user32561: 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.

user11144: Since there are 10^umpty possible chess positions (an obvious upper bound is 13^64, and estimates are 10^45+-5), 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 (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) 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 and the analog of your value could readily be extracted.

32561: <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>
 11144: <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>
