Main
Date: 27 Nov 2007 06:55:49
From: Berkeley Brett
Subject: Disproof: Possible Number of Chess Games
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
bookk, in my view.

http://mathworld.wolfram.com/

Your feedback is most welcome....

--
David Brett Richardson
http://www.100bestwebsites.org/
"The 100 finest sites on the Web, all in one place!"
Widely-watched non-profit ranking of top Internet sites




 
Date: 28 Nov 2007 12:33:02
From: Stephan Bird
Subject: Re: Disproof: Possible Number of Chess Games
On Wed, 28 Nov 2007 11:30:35 -0600 in [email protected],
Kenneth Sloan <[email protected] > wrote:

> so, it's more important for the answer to be interesting than for it to
> be correct?

Well, as we've shown, the actual number of games is infinite, so it's
probably more interesting for the answer to be 'interesting' rather than
correct. (Or something like that)

Stephan
--
Stephan Bird MChem(Hons) AMRSC
Currently in Caernarfon, Wales


 
Date: 28 Nov 2007 07:50:55
From: [email protected]
Subject: Re: Disproof: Possible Number of Chess Games
Although this is a degenerate example, it is correct: the number of
possible chess *games* is infinite, as is easily demonstrated by one
of many possible examples. Consider the sequence

1. Nf3 Nf6
2. Ng1 Ng8

Now repeat this sequence any arbitrary number of times. Though the
number of positions generated is just a few, the number of *games*
possible is as many as you wish, even though the games are not exactly
thrillers.

A draw can be claimed by repetition or the 50 move rule under this
scenario, but that is an option, not a requirement, as I understand
the rules (otherwise my example is incorrect).



  
Date: 29 Nov 2007 10:24:27
From: Ray Johnstone
Subject: Re: Disproof: Possible Number of Chess Games
On Wed, 28 Nov 2007 07:50:55 -0800 (PST), "[email protected]"
<[email protected] > wrote:

>Although this is a degenerate example, it is correct: the number of
>possible chess *games* is infinite, as is easily demonstrated by one
>of many possible examples. Consider the sequence
>
>1. Nf3 Nf6
>2. Ng1 Ng8
>
>Now repeat this sequence any arbitrary number of times. Though the
>number of positions generated is just a few, the number of *games*
>possible is as many as you wish, even though the games are not exactly
>thrillers.
>
>A draw can be claimed by repetition or the 50 move rule under this
>scenario, but that is an option, not a requirement, as I understand
>the rules (otherwise my example is incorrect).
That is why I wrote:
A "game" must here be defined as a legal sequence of moves ending in
mate, stalemate or a draw by obligatory imposition of the three-fold
repetition and fifty-move laws so that the result is beyond dispute.

[email protected]
www.iinet.com.au/~ray


   
Date: 29 Nov 2007 11:42:14
From: David Richerby
Subject: Re: Disproof: Possible Number of Chess Games
Ray Johnstone <[email protected] > wrote:
> A "game" must here be defined as a legal sequence of moves ending in
> mate, stalemate or a draw by obligatory imposition of the three-fold
> repetition and fifty-move laws so that the result is beyond dispute.

Giving the players the option of not claiming possible draws does not
put the result into dispute. Under the actual laws of chess, these
are all legal games, where `**' means any one of `Black resigns 1-0',
`White resigns 0-1' and `Draw agreed 1/2-1/2':

**
1.Nf3 **
1.Nf3 Nf6 **
1.Nf3 Nf6 2.Ng1 **
1.Nf3 Nf6 2.Ng1 Ng8 **
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 **
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 **
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 **
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 **
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 **
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 **
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 6.Ng1 **
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 6.Ng1 {Black announces
his intention to play 6... Ng8. Draw by repetition} 1/2-1/2
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 6.Ng1 Ng8 **
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 6.Ng1 Ng8 {White
claims a draw by repetition} 1/2-/12
1.Nf3 Nf6 2.Ng1 Ng8 3.Nf3 Nf6 4.Ng1 Ng8 5.Nf3 Nf6 6.Ng1 Ng8 {White
announces his intention to play 7.Nf3. Draw by repetition} 1/2-1/2

and so on. It's just that there are infinitely many of them, so the
question is trivial and boring without the supposition that draws will
be claimed when available.


Dave.

--
David Richerby Mentholated Dish (TM): it's like
www.chiark.greenend.org.uk/~davidr/ a fine ceramic dish but it's
invigorating!


  
Date: 28 Nov 2007 16:29:19
From: David Richerby
Subject: Re: Disproof: Possible Number of Chess Games
[email protected] <[email protected] > wrote:
> Although this is a degenerate example, it is correct: the number of
> possible chess *games* is infinite [...] A draw can be claimed by
> repetition or the 50 move rule under this scenario, but that is an
> option, not a requirement, as I understand the rules (otherwise my
> example is incorrect).

You are absolutely correct that the number of possible chess games is
infinite because draws by repetition or the fifty-move rule have to be
claimed and are not automatic.

In order to generate a non-trivial problem, one supposes that either
player will claim a draw as soon as he has the opportunity. One then
asks how many of these `interesting' games there are.


Dave.

--
David Richerby Hilarious Projector (TM): it's like a
www.chiark.greenend.org.uk/~davidr/ 16mm film projector but it's a bundle
of laughs!


   
Date: 28 Nov 2007 11:30:35
From: Kenneth Sloan
Subject: Re: Disproof: Possible Number of Chess Games
David Richerby wrote:
> [email protected] <[email protected]> wrote:
>> Although this is a degenerate example, it is correct: the number of
>> possible chess *games* is infinite [...] A draw can be claimed by
>> repetition or the 50 move rule under this scenario, but that is an
>> option, not a requirement, as I understand the rules (otherwise my
>> example is incorrect).
>
> You are absolutely correct that the number of possible chess games is
> infinite because draws by repetition or the fifty-move rule have to be
> claimed and are not automatic.
>
> In order to generate a non-trivial problem, one supposes that either
> player will claim a draw as soon as he has the opportunity. One then
> asks how many of these `interesting' games there are.
>
>
> Dave.
>

so, it's more important for the answer to be interesting than for it to
be correct?



--
Kenneth Sloan [email protected]
Computer and Information Sciences +1-205-932-2213
University of Alabama at Birmingham FAX +1-205-934-5473
Birmingham, AL 35294-1170 http://KennethRSloan.com/


    
Date: 01 Dec 2007 18:59:39
From: Guy Macon
Subject: Re: Guy Macon on the longest possible chess game



Kenneth Sloan wrote:
>
>David Richerby wrote:
>
>> [email protected] <[email protected]> wrote:
>>
>>> Although this is a degenerate example, it is correct: the number of
>>> possible chess *games* is infinite [...] A draw can be claimed by
>>> repetition or the 50 move rule under this scenario, but that is an
>>> option, not a requirement, as I understand the rules (otherwise my
>>> example is incorrect).
>>
>> You are absolutely correct that the number of possible chess games is
>> infinite because draws by repetition or the fifty-move rule have to be
>> claimed and are not automatic.
>>
>> In order to generate a non-trivial problem, one supposes that either
>> player will claim a draw as soon as he has the opportunity. One then
>> asks how many of these `interesting' games there are.
>
>so, it's more important for the answer to be interesting than for it to
>be correct?

David Richerby correctly anwered to question asked, noted that the
question was not interesting, and then posed a related and far more
interesting question. I fail to see on what basis you have a problem
with that.

BTW, do you have an answer (or at least an upper limit) to David
Richerby's question?


--
Guy Macon
<http://www.guymacon.com/ >



     
Date: 02 Dec 2007 02:04:48
From: Phil Carmody
Subject: Re: Guy Macon on the longest possible chess game
Guy Macon <http://www.guymacon.com/ > writes:
[SNIP - whatever]

If you wish to be taken seriously on this usenet lark,
then don't munge subject lines so. Narcissism is neither
big nor clever.

It's tempting to re-title the thread "his experience
sucking donkey balls", just to see what you'd do with
it.

Phil
--
Dear aunt, let's set so double the killer delete select all.
-- Microsoft voice recognition live demonstration


      
Date: 02 Dec 2007 11:44:17
From: Guy Macon
Subject: Re: Guy Macon on the longest possible chess game


Phil Carmody wrote:

>If you wish to be taken seriously on this usenet lark,
>then don't munge subject lines so. Narcissism is neither
>big nor clever.

Thanks for bringing this up. I welcome criticism and am
very much willing to change my behavior to suit your
preferences where appropriate. May I explain why I acted
as I did? It may be that you will see what I am getting
at and agree -- or at least be willing to tolerate two
unnecessarily words in a subject line.

As for my general reason for changing the subject line, the
original was incorrect and labeled as such by the original
poster in the thread. I believe that it was proper to
change it from the number of possible games (which is not
what he posted about) to what he did post about (the number
of possible positions) or to what another poster brought
up in reply (the longest possible game being infinitely long
and thus the number of possible games being infinite). If
your objection is to any such change of subject lines, I am
open to hearing your arguments and will do as you wish if
the arguments are compelling -- but I suspect that the real
objection is to the first two words in the changed subject
line, not changed subject lines in general.

As for my reason for putting my name in the subject line,
it is in response to a situation that you most likely are
not aware of. A while back, an individual in one of the
Linux newsgroups came to the incorrect conclusion that one
of his many anonymous enemies was, in fact, me. As far as
I can tell, his only reason for thinking that is the fact
that I was posting through Supernews -- the largest news
service as far as generating text-based messages goes.
I switched to posting from Giganews, but that didn't help.

This individual then decided to work toward his stated goal
of making me unhirable by salting Google with material that
would drive away potential employers. His chosen methods
included posting copies of my resume modified to make me
look bad, posts from nonexistent employers claiming I was
fired for arson, accusations of breaking laws that don't
exist, and impersonating me. These posts to Usenet ended
up on the many websites that repost content from newsgroups.
(Your name has ended up on such websites as well; see
http://www.google.com/search?Phil+Carmody -- but without
any obvious forgeries.)

While I am employed at the moment, I may be looking for
work some time in the future, and I have a handicapped
wife who needs a lot of expensive medical care. Because
of this I decided to attempt to drive the forgeries lower
in the Google search results. Alas, the Google search
algorithm puts posts with a person's name in the title
much higher than posts by that person, and thus the only
practical way to accomplish that goal is to put the name
in the subject line. I would prefer not to do so, because
it may make others see what you saw -- narcissism -- but
given the alternative I would prefer to have someone think
me a narcissist rather than think me a person fired for
arson.

>It's tempting to [snip] just to see what you'd do with it.

I would very much prefer that you not do so. I certainly
wouldn't do such a thing to you, after seeing the damage
that can be done to a person's online reputation firsthand.

--
Guy Macon
<http://www.guymacon.com/ >



       
Date: 03 Dec 2007 11:13:16
From: David Richerby
Subject: Re: Guy Macon on the longest possible chess game
Guy Macon <http://www.guymacon.com/ > wrote:
> I believe that it was proper to change it from the number of
> possible games (which is not what he posted about) to what he did
> post about (the number of possible positions) or to what another
> poster brought up in reply (the longest possible game being
> infinitely long and thus the number of possible games being
> infinite).

But the thread isn't about the longest possible game: it's about the
number of possible games, on the assumption that draws will be claimed
ASAP. I realise that knowing the longest possible game under that
assumption is useful but it's hardly the whole answer to the question.
So your subject line is no better than the original.

> As for my reason for putting my name in the subject line, it is in
> response to a situation that you most likely are not aware of.
> [viz: large numbers of forged postings]

I'm sorry you're being attacked. But thank you so much for making
your problem our problem as well. How about, instead of engaging in
pissing contests with an idiot, you just inform potential employers
that there's a lot of forged material in your name on the net?

Your response to the situation means that there are two major sources
of information about you on the net. One, which is forged, makes you
look bad. The other, which is not forged, makes you look like an
immature narcissist. I wouldn't want to have to explain the latter to
a potential employer.


Dave.

--
David Richerby Portable Microsoft Gerbil (TM):
www.chiark.greenend.org.uk/~davidr/ it's like a children's pet that's
really hard to use but you can take
it anywhere!


       
Date: 02 Dec 2007 11:10:09
From: Kenneth Sloan
Subject: Re: Kenneth Sloan on Guy Macon on the longest possible subject line
Sounds like a great justification.

Can I play, too?

Guy Macon wrote:
> Phil Carmody wrote:
>
>> If you wish to be taken seriously on this usenet lark,
>> then don't munge subject lines so. Narcissism is neither
>> big nor clever.
>
> Thanks for bringing this up. I welcome criticism and am
> very much willing to change my behavior to suit your
> preferences where appropriate. May I explain why I acted
> as I did? It may be that you will see what I am getting
> at and agree -- or at least be willing to tolerate two
> unnecessarily words in a subject line.
>
> As for my general reason for changing the subject line, the
> original was incorrect and labeled as such by the original
> poster in the thread. I believe that it was proper to
> change it from the number of possible games (which is not
> what he posted about) to what he did post about (the number
> of possible positions) or to what another poster brought
> up in reply (the longest possible game being infinitely long
> and thus the number of possible games being infinite). If
> your objection is to any such change of subject lines, I am
> open to hearing your arguments and will do as you wish if
> the arguments are compelling -- but I suspect that the real
> objection is to the first two words in the changed subject
> line, not changed subject lines in general.
>
> As for my reason for putting my name in the subject line,
> it is in response to a situation that you most likely are
> not aware of. A while back, an individual in one of the
> Linux newsgroups came to the incorrect conclusion that one
> of his many anonymous enemies was, in fact, me. As far as
> I can tell, his only reason for thinking that is the fact
> that I was posting through Supernews -- the largest news
> service as far as generating text-based messages goes.
> I switched to posting from Giganews, but that didn't help.
>
> This individual then decided to work toward his stated goal
> of making me unhirable by salting Google with material that
> would drive away potential employers. His chosen methods
> included posting copies of my resume modified to make me
> look bad, posts from nonexistent employers claiming I was
> fired for arson, accusations of breaking laws that don't
> exist, and impersonating me. These posts to Usenet ended
> up on the many websites that repost content from newsgroups.
> (Your name has ended up on such websites as well; see
> http://www.google.com/search?Phil+Carmody -- but without
> any obvious forgeries.)
>
> While I am employed at the moment, I may be looking for
> work some time in the future, and I have a handicapped
> wife who needs a lot of expensive medical care. Because
> of this I decided to attempt to drive the forgeries lower
> in the Google search results. Alas, the Google search
> algorithm puts posts with a person's name in the title
> much higher than posts by that person, and thus the only
> practical way to accomplish that goal is to put the name
> in the subject line. I would prefer not to do so, because
> it may make others see what you saw -- narcissism -- but
> given the alternative I would prefer to have someone think
> me a narcissist rather than think me a person fired for
> arson.
>
>> It's tempting to [snip] just to see what you'd do with it.
>
> I would very much prefer that you not do so. I certainly
> wouldn't do such a thing to you, after seeing the damage
> that can be done to a person's online reputation firsthand.
>
Kenne

--
Kenneth Sloan [email protected]
Computer and Information Sciences +1-205-932-2213
University of Alabama at Birmingham FAX +1-205-934-5473
Birmingham, AL 35294-1170 http://KennethRSloan.com/


     
Date: 01 Dec 2007 14:57:09
From: Kenneth Sloan
Subject: Re: Guy Macon on the longest possible chess game
Guy Macon wrote:
> Kenneth Sloan wrote:
>> David Richerby wrote:
>>
>>> [email protected] <[email protected]> wrote:
>>>
>>>> Although this is a degenerate example, it is correct: the number of
>>>> possible chess *games* is infinite [...] A draw can be claimed by
>>>> repetition or the 50 move rule under this scenario, but that is an
>>>> option, not a requirement, as I understand the rules (otherwise my
>>>> example is incorrect).
>>> You are absolutely correct that the number of possible chess games is
>>> infinite because draws by repetition or the fifty-move rule have to be
>>> claimed and are not automatic.
>>>
>>> In order to generate a non-trivial problem, one supposes that either
>>> player will claim a draw as soon as he has the opportunity. One then
>>> asks how many of these `interesting' games there are.
>> so, it's more important for the answer to be interesting than for it to
>> be correct?
>
> David Richerby correctly anwered to question asked, noted that the
> question was not interesting, and then posed a related and far more
> interesting question. I fail to see on what basis you have a problem
> with that.
>
> BTW, do you have an answer (or at least an upper limit) to David
> Richerby's question?
>
>

Your question is correct - but not interesting.

--
Kenneth Sloan [email protected]
Computer and Information Sciences +1-205-932-2213
University of Alabama at Birmingham FAX +1-205-934-5473
Birmingham, AL 35294-1170 http://KennethRSloan.com/


    
Date: 29 Nov 2007 11:29:53
From: David Richerby
Subject: Re: Disproof: Possible Number of Chess Games
Kenneth Sloan <[email protected] > wrote:
> David Richerby wrote:
>> In order to generate a non-trivial problem, one supposes that
>> either player will claim a draw as soon as he has the opportunity.
>> One then asks how many of these `interesting' games there are.
>
> so, it's more important for the answer to be interesting than for it
> to be correct?

It's not incorrect as long as you state the problem you're solving
correctly: `the number of possible chess games, assuming that both
players will claim a draw as soon as they have that option, is N.'


Dave.

--
David Richerby Adult Peanut (TM): it's like a roasted
www.chiark.greenend.org.uk/~davidr/ nut that you won't want the children
to see!


 
Date: 28 Nov 2007 11:26:05
From: Ray Johnstone
Subject: Re: Disproof: Possible Number of Chess Games
On Tue, 27 Nov 2007 06:55:49 -0800 (PST), Berkeley Brett
<[email protected] > wrote:

I've written a note about the number of chess games at:
http://members.iinet.net.au/~ray/Chessgames.htm
[email protected]
www.iinet.com.au/~ray


 
Date: 27 Nov 2007 16:54:01
From: Anders Thulin
Subject: Re: Disproof: Possible Number of Chess Games
Berkeley Brett wrote:

> Assuming that is a reasonable estimate, this number is far, far
> smaller than 10^(120)!!

So ... you're not computing the same number. Did you check out
the reference to Peterson 1966 to see what he was referring to?

The number of different chess positions is generally stated
to be on the order of 2*10^43 (no promoted pieces), or the order of
10^50 (promoted pieces), or 10^?? (including promoted pieces,
castling state, e.p. state and 50-move clock state.)
They are, of course, approximations.

The number that Peterson mentioned was the number of
different positions in all games of a length of 40 moves. That
is not clearly one of the numbers above. (The number is just
mentioned, without any source reference or indication of how
it was computed.)

--
Anders Thulin anders*thulin.name http://www.anders.thulin.name/


 
Date: 27 Nov 2007 08:41:42
From: Berkeley Brett
Subject: Re: Disproof: Possible Number of Chess Games
Thanks to both David and Mike for their feedback.

My confusion occurred to me just after posting the message -- just as
I often see a Chess blunder milliseconds after making it!

Oh well! Sometimes our mistakes can also be instructive and valuable!

--
David Brett Richardson
http://www.100bestwebsites.org/
"The 100 finest sites on the Web, all in one place!"
Widely-watched non-profit ranking of top Internet sites



 
Date: 27 Nov 2007 15:19:48
From: Mike Robson
Subject: Re: Disproof: Possible Number of Chess Games
Le 27-11-2007, Berkeley Brett <[email protected] > a �crit�:
> 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
>
[...]

> Your feedback is most welcome....
>
> --
> David Brett Richardson
> http://www.100bestwebsites.org/
> "The 100 finest sites on the Web, all in one place!"
> Widely-watched non-profit ranking of top Internet sites

Quite right. People often sloppily confuse the number of possible games
(for which 10^120 looks like a conservative estimate) with the number
of possible board positions (which is much smaller because the same
position can be reached in a vast number of different ways.)

I have seen "between 10^30 and 10^40" quoted as an approximation to the number
of possible board positions. When I tried to calculate a better estimate
I decided it was near to the bottom of that range (~10^39) but I
no longer have more than a hazy memory of how I came to that conclusion.

Mike.



















 
Date: 27 Nov 2007 15:56:14
From: David Richerby
Subject: Re: Disproof: Possible Number of Chess Games
Berkeley Brett <[email protected] > wrote:
> 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.)

Your argument that 13^64 is an upper bound on the number of ways to
arrange the pieces on the board is correct. However, this in no way
proves that the total number of games cannot be more than 13^64. Many
positions can be reached in more than one way. For example, the
position after 1.e4 e5 can be reached by all kinds of routes,
including but not limited to,

1.e4 e5
1.e3 e6 2.e4 e5
1.Nf3 Nc6 2.Ng1 Nb8 3.e4 e5

Thus, the number of possible games dramatically exceeds the number of
possible positions.


Dave.

--
David Richerby Accelerated Permanent Hat (TM): it's
www.chiark.greenend.org.uk/~davidr/ like a hat but it'll be there for ever
and it's twice as fast!


 
Date: 27 Nov 2007 07:18:46
From: Berkeley Brett
Subject: Re: Disproof: Possible Number of Chess Games
Ah, of course, this is the number of possible Chess POSITIONS --not
Chess GAMES!

Nevermind!

--
Brett
http://www.100bestwebsites.org/
"The 100 finest sites on the Web, all in one place!"
Widely-watched non-profit ranking of top Internet sites


  
Date: 27 Nov 2007 16:35:24
From: David Richerby
Subject: Re: Disproof: Possible Number of Chess Games
Berkeley Brett <[email protected] > wrote:
> Ah, of course, this is the number of possible Chess POSITIONS --not
> Chess GAMES!

Gaaaaah, this whole thread is a pile of confusion about the difference
between positions and games. :-) I've filled in Mathworld's comment
form to point out that their page is incorrect.


Dave.

--
David Richerby Disgusting Laptop Goldfish (TM): it's
www.chiark.greenend.org.uk/~davidr/ like a fish that you can put on your
lap but it'll turn your stomach!