Hello all:
How many possible Chess games are there?
Well, I think I've come up with an easy, intuitive disproof of a
number one often sees. Perhaps some of the mathematically savvy
readers of this group can give their opinion of my claim. (Thanks in
advance, should you choose to do so.)
Here's a number I often see:
"In a game of 40 moves, the number of possible board positions P(40)
is at least 10^(120) according to Peterson (1996)...."
http://mathworld.wolfram.com/Chess.html
(other estimates are given in the article hyperlinked above)
10^(120) is a 1 followed by 120 zeros
For comparison, one estimate of the number of particles in the
observable universe is 10^(80) -- a much, much smaller number.
(
http://varatek.com/scott/bnum.html )
Well, the 10^(120) estimate is, I think, demonstrably too large (by
far). Here's my reason for thinking so:
In a game of Chess, a square can be in only one of 13 states. It is
either
1) empty,
2) occupied by one of the 6 white pieces (King, Queen, Rook, Bishop,
Knight, Pawn), or
3) occupied by one of the 6 black pieces (King, Queen, Rook, Bishop,
Knight, Pawn).
Since there are 64 squares, this means there are 13^(64) imaginable
states of the Chessboard.
Of course, 13^(64) is itself an overestimate, since there are not
enough pieces to "populate" all the squares with all possible
combinations (and even if there were, not all possible combinations
are legal).
But what we can tell from this consideration is that any number that
is GREATER than 13^(64) is obviously too high!
Now, just how big is 13^(64)? Well, my Microsoft Windows calculator
gives the estimate
1.96053...e+71 (a 72-digit number)
Assuming that is a reasonable estimate, this number is far, far
smaller than 10^(120)!!
Q.E.D.
If my disproof is sound, may I dub the number 13^(64) "Caissa's
Constant," being a fixed overestimate of the number of possible Chess
positions.
By the way, Mathworld, is an excellent resource for all subjects
mathematical (and all subjects Mathematica!) -- it is well worth a
bookmark, in my view.
http://mathworld.wolfram.com/
Your feedback is most welcome....
--
David Brett Richardson
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