Answers.Body: <p>This answer is extended from my comment to Ross Milikan's answer to your question <a href="https://chess.stackexchange.com/questions/5769/numerical-fen-writing">Numerical FEN writing</a>.</p>
  
  <p>There is a <a href="http://kirill-kryukov.com/chess/nulp/" rel="nofollow noreferrer">website</a> devoted to counting the number of legal positions up to equivalence with given number of pieces. Currently they have results for positions with 2 to 8 pieces. </p>
  
  <pre><code>2:                     462
  3:                 368,079
  4:             125,246,598
  5:          25,912,594,054
  6:       3,787,154,440,416
  7:     423,836,835,667,331
  8:  38,176,306,877,748,245
  </code></pre>
  
  <p>With their definition, a position is legal if:</p>
  
  <ul>
  <li>It has exactly one white and one black king</li>
  <li>There are no pawns on the 1st and 8th lines</li>
  <li>Side to move is not giving a check</li>
  </ul>
  
  <p>Note that with these definitions, some positions could be legal even if they cannot be achieved in a real game. Edit: As @Pimgd pointed out in the comments, an example is a position with white pawns on a2, b2 and a3.</p>
  
  <p>Two positions are considered equivalent if they are essentially the same, in this case, one position can be obtained from another by combining the following operations:</p>
  
  <ul>
  <li>Any position: Swapping the colours of the pieces, the side to move, and mirroring the position around the horizontal line such that the square a1 becomes a8 etc.</li>
  <li>Positions with no castling rights: mirroring the position around the vertical line such that the square a1 becomes h1 etc.  </li>
  <li>Pawnless positions with no castling rights: mirroring the position around the horizontal line such that a1 becomes a8 etc.</li>
  <li>Pawnless positions with no castling rights: Mirroring the board diagonally</li>
  </ul>
  
  <p>Depending on the existence of pawns and castling rights, each "unique" position thus corresponds to 2 to 16 different positions.</p>
  
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2014-07-16T12:13:39.927Z
Answers.Text: "I cannot think of an example of such a position with only 8 pieces on board." FEN:(4k3/8/8/8/8/P7/PP6/4K3 w KQkq - 0 1) - basically, 3 white pawns on a2, b2, a3. That's a position you can't achieve naturally with only 5 pieces.
Answers.UserDisplayName: Pimgd
Answers.ContentLicense: CC BY-SA 3.0
Answers.CreationDate: 2014-07-16T10:44:47.970Z
Answers.LastActivityDate: 2014-07-16T13:16:46.190Z
Answers.LastEditDate: 2017-04-13T12:48:53.737Z
Answers.UserDisplayName: JiK
Body: <p><a href="https://chess.stackexchange.com/questions/5592/positions-in-chess-games">What is the number of legal positions in a chess game?</a></p>
  
  <p>This is a question asked earlier on the site. Now I am wondering, what can be the number of positions for a certain number of pieces. Like 8 pieces for example.</p>
  
  <p>Is there a top bar formula like in that case? Thank you!</p>
  
Tags: <pieces><game-complexity>
Title: Number of positions with specifically a number of pieces
ThreadText: # Number of positions with specifically a number of pieces
  
  (Asked by: username_0 on 2014-07-16)
  
  [What is the number of legal positions in a chess game?](https://chess.stackexchange.com/questions/5592/positions-in-chess-games)
  
  This is a question asked earlier on the site. Now I am wondering, what can be the number of positions for a certain number of pieces. Like 8 pieces for example.
  
  Is there a top bar formula like in that case? Thank you!
  
  **Tags:** pieces, game-complexity
  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2014-07-16. Score: 9 | Accepted answer
  
  This answer is extended from my comment to Ross Milikan's answer to your question Numerical FEN writing.
  There is a website devoted to counting the number of legal positions up to equivalence with given number of pieces. Currently they have results for positions with 2 to 8 pieces. 
  2:                     462
  3:                 368,079
  4:             125,246,598
  5:          25,912,594,054
  6:       3,787,154,440,416
  7:     423,836,835,667,331
  8:  38,176,306,877,748,245
  
  With their definition, a position is legal if:
  
  It has exactly one white and one black king
  There are no pawns on the 1st and 8th lines
  Side to move is not giving a check
  
  Note that with these definitions, some positions could be legal even if they cannot be achieved in a real game. Edit: As @username_2 pointed out in the comments, an example is a position with white pawns on a2, b2 and a3.
  Two positions are considered equivalent if they are essentially the same, in this case, one position can be obtained from another by combining the following operations:
  
  Any position: Swapping the colours of the pieces, the side to move, and mirroring the position around the horizontal line such that the square a1 becomes a8 etc.
  Positions with no castling rights: mirroring the position around the vertical line such that the square a1 becomes h1 etc.  
  Pawnless positions with no castling rights: mirroring the position around the horizontal line such that a1 becomes a8 etc.
  Pawnless positions with no castling rights: Mirroring the board diagonally
  
  Depending on the existence of pawns and castling rights, each "unique" position thus corresponds to 2 to 16 different positions.
  
  > **Comments:**
  > **username_2**: "I cannot think of an example of such a position with only 8 pieces on board." FEN:(4k3/8/8/8/8/P7/PP6/4K3 w KQkq - 0 1) - basically, 3 white pawns on a2, b2, a3. That's a position you can't achieve naturally with only 5 pieces.  
  
  ---
  
  