user3207: There have been several questions regarding the possible number of chess games. for example; Database of every possible move in chess. However,can there be an estimate on the number of "allowable" chess positions?
For another example, White pawns cannot occupy the first row and Black pawns cannot occupy the eighth row. If a given chess piece is occupying a square, then another chess piece cannot occupy that. Using simple elimination rules coded, can we have an estimate on the number of unique positions that chess pieces can have since they would be reducing in number? An upper bound on positions before applying any elimination rules would be the sum of (64 C 32 + 64 C 32 ... + 64 C 2). This is less than 10^20. With elimination rules, they should be significantly lower.
Any ideas on how elimination rules are written for chess pieces?

user16966: It is really difficult to find out legal positions but total positions possible is 13^64 , assuming each piece can be positioned in every  available square

user16908: Old question, but for people who just want a quick estimate they can do with basic math this is an easy way to go about it
If we consider all 32 pieces, we can rearrange them P(64,32) or 64 permute 32 ways. If we then consider 31 pieces, we do the same as above with P(64,31) but this time we must factor in the choice of 31 out of the 32 pieces so we multiply P(64,31) by C(32,31) or 32 choose 31. If we continue along this manner we have the sum from n =2 to 32 of P(64,n)C(32,n) which gives 1.24e54 which is only a few orders of magnitude larger than the number given by Vicor Allis.
Note that in the assumptions, we allowed illegal positions, and double or triple counted certain positions by treating each piece as distinct. But the number is reasonable compared to that above and more importantly is easily attainable with pretty basic math.

user1485: Upper bounds on the number of allowable positions are discussed on the Shannon number  Wikipedia page.
There is an accepted upper bound of 1.8x10^46 by Shirish Chinchalkar, and a somewhat accurate estimate of 4x10^44 by John Tromp.
By comparison, the number of atoms on earth is about 10^50.

3207: <p>There have been several questions regarding the possible number of chess games. for example; <a href="https://chess.stackexchange.com/questions/2422/database-of-every-possible-move-in-chess">Database of every possible move in chess</a>. However,can there be an estimate on the number of &quot;allowable&quot; chess positions?</p>
<p>For another example, White pawns cannot occupy the first row and Black pawns cannot occupy the eighth row. If a given chess piece is occupying a square, then another chess piece cannot occupy that. Using simple elimination rules coded, can we have an estimate on the number of unique positions that chess pieces can have since they would be reducing in number? An upper bound on positions before applying any elimination rules would be the sum of (64 C 32 + 64 C 32 ... + 64 C 2). This is less than 10^20. With elimination rules, they should be significantly lower.</p>
<p>Any ideas on how elimination rules are written for chess pieces?</p>
 16966: <p>It is really difficult to find out legal positions but total positions possible is <code>13^64</code> , assuming each piece can be positioned in every  available square</p>
 16908: <p>Old question, but for people who just want a quick estimate they can do with basic math this is an easy way to go about it</p>
<p>If we consider all 32 pieces, we can rearrange them P(64,32) or 64 permute 32 ways. If we then consider 31 pieces, we do the same as above with P(64,31) but this time we must factor in the choice of 31 out of the 32 pieces so we multiply P(64,31) by C(32,31) or 32 choose 31. If we continue along this manner we have the sum from n =2 to 32 of P(64,n)C(32,n) which gives <strong>1.24e54</strong> which is only a few orders of magnitude larger than the number given by Vicor Allis.</p>
<p>Note that in the assumptions, we allowed illegal positions, and double or triple counted certain positions by treating each piece as distinct. But the number is reasonable compared to that above and more importantly is easily attainable with pretty basic math.</p>
 1485: <p>Upper bounds on the number of allowable positions are discussed on the <a href="http://en.wikipedia.org/wiki/Shannon_number" rel="nofollow noreferrer">Shannon number</a>  Wikipedia page.</p>
<p>There is an accepted upper bound of 1.8x10^46 by Shirish Chinchalkar, and a somewhat accurate estimate of 4x10^44 by <a href="https://github.com/tromp/ChessPositionRanking" rel="nofollow noreferrer">John Tromp</a>.</p>
<p>By comparison, the number of atoms on earth is about 10^50.</p>
