![]() | This article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
Hmmm... well, it occured to me that the total number of posible board positions couldn't be more than 64!/32!... that is, the total number of ways of arranging 32 distinguishable pieces on a board of 64 spaces. But I didn't consider the affect of promoting pawns, and all the various choices that might be available in that case, and the extra board positions that this would create... still though, if promoting pawns was disallowed, it would limit the positions to 64!/32! (as well as noticably changing the dynamics of the game)... just a curious thought. (The Swami 08:36, 17 August 2005 (UTC))
There are several possible problems with this page. First, "One novemtriginillion is known as the Shannon number." It seems like the Shannon Number is actually 90040, which is 1.478...×10118, for which 10120 is a poor approximation. If 90040 is indeed the Shannon Number, then that doesn't really belong in this article. Also, for the number of atoms in the universe: can that figure really be estimated with so much precision? Ardric47 05:18, 6 April 2006 (UTC)
I restored the number 10^120, with a quote from the paper. Sure, it's an approximation, but it's Shannon's number so we should use Shannon's approximation, not Ardric47's approximation! Gdr 19:55, 13 May 2007 (UTC)
I think that Shannon knew that some games might last 41 moves, which gives 90041=1.3x10121 so he just rounded it off to 10120. However it is true that for him to say that 90040 "will be 10120 variations" is pretty sloppy mathematics. Was he using a slide rule at the time? 199.125.109.62 14:42, 6 June 2007 (UTC)
This article is about the Shannon number. Now that it has been decided that the Shannon number is not 10^120, shouldn't the article be moved back to Shannon number? Melchoir 01:49, 8 April 2006 (UTC)
Due to confusing redirects, this page has been mirrored (and then added to by me...) at Talk:10^120 (number). Ardric47 21:14, 8 April 2006 (UTC)
This article refers to this site when mentioning the lower bound of an estimate of all the atoms in the universe. However, the following calculation on that page is incorrect: . The correct answer is , not . This, in turn, affects the subsequent calculation and results in a different estimate of atoms in the universe.
I cut this text from the article:
because it appears (considered very charitably) merely to be a restatement of Shannon's estimate from the first sentence of the article, adding nothing to the article, which is not a general article on combinatorial estimates relating to chess. (Considered uncharitably, with "unique games" taken literally, the claim is of course quite false.) Gdr 10:31, 25 September 2008 (UTC)
How in the world can a large number about chess variants be an analog to a joking reference about how close you are to a mathematician? I don't understand. Practicality (talk) 15:27, 2 February 2010 (UTC)
In a 2006 posting to newsgroup rec.games.chess.computer Will Entriken proved an upper bound of 23937533792747905898433845980097921846050276105440 on the number of chess diagrams (roughly 2^164). The following Haskell program, if correct, improves this to just under 2^155 for the number of positions, which includes side-to-move, castling, and en-passant info.
import Control.Monad
import System(getArgs)
import Array
armies np ir = do -- max number of pawns and initial rooks
q <- [0..1+np]
let prom1 = max (q-1) 0
r <- [0..ir+np-prom1]
let prom2 = prom1 + max (r-ir) 0
b <- [0..2+np-prom2]
let prom3 = prom2 + max (b-2) 0
n <- [0..2+np-prom3]
let proms = prom3 + max (n-2) 0
guard $ proms <= np
p <- [0..np-proms]
return (q+r+b+n, p, proms, fac q*fac r*fac b*fac n*fac p)
fac :: Int -> Integer
fac n = fac64!n where fac64 = listArray (0,64) (scanl (*) 1 [1..64])
count fixp pspace fixwr fixbr = sum $ do
let np = 8-fixp
(wpcs,wp,wproms,wprod) <- armies np (2-fixwr)
let wpx = np-wp-wproms -- white p captured
(bpcs,bp,bproms,bprod) <- armies np (2-fixbr)
let bpx = np-bp-bproms -- black p captured
let caps = 30-2*fixp-fixwr-fixbr-wp-bp-wpcs-bpcs
guard $ wproms <= bpx + caps
guard $ bproms <= wpx + caps
let space = 62-4*fixp-fixwr-fixbr-wp-bp
return $ (fac pspace `div` fac (pspace-wp-bp)) *
(fac space `div` fac (space-wpcs-bpcs)) `div` (wprod * bprod)
count0 (mul,ek) = mul * count 0 (46+ek) 0 0
-- Diagrams
main0 = print . sum $ map count0 [(212,2),(1448,1),(1952,0)]
-- 22124621884617108585387385940828998876019391612
-- multiplier, edge kings, fixed white rooks, fixed black rooks
countep (mul,ek,fwr,fbr) = mul *
(count 0 (46+ek) fwr fbr + 14 * count 1 (42+ek) fwr fbr)
-- Positions
main = print . (* 2) . sum $ map countep
[(1,2,2,2),(4,2,1,2),(4,2,1,1),
(2*9,2,0,2),(2*43,1,0,2),(4*11,2,0,1),(4*44,1,0,1),
(212,2,0,0),(1448,1,0,0),(1952,0,0,0)]
-- 45193640626062205213735739171550309047984050718
--Tromp (talk) 17:32, 5 May 2010 (UTC)