Answers.Body: <p>Since there are 10^umpty possible chess positions <a href="https://chess.stackexchange.com/questions/5592/what-is-the-number-of-legal-positions-in-a-chess-game">(an obvious upper bound is 13^64, and estimates are 10^45+-5)</a>, even a supercomputer is useless.</p>
  <p>The following &quot;Monte Carlo&quot; attempt is a feasible looking way to give an approximate answer, though:</p>
  <ul>
  <li>Randomly generate a position (already nontrivial: how to generate truely random material in the sense that it's the correct proportion of legal positions?)</li>
  <li>Make a <a href="https://chess.stackexchange.com/questions/35602/which-methods-can-be-used-to-prove-that-a-position-is-illegal">flimsy legality test</a> (material, impossible checks - a total one is very hard!)</li>
  <li>Test whether it's a mate.</li>
  <li>Rinse and repeat.</li>
  </ul>
  <p>Assume that you test p positions, m are mate and there is a total t of legal chess positions, your answer is approximately t*m/p. I only fear that m/p is very low (for very full boards, <a href="https://ulthiel.com/math/other/endgames/" rel="nofollow noreferrer">in light positions things are different</a>) and you again need a far too high p to get meaningful stats.</p>
  <p>I strongly advise to make first tests on smaller board. For example, <a href="http://kirr.homeunix.org/chess/3x4-chess/" rel="nofollow noreferrer">3x4 chess is tablebase-solved</a> and the analog of your value could readily be extracted.</p>
  
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Answers.CreationDate: 2022-05-24T07:27:04.550Z
Answers.LastActivityDate: 2022-05-24T07:27:04.550Z
Answers.UserDisplayName: Hauke Reddmann
Body: <p>How many different checkmate positions are there? A Super-Computer problem. Not with any conditions or hypothetical situations, just simply the number of checkmates that can be arrived at in a properly played game.</p>
  
Tags: <checkmate>
Title: Number of checkmate positions
ThreadText: # Number of checkmate positions
  
  (Asked by: username_0 on 2022-05-23)
  
  How many different checkmate positions are there? A Super-Computer problem. Not with any conditions or hypothetical situations, just simply the number of checkmates that can be arrived at in a properly played game.
  
  **Tags:** checkmate
  
  
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  ## 1 answers
  
  ### Answer by username_1 on 2022-05-24. Score: 1
  
  Since there are 10^umpty possible chess positions [(an obvious upper bound is 13^64, and estimates are 10^45+-5)](https://chess.stackexchange.com/questions/5592/what-is-the-number-of-legal-positions-in-a-chess-game), even a supercomputer is useless.
  
  The following "Monte Carlo" attempt is a feasible looking way to give an approximate answer, though:
  
  * Randomly generate a position (already nontrivial: how to generate truely random material in the sense that it's the correct proportion of legal positions?)
  * Make a [flimsy legality test](https://chess.stackexchange.com/questions/35602/which-methods-can-be-used-to-prove-that-a-position-is-illegal) (material, impossible checks - a total one is very hard!)
  * Test whether it's a mate.
  * Rinse and repeat.
  
  Assume that you test p positions, m are mate and there is a total t of legal chess positions, your answer is approximately t\*m/p. I only fear that m/p is very low (for very full boards, [in light positions things are different](https://ulthiel.com/math/other/endgames/)) and you again need a far too high p to get meaningful stats.
  
  I strongly advise to make first tests on smaller board. For example, [3x4 chess is tablebase-solved](http://kirr.homeunix.org/chess/3x4-chess/) and the analog of your value could readily be extracted.
  
  
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