w KQkq - 0 1"]
  
  This is a well-known position: both sides are stalemated (if it's their move). It's the lightest position with this property.
  Devise the lightest positions where each piece of each side has exactly n moves. (So this would be n=0, you may go up to n=8, which is of course solved by putting only the kings into the free field.)
  "Lightest" is defined in the standard way: first piece count, rather pawns than figures, rather black than white.
  Experts, please hold you back, it's a very easy puzzle. :-)
  
  **Tags:** problems, puzzles, records, mathematics
  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2021-04-16. Score: 3 | Accepted answer
  
  It is true that for larger numbers of moves it becomes easier. From the following solutions, the ones from n=2 to n=8 are clearly optimal. The solution for n=1 looks good, but it may not be optimal.
  All pieces have 1 move
  [FEN "8/8/8/1pp5/8/K7/8/k7 w - - 0 1"]
  
  2 moves (found by username_2)
  [FEN "2k5/K1p5/8/8/8/8/8/8 w - - 0 1"]
  
  3 moves
  [FEN "8/8/8/8/8/8/K7/2k5 w KQkq - 0 1"]
  
  4 moves
  [FEN "8/8/8/8/8/K7/8/2k5 w KQkq - 0 1"]
  
  5 moves
  [FEN "8/8/8/8/8/K7/8/3k4 w KQkq - 0 1"]
  
  6 moves
  [FEN "8/8/8/8/8/1K6/3k4/8 w KQkq - 0 1"]
  
  7 moves
  [FEN "8/8/8/8/1K6/8/3k4/8 w KQkq - 0 1"]
  
  8 moves
  [FEN "8/8/8/8/1K6/8/4k3/8 w KQkq - 0 1"]
  
  > **Comments:**
  > **username_0**: I totally overlooked the 2 move with 3 pieces solution. Perfect! 1 move can also be done with 2 black pawns, but that is hardly a real improvement.  
  > **username_2**: I think I've found a 3 unit solution for 2 moves: 8/8/8/8/8/8/k1P5/2K5 b - - 0 1. It is optimal as it can not be achieved with only 2 kings.  
  > **username_1**: @HaukeReddmann I found the solution with 2 black pawns, which is better according to the given definition of "lightness", so I updated the answer.  
  > **username_1**: @username_2 good find! I added it to the answer (or a symmetric one, because "rather black than white" according to the question).  
  > **username_0**: @username_2: And I overlooked that too...BTW, "rather black than white material" is sort of problem chess standard.  
  
  ---
  
  