How many games of chess are there?

[Guest]

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The only estimate I know of is Littlewood's. He deduced an upper bound
of 10^10^70.5.Has anyone produced a reasonable lower bound?

 

[Guest]

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"Ray Johnstone" <ray@iinet.com.au> wrote in message
news:bntci1htn4k7o05isqaqdngm02ej5ernp1@4ax.com...
> The only estimate I know of is Littlewood's. He deduced an upper bound
> of 10^10^70.5.Has anyone produced a reasonable lower bound?

Why is there an upper bound? There are a limited number of pieces and a
limited number of squares, surely someone could work out every possible
permutation from there and then just strike off all the illegal positions?

 

[Guest]

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Ray Johnstone wrote:

>The only estimate I know of is Littlewood's. He deduced an upper
>bound of 10^10^70.5.

In _Programming a Computer for Playing Chess_ (found in Levy's
_Computer Chess Compendium_) Shannon estimates 64!/(32!*8!^2*2!^6),
or roughly 10^43.

Also see:
http://mathworld.wolfram.com/Chess.html
http://www.cs.berkeley.edu/~flab/chess/statistics-positions.html
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A006494
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007545
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A019319
http://www.combinatorics.org/Volume_11/Abstracts/v11i2a4.html
http://anduin.eldar.org/~problemi/papers.html
http://ite.pubs.informs.org/Vol2No2/ChlondToase/

In my opinion, all of the above calculations are wrong. Because a
game of chess can be infinitely long, there are an infinite number
of possible games.

---------------------------------------------------------

LONGEST POSSIBLE CHESS GAME CALCULATION:
By Guy Macon <http://www.guymacon.com/>

Here is my calculation for the longest possible chess
game before one of the players can claim a draw under
FIDE rules. (If nobody claims a draw the players can
continue to push pieces back and forth forever and the
game is infinitely long - and boring to calculate.) See
<http://www.fide.com/official/handbook.asp?level=EE101>

Note to the non-chessplayer: a "ply" is a black piece
changing position or a white piece changing position.
Chessplayers call ten black piece moving and ten white
pieces moving "ten moves"/"twenty plies." Non-chess-players
often call the same sequence "twenty moves."

To calculate the longest possible chess game before one
of the players can claim a draw under FIDE rules:

Start with 32 chess pieces.

Move 100 plies, avoiding repeating positions.

On ply 100, move a pawn or make a capture.

Repeat N times until you make the last capture that
leaves 2 kings.

So how big is N?

There are 30 100-ply sequences ending with a capture.

There are 96 100-ply sequences ending with a pawn move.

8 of these sequences end with a pawn move that is also a capture.

1 of those sequences is only 98 plies long so that black can start
taking his turn moving pawns and making captures.

Assuming FIDE rules, that comes to a total of
((100*(30+96-8))-1)=11799 plies until the game is over.

- - - - - - - - - - - - - - -

Here is one way of reaching the maximum number of moves
in a chess game (see calculation above). Assume that
each ply described has 99 or (in one case) 98 "wasted"
plies between the plies described..

White advances his A,C,E,G pawns as far as they will go.

Black advances his B,D,F,H pawns as far as they will go.

White captures every black piece except 8 pawns, 2 knights,
1 bishop, one rook, and a King.

The white pawns on A,C,E,G capture the knights, bishops and rook,
passing and freeing the black pawns blocking them.

The now-unblocked black pawns move forward, promote, and move into
position to be taken by the white pawns on B,D,F,H, unblocking the
black pawns on B,D,F,H.

The now-unblocked black pawns move forward, promote, and are taken.

Black now only has a king. (Here is the lone 98 ply sequence) The
black king captures something, and the game continues a capture
by black on every 100th ply. When black captures the last white
piece, there are only the two kings left and the game is a draw
after exactly (and no more than) 11799 plies.

Comments/corrections are welcome.

Guy Macon
<a href="http://www.guymacon.com/">
http://www.guymacon.com/
</a>

---------------------------------------------------------

BTW, here is how to generate an infinite number of games:
Push pieces back and forth 1,000 times, then checkmate.
Push pieces back and forth 1,001 times, then checkmate.
Push pieces back and forth 1,002 times, then checkmate.
Push pieces back and forth 1,003 times, then checkmate.
Push pieces back and forth 1,004 times, then checkmate.
....

 

[Guest]

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Lee Harris wrote:

>Why is there an upper bound? There are a limited number of pieces and a
>limited number of squares, surely someone could work out every possible
>permutation from there and then just strike off all the illegal positions?

Number of positions and number of games are two different questions.

They do have one thing in common; many smart people making many
calculations and coming up with many different answers.

 

[Guest]

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Guy Macon <http://www.guymacon.com> wrote:
> Ray Johnstone wrote:
>> The only estimate I know of is Littlewood's. He deduced an upper
>> bound of 10^10^70.5.
>
> In _Programming a Computer for Playing Chess_ (found in Levy's
> _Computer Chess Compendium_) Shannon estimates 64!/(32!*8!^2*2!^6),
> or roughly 10^43.

I don't have a copy of the book to hand but that looks like an estimate of
the number of positions, not games.


Dave.

--
David Richerby Natural Flammable Monk (TM): it's
www.chiark.greenend.org.uk/~davidr/ like a man of God but it burns really
easily and it's completely natural!

 

[Guest]

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Lee Harris <leeh@medphysics.leeds.ac.uk> wrote:
> Ray Johnstone <ray@iinet.com.au> wrote:
>> The only estimate I know of is Littlewood's. He deduced an upper bound
>> of 10^10^70.5.Has anyone produced a reasonable lower bound?
>
> Why is there an upper bound?

To say that something is an upper bound for a quantity just means that the
quantity can't possibly be bigger than that.


> There are a limited number of pieces and a limited number of squares,
> surely someone could work out every possible permutation from there and
> then just strike off all the illegal positions?

In theory, yes. In practise, this requires an infeasible amount of
computation so we have to estimate.


Dave.

--
David Richerby Cyber-Ghost (TM): it's like a
www.chiark.greenend.org.uk/~davidr/ haunting spirit that exists only in
your computer!

 

[Guest]

Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

Guy Macon wrote:


> In _Programming a Computer for Playing Chess_ (found in Levy's
> _Computer Chess Compendium_) Shannon estimates 64!/(32!*8!^2*2!^6),
> or roughly 10^43.

But that is not games, those are 'positions' ... or rather ways of
placing chess pieces on a chess board without paying too close attention
to rules.

Games are one lousy position after another, and so there should be
considerably more games ...


--
Anders Thulin ath*algonet.se http://www.algonet.se/~ath

 

[Guest]

Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

On Tue, 13 Sep 2005 09:40:48 +0000, Guy Macon
<http://www.guymacon.com> wrote:

>
>
>
>Ray Johnstone wrote:
>
>>The only estimate I know of is Littlewood's. He deduced an upper
>>bound of 10^10^70.5.
>
>In _Programming a Computer for Playing Chess_ (found in Levy's
>_Computer Chess Compendium_) Shannon estimates 64!/(32!*8!^2*2!^6),
>or roughly 10^43.
>
>Also see:
>http://mathworld.wolfram.com/Chess.html
>http://www.cs.berkeley.edu/~flab/chess/statistics-positions.html
>http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A006494
>http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007545
>http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A019319
>http://www.combinatorics.org/Volume_11/Abstracts/v11i2a4.html
>http://anduin.eldar.org/~problemi/papers.html,
>http://ite.pubs.informs.org/Vol2No2/ChlondToase/
>
>In my opinion, all of the above calculations are wrong. Because a
>game of chess can be infinitely long, there are an infinite number
>of possible games.
Thanks for the advice, very helpful. But I think you are wrong about
chess being infinite.The fifty-move rule declares a game drawn if
there have been 50 moves without a capture or a pawn move. The tree
stops branching.

 

[Guest]

Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

Ray Johnstone wrote:

> Thanks for the advice, very helpful. But I think you are wrong about
> chess being infinite.The fifty-move rule declares a game drawn if
> there have been 50 moves without a capture or a pawn move. The tree
> stops branching.

I read somewhere that there are certain rare position, where
the analysis has revealed that one side can force a win, but
not within the constraints of the 50 move rule. This book
also said that consequently the 50 move rule is replaced by
100 move rule in these circumstances. I would welcome it, if
any expert cares to shed more light on this ??

Cheers,

Jyrki Lahtonen, Turku, Finland

 

[Guest]

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Jyrki Lahtonen wrote:
>
>Ray Johnstone wrote:
>
>> Thanks for the advice, very helpful. But I think you are wrong about
>> chess being infinite.The fifty-move rule declares a game drawn if
>> there have been 50 moves without a capture or a pawn move. The tree
>> stops branching.
>
>I read somewhere that there are certain rare position, where
>the analysis has revealed that one side can force a win, but
>not within the constraints of the 50 move rule. This book
>also said that consequently the 50 move rule is replaced by
>100 move rule in these circumstances. I would welcome it, if
>any expert cares to shed more light on this ??

In the FIDE Laws of Chess, published in 1984 and 1988, the 50-move
rule was extended to 75 moves for the following positions:

King + Rook + Bishop against King + Rook;
King + 2 Knights against King + pawn;
King + Queen + pawn one square from promotion against King + Queen;
King + Queen against King + 2 Knights;
King + Queen against King + 2 Bishops;
King + 2 Bishops against King + Knight

In 1992 during the FIDE Congress in Manila the Rules Committee
suggested establishing one rule for all endings: 50 moves, and
the General Assembly of FIDE approved the change. In 1996 the
Congress in Yerevan revisited the decision and kept the 1992
rules. There are currently no exceptions to the 50-move rule.

Source: _An Arbiter's Notebook_,
by International Arbiter Geurt Gijssen

The Archives of _An Arbiter's Notebook_ are really quite
interesting. You can find them here:
http://www.chesscafe.com/archives/archives.htm#An%20Arbiter's%20Notebook


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

 

[Guest]

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Jyrki Lahtonen wrote:

> 100 move rule in these circumstances. I would welcome it, if
> any expert cares to shed more light on this ??

From the latest edition of "Artificial Intelligence: A Modern Approach
(2nd Edition)", by Stuart J. Russell & Peter Norvig:

"Remarkable work by Ken Thompson and Lewis Stiller (1992) solved all
five-piece and some six-piece chess endgames, making them available on
the Internet. Stiller discovered one case where a forced mate existed
but required 262 moves; this caused some consternation because the rules
of chess require some “progress” to occur within 50 moves."

Here's the quote, rather cryptic:
Stiller, L. (1992). KQNKRR. ICCA Journal, Vol. 15, No. 1, pp. 16-18. (N)

ciao!
S

 

[Guest]

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Jyrki Lahtonen <lahtonen@utu.fi> wrote:
> I read somewhere that there are certain rare position, where the
> analysis has revealed that one side can force a win, but not within the
> constraints of the 50 move rule. This book also said that consequently
> the 50 move rule is replaced by 100 move rule in these circumstances. I
> would welcome it, if any expert cares to shed more light on this ??

There are positions such as rook and bishop vs rook where there are forced
mates that take more than 50 moves and even some positions (with other
material) where it can take over 200 moves to force mate even with optimal
play. This sort of thing was discovered through endgame tablebases and
two nice examples, due to Stiller, are at

http://www.xs4all.nl/~timkr/chess/perfect.htm

For a time, the 50-move rule in the FIDE laws was extended to deal with
cases like these but, in practice, these things come very infrequently and
the law was changed back to 50 moves in all circumstances. In many of
these forced mates, the first hundred moves look completely random anyway
so there's very little chance of a human being able to reproduce them.


Dave.

--
David Richerby Frozen Incredible Gnome (TM): it's
www.chiark.greenend.org.uk/~davidr/ like a smiling garden ornament but
it'll blow your mind and it's frozen
in a block of ice!

 

[Guest]

Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

Guy Macon wrote:
There are currently no exceptions to the 50-move rule.
>
Thanks. Much appreciated. Apparently the book I read was
printed in the late 80s

Cheers,

Jyrki

 

[Guest]

Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

In article <038di1dsrr4tq6eg9614m6hqpodcpbub0t@4ax.com>,
Ray Johnstone <ray@iinet.com.au> wrote:

> On Tue, 13 Sep 2005 09:40:48 +0000, Guy Macon
> <http://www.guymacon.com> wrote:

> >In my opinion, all of the above calculations are wrong. Because a
> >game of chess can be infinitely long, there are an infinite number
> >of possible games.

> Thanks for the advice, very helpful. But I think you are wrong about
> chess being infinite.The fifty-move rule declares a game drawn if
> there have been 50 moves without a capture or a pawn move. The tree
> stops branching.

You may have missed the part where he wrote,

If nobody claims a draw the players can
continue to push pieces back and forth forever and the
game is infinitely long

The point being one player or the other must *claim* the draw,
it is not enforced automatically by anyone.

--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)

 

[Guest]

Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

Gerry Myerson a écrit :
> In article <038di1dsrr4tq6eg9614m6hqpodcpbub0t@4ax.com>,
> Ray Johnstone <ray@iinet.com.au> wrote:
>
>
>>On Tue, 13 Sep 2005 09:40:48 +0000, Guy Macon
>><http://www.guymacon.com> wrote:
>
>
>>>In my opinion, all of the above calculations are wrong. Because a
>>>game of chess can be infinitely long, there are an infinite number
>>>of possible games.
>
>
>>Thanks for the advice, very helpful. But I think you are wrong about
>>chess being infinite.The fifty-move rule declares a game drawn if
>>there have been 50 moves without a capture or a pawn move. The tree
>>stops branching.
>
>
> You may have missed the part where he wrote,
>
> If nobody claims a draw the players can
> continue to push pieces back and forth forever and the
> game is infinitely long
>
> The point being one player or the other must *claim* the draw,
> it is not enforced automatically by anyone.
>
You hve never participated in a real tournament, have you? Referees do
intervene, you know... If only to be able to draw the next round...

 

[Guest]

Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

Content-Transfer-Encoding: 8Bit


Ray Johnstone wrote:
>
>Guy Macon <http://www.guymacon.com> wrote:
>
>>Because a game of chess can be infinitely long, there are an
>>infinite number of possible games.
>
>Thanks for the advice, very helpful. But I think you are wrong about
>chess being infinite.The fifty-move rule declares a game drawn if
>there have been 50 moves without a capture or a pawn move.

No it doesn't. Go look at
http://www.fide.com/official/handbook.asp?level=EE101
section 9.3. Note the phrase "The game is drawn, upon a correct
claim by the player having the move." He does not have to claim
a draw. They can play on forever.

-------------------------------------------------------------

denis feldmann wrote:

>Gerry Myerson a écrit :
>>
>> You may have missed the part where he wrote,
>>
>> If nobody claims a draw the players can
>> continue to push pieces back and forth forever and the
>> game is infinitely long
>>
>> The point being one player or the other must *claim* the draw,
>> it is not enforced automatically by anyone.
>>
>You have never participated in a real tournament, have you?

Plenty of them. FIDE and USCF. (We are, of course, discussing
FIDE rules. I dare anyone to repeat my calculations under USCF
rules; the USCF rules are rather unclear in many areas.)

>Referees do intervene, you know...

I don't know what you consider a "real tournament", but FIDE
tournaments have Arbiters, not Referees.

>Referees do intervene, you know...
>If only to be able to draw the next round...

No they don't. To do so would be against the rules.
Look at http://www.fide.com/official/handbook.asp?level=EE101
section 13.6. Note the phrase "The arbiter must not intervene
in a game except in cases described by the Laws of Chess." Now
try to find a place in the Laws of Chess wher an arbiter is
allowed to claims a 50-move draw when neither player has made
such a claim.


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

 

[Guest]

Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

denis feldmann <denis-feldmann@club-internet.fr> writes in article <4326ad62$0$308$7a628cd7@news.club-internet.fr> dated Tue, 13 Sep 2005 12:43:52 +0200:
>Gerry Myerson a écrit :
>> The point being one player or the other must *claim* the draw,
>> it is not enforced automatically by anyone.
>>
>You hve never participated in a real tournament, have you? Referees do
>intervene, you know... If only to be able to draw the next round...

No good TD would call the game a draw unless there was no possibility of a
win by either side. If a game takes too long, he might order it adjourned
(suspended) and post pairings for the next round as if it were a draw, but
the final score would have to include the actual result of that game.

I have played in tournaments where this has happened.

--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.

 

[skeptic]

Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

"denis feldmann" <denis-feldmann@club-internet.fr>

> You hve never participated in a real tournament, have you? Referees do
> intervene, you know... If only to be able to draw the next round...

But that's not the point; the question is whether there is something in the
laws of chess that FORCE the game to be a draw by the 50-move-rule.

The answer is "no", as, indeed, one of the players has to claim a draw. So
the players could go on playing, say, 1. Nf3 Nf6 2. Nb1 Ng8 3. Nf3 ...
forever; so there is an infinite number of games. (At least the one ending
in a draw after the first move, after the second move... after the nth
move... in this sequence).

The reason most "how many games are there?" calculations pretend that the
50-move-rule is "automatically"in effect, so to speak, is that if it is not,
the answer is trivially "an infinite number".

Incidentally, on the "50 moves rule optional" reading of the laws of chess,
there seems to be nothing in the rules of chess that forbids the possiblity
of an infinitely long game (such as the sequence of knight moves above going
on forever). If that is allowed, then there is an uncountable infinite
number of games; if it is not (and we assume that, even without the
50-move-law, every game has a finite number of moves).

Proof is left as excercise to the reader

By

 

[Guest]

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Skeptic <apilpel1@nyc.rr.com> wrote:
> denis feldmann <denis-feldmann@club-internet.fr>
>> You hve never participated in a real tournament, have you? Referees do
>> intervene, you know... If only to be able to draw the next round...

In all the tournaments I've played in, the mechanism used to prevent games
going on arbitrarily long is the clock, not the arbiter.


> The answer is "no", as, indeed, one of the players has to claim a draw.
> So the players could go on playing, say, 1. Nf3 Nf6 2. Nb1 Ng8 3. Nf3
> ... forever; so there is an infinite number of games. (At least the one
> ending in a draw after the first move, after the second move... after
> the nth move... in this sequence).

(These would be draws by agreement, for the first five moves, draws by
agreement or claim of threefold repetition from the sixth to 49th and by
agreement or claim by the 50-move rule thereafter but that doesn't matter
as they're still distinct games.)


> The reason most "how many games are there?" calculations pretend that
> the 50-move-rule is "automatically"in effect, so to speak, is that if it
> is not, the answer is trivially "an infinite number".

From a practical point of view, it's a good idea to treat the 50-move rule
as mandatory. You should always assume that your opponent will play the
best possible move and that includes claiming any draw available if they
are in a worse position. From the point of view of trying to determine
whether chess is a forced win for either player or a forced draw, the
50-move rule and threefold repetition rule are just bounded ways of saying
``infinite plays are drawn''.

Admittedly, this is slightly at odds with the idea of counting the number
of possible games because a pair of beginners might shuffle their knights
for a hundred moves before one of them remembered the 50-move rule and
another twenty-five while they argued about it, while Kramnik and Leko
would just claim by repetition at White's fifth.


> Incidentally, on the "50 moves rule optional" reading of the laws of
> chess, there seems to be nothing in the rules of chess that forbids the
> possiblity of an infinitely long game (such as the sequence of knight
> moves above going on forever).

The use of clocks is explicitly mentioned in the rules. On the other
hand, I don't think there's anything that says that the clock must be set
to indicate that the game is over at any point.


Dave.

--
David Richerby Cheese Wine (TM): it's like a vintage
www.chiark.greenend.org.uk/~davidr/ Beaujolais that's made of cheese!

 

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Guy Macon wrote:

> In my opinion, all of the above calculations are wrong. Because a
> game of chess can be infinitely long, there are an infinite number
> of possible games.

I don't think a game that repeats sequences wholesale (including a
return to the same sequence start point) could be considered a new game,
and since this would be sure to happen in an infinitely long game then
that game would be "just another game", and nothing special from the
point of view of creating fresh and unique instances of games.

Pete.
--
Peter Clinch Medical Physics IT Officer
Tel 44 1382 660111 ext. 33637 Univ. of Dundee, Ninewells Hospital
Fax 44 1382 640177 Dundee DD1 9SY Scotland UK
net p.j.clinch@dundee.ac.uk http://www.dundee.ac.uk/~pjclinch/

 

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Peter Clinch <p.j.clinch@dundee.ac.uk> wrote:
> Guy Macon wrote:
>> In my opinion, all of the above calculations are wrong. Because a
>> game of chess can be infinitely long, there are an infinite number
>> of possible games.
>
> I don't think a game that repeats sequences wholesale (including a
> return to the same sequence start point) could be considered a new game,
> and since this would be sure to happen in an infinitely long game then
> that game would be "just another game", and nothing special from the
> point of view of creating fresh and unique instances of games.

A game that follows another one but inserts a few repetitions is certainly
a different game! It's not a very interesting different game but it's not
the same game as the one without the repetitions. In particular, the
original game might have been won by White but Black might have missed a
claim for draw by repetition in the new one.


Dave.

--
David Richerby Voodoo Laser (TM): it's like an
www.chiark.greenend.org.uk/~davidr/ intense beam of light that has
mystical powers!

 

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| David Richerby wrote:
|> Skeptic wrote:
|>> denis feldmann wrote:
|>> You hve never participated in a real tournament, have you? Referees do
|>> intervene, you know... If only to be able to draw the next round...

| In all the tournaments I've played in, the mechanism used to prevent games
| going on arbitrarily long is the clock, not the arbiter.
-----[-----snipped-----]-----

The clock doesn't stop an arbitrarily long game. The clock gets reset
(as I understand it) when both players have made 40 or 50 moves, depending
upon the tournament rules. ______________________________________Gerard S.

 

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"Skeptic" <apilpel1@nyc.rr.com> wrote in message
news:u0NVe.3603$Ap4.360262@twister.nyc.rr.com...
>
> Incidentally, on the "50 moves rule optional" reading of the
> laws of chess, there seems to be nothing in the rules of
> chess that forbids the possiblity of an infinitely long game
> (such as the sequence of knight moves above going
> on forever). If that is allowed, then there is an uncountable
> infinite number of games; if it is not (and we assume that,
> even without the 50-move-law, every game has a finite
> number of moves).

By "uncountable infinite" do you mean Aleph-1? I can't see why the number of
games isn't Aleph-0.

--
John Rowland - Spamtrapped
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Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

In article <11id7l1glhlupc9@corp.supernews.com>,
Guy Macon <http://www.guymacon.com> wrote:

>
> In my opinion, all of the above calculations are wrong. Because a
> game of chess can be infinitely long, there are an infinite number
> of possible games.
>

The most basic point is that, to do these calculations, you're going to
assume something simple like "In any game that goes 200 moves or more
without a pawn advance or capture, one side will claim a draw." Of
course, you'll get different answers assuming different numbers from
200. For this reason, I think people often pick 50 moves, since that's
the earliest a draw can be claimed. You can also add in that a draw will
be claimed when possible if there's a three-fold rep, but that makes
calculations pretty hard, I'd bet.

--Harold Buck


"I used to rock and roll all night,
and party every day.
Then it was every other day. . . ."
-Homer J. Simpson

 

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Archived from groups: rec.games.chess.computer,sci.math,rec.games.chess.misc,rec.games.board (More info?)

"John Rowland" <johnr@journeyflow.spamspam.demon.co.uk> writes:

> "Skeptic" <apilpel1@nyc.rr.com> wrote in message
> news:u0NVe.3603$Ap4.360262@twister.nyc.rr.com...
>>
>> Incidentally, on the "50 moves rule optional" reading of the
>> laws of chess, there seems to be nothing in the rules of
>> chess that forbids the possiblity of an infinitely long game
>> (such as the sequence of knight moves above going
>> on forever). If that is allowed, then there is an uncountable
>> infinite number of games; if it is not (and we assume that,
>> even without the 50-move-law, every game has a finite
>> number of moves).
>
> By "uncountable infinite" do you mean Aleph-1? I can't see why the number of
> games isn't Aleph-0.

If an infinitely long game is considered a game, then just think that
we have gotten into a situation where there is a possibility for a
circular move sequence A, and a circular move sequence B (where
circular means that the board is not changed after the sequence).

Now I can compose continuations consisting of an arbitrary infinite
sequence of A or B. That's Aleph-1.

--
David Kastrup, Kriemhildstr. 15, 44793 Bochum