A006494 -id:A006494 - cp's OEIS frontend

A048987 Number of possible chess games at the end of the n-th ply.

1, 20, 400, 8902, 197281, 4865609, 119060324, 3195901860, 84998978956, 2439530234167, 69352859712417, 2097651003696806, 62854969236701747, 1981066775000396239, 61885021521585529237, 2015099950053364471960

Offset: 0

S. J. Edwards

Does not include games which end in fewer than n plies.

According to the laws of chess, the "50-move rule" and "draw by 3-fold repetition" do not prevent infinite games because they require an appeal by one of the players, but the "75-move rule" introduced on Jul 01 2014 is automatic and makes chess finite. - François Labelle, Mar 30 2015

A. Bertilsson, Distributed Perft Project [also found a(11)]
James Grime and Brady Haran, How many chess games are possible? (Numberphile video), 2015.
Jakub Kowalski, Marek Szykuła, Experimental Studies in General Game Playing: An Experience Report, University of Wrocław, (Poland, 2020).
F. Labelle, Statistics on chess games
T. Sillke, number of legal chess positions
R. P. Stanley, Extremal [Chess] Problems
Wikipedia, Shannon number
Index entries for sequences related to number of chess games

Cf. A006494, A079485, A083276, A019319, A285873 (no queens), A285874 (no rooks), A285875 (no knights), A285876 (no bishops), A285877 (no pawns), A285878 (pawns and king).

a(10) from Richard Bean, Jun 02 2003
a(11) from François Labelle, Jul 25 2004, who thanks Joost de Heer for providing computer time
a(12) from Paul Byrne on Oct 24 2006, verified by S. J. Edwards on Apr 25 2011
a(13) from Paul Byrne on Nov 08 2011, verified by S. J. Edwards on Oct 03 2012
a(14) from Richard Bean on Jul 24 2018, value from Peter Osterlund and Ankan Banerjee
a(15) from Paul Barnett on Jul 04 2022, value from wikipedia page "Shannon number"

A019319 Number of possible chess diagrams after n plies.

Original entry on oeis.org

1, 20, 400, 5362, 71852, 815677, 9260610, 94305342, 958605819, 8866424380, 81766238574, 692390232505

Offset: 0

Bernd Schwarzkopf (schwarzkopf(AT)uni-duesseldorf.de)

Definition: position = position with castling and en passant information, diagram = position without castling and en passant information.
Even though the sequence may be infinite (if none of the rules for draw is ever invoked by any of the players), the sequence becomes constant from a given rank n on, since it is increasing (I conjecture - even though some positions available at the n-th move might not be available on the (1+n)-th move) and bounded, thus it has a limit. The challenge is now to find this limit (or at least nontrivial upper bounds) and the rank from which on the sequence becomes constant. - M. F. Hasler, Feb 15 2008
The sequence became finite on Jul 01 2014 with the introduction of a new draw rule which is automatic (the 75-move rule). About Hasler's second challenge, a chess problem by L. Ceriani and K. Fabel shows that at least one position is visited for the first time at ply 366. - François Labelle, Apr 01 2015

Bernd Schwarzkopf, Die ersten Züge (The First Moves), Problemkiste (No. 92, April 1994, p. 142-143).

L. Ceriani, K. Fabel, Chess problem: Non-unique proof game in 183 moves, Am Rande des Schachbretts, 1947
F. Labelle, Statistics on chess positions
Eric Weisstein's World of Mathematics, Chess
Index entries for sequences related to number of chess games

Cf. A083276, A048987, A090051, A006494, A079485, A019319.

More terms from Richard Bean, Jun 02 2002
a(6)-a(8) from François Labelle, Jan 19 2004
a(9)-a(10) from Arkadiusz Wesolowski, Jan 04 2012
a(11) from François Labelle, Jan 16 2017

A079485 Number of chess games that end in checkmate after exactly n plies.

Original entry on oeis.org

0, 0, 0, 0, 8, 347, 10828, 435767, 9852036, 400191963, 8790619155, 362290010907, 8361091858959, 346742245764219

Offset: 0

N. J. A. Sloane, Jan 17 2003

a(6) was computed in 1897.

			a(n) = 0 for n < 4 and a(4) = 8, because the earliest checkmate can occur at move 2, e.g., with 1.f3 e5, 2.g4 Qh4#. The 8 = 2*2*2 possibilities counted in a(4) all correspond to essentially the same mate, with (1) the alternative 1.f4, (2) the alternative 1...e6, (3) inversed order of white's moves. - _M. F. Hasler_, Mar 03 2022

H. Simpson, Chess Review, Jan-Feb 1982.

F. Labelle, Statistics on chess games
R. P. Stanley, Extremal [Chess] Problems
M. Willey, Iowa Chess Homepage.
World Chess Federation, Laws of Chess: For competitions starting on or after 1 July 2014
J. Young, Opening Tree Statistics.
Index entries for sequences related to number of chess games

Cf. A006494, A048987, A089956.

More terms from Richard Bean, Jun 08 2003
a(9) and a(10) from François Labelle, Jan 12 2004
a(11) from François Labelle, Jul 25 2004, who thanks Joost de Heer for providing computer time.
Link to Extremal [Chess] Problems of R. P. Stanley corrected by Johannes W. Meijer, Mar 01 2009
a(12) from François Labelle, Mar 04 2012
a(13) from François Labelle, Aug 15 2017

A157851 Number of possible Fischer Random Chess games at the end of the n-th ply.

Original entry on oeis.org

960, 18882, 371766, 8224968, 181106056, 4433048830, 107637760217, 2854198413886, 75006431287937

Offset: 0

Johannes W. Meijer & Richard Pijl (richard.pijl(AT)telenet.be), Mar 07 2009, Feb 25 2010

Fischer Random Chess is also called Chess960 because the number of different initial positions is 960.
The number of possible games at the end of the n-th ply is the sum of all possible games on all 960 boards with a different initial position.
The number of possible first moves for white depends on the following three factors:
a) The eight pawns.
b) The positions of the two knights. If they are on a1 and/or h1 the number of possible moves reduces from 20 to 18 or 19. On the 960 boards there are 240 boards with a knight on a1. Looking more closely at the positions of the second knight on these 240 boards reveals that 36 knights can be found on b1, d1, f1 and h1 and 32 knights can be found on c1, e1 and g1, something that can be proved with some simple combinatorics.
c) The possibility of castling. On the 960 boards there are 72 boards with a king on d1 and a rook on c1 and there are 90 boards with a king on f1 and a rook on g1. Both positions allow castling under the Fischer Random Chess rules.
These three factors lead to the following partition of the 960 boards (K = King; R = Rook; N = Knight; NoN = No Knight; NoC = No castling allowed): 454 (NoNa1+NoNh1+NoC), 162 (Na1+NoNh1+NoC), 160 (Nh1+NoNa1+NoC), 34 (NoNa1+NoNh1+Kf1+Rg1), 28 (NoNa1+NoNh1+Kd1+Rc1), 28 (Nh1+NoNa1+Kf1+Rg1), 22 (Na1+Nh1+NoC), 22 (Na1+NoNh1+Kd1+Rc1), 20 (Na1+NoNh1+Kf1+Rg1), 16 (Nh1+NoNa1+Kd1+Rc1), 8 (Na1+Nh1+Kf1+Rg1), 6 (Na1+Nh1+Kd1+Rc1).
The first three terms of the sequence can be calculated in a straightforward way, see the examples. The values of a(1) and a(2) were confirmed by Richard Pijl with his Fischer Random Chess playing chess engine The Baron, see the links. He also determined the values of a(3), a(4) and a(5).
The Baron 3.41 now gives different values for a(3)-a(6), confirmed by my own chess engine. - François Labelle, Dec 05 2017

			a(0) = 4 (Bishop) * 4 (Bishop) * 6 (Queen) * 10 (Knights) * 1 (King and Rooks) = 960.
a(1) = 36*18 + 204*19 + 204*19 + 516*20 + 90 + 72 = 18882.
a(2) = 22*18^2 + (162+160+8+6)*19^2 + (454+28+22+20+16)*20^2 + (34+28)*21^2 = 371766.

Chessgames.com, Fischerandom Chess Generator
Chessvariants.com, Fischer Random Chess
Richard Pijl, The Baron
Wikipedia, Chess960

Cf. Chess: A006494, A048987, A079485.
Cf. Go: A007565, A048289.
Cf. Checkers: A133046, A133047.

import chess
def A157851(n, b = None): return (b.legal_moves.count() if b else 960) if not n else sum(b.push(m) or A157851(n-1, b)+(not b.pop()) for m in b.legal_moves) if b else sum(A157851(n-1, chess.Board.from_chess960_pos(s)) for s in range(960)) # (For illustration, slow for n > 3.) - M. F. Hasler, Apr 25 2023

Corrected and edited by Johannes W. Meijer, Feb 25 2010, Mar 03 2010
a(6) added by Richard Pijl (richard.pijl(AT)telenet.be). - Johannes W. Meijer, May 29 2010
a(3)-a(6) corrected by François Labelle, Dec 05 2017
a(7)-a(8) from François Labelle, Jan 18 2018

A007545 Number of chess games with n plies (another version).

Original entry on oeis.org

1, 20, 400, 8902, 197742, 4897256, 120921506, 3284294545, 88867026005

Offset: 0

Ken Thompson (ken(AT)google.com)

Up to a(6) this is the number of chess games with all legality constraints removed, even allowing the king to be captured. (This differs from the number of positions in suicide chess, as suicide chess contains compulsory captures.) - Richard Bean, Jun 18 2003

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eric Weisstein's World of Mathematics, Chess
Index entries for sequences related to number of chess games

Cf. A006494, A048987.

I am not sure of the precise rules that were used to compute these numbers. A006494 and A048987 are the preferred versions of this sequence. - N. J. A. Sloane.

A114037 Number of games of Gothic chess after n plies plus number of games that terminate (i.e., mate) in fewer than n plies.

Original entry on oeis.org

1, 28, 784, 25283, 808984, 28946187, 1025229212, 39532257395, 1509030960338

Offset: 0

N. J. A. Sloane, based on email from Ed Trice, Feb 01 2006

nonn

Gothic chess is a variant played on a board with 80 squares.

Gothic Chess Federation, Home Page
Gothic Chess Forum, Source

Cf. A006494.

A366476 Number of possible chmess games at the end of the n-th ply.

Original entry on oeis.org

1, 25, 625

Offset: 0

N. J. A. Sloane, Oct 26 2023

Chmess is an imaginary game very similar to chess, with the exception that the king can move two squares in any direction instead of just one. To get a(1) = 25 I am assuming the king can jump over an occupied square, but is not allowed to move out and back (which would give a(1) = 26).

Daniel C. Dennett, Higher-order truths about chmess, Topoi 25 (1-2):39-41 (2006).
Nigel Warburton, Review of "I've Been Thinking" by Daniel C. Dennett (Allen Lane, 2023), TLS #6290 (Oct 20, 2023), pp. 8-9.

Daniel C. Dennett, Higher-order truths about chmess, Topoi 25 (1-2):39-41 (2006).
N. J. A. Sloane and others, Anyone for Chmess?, Conversation in the SeqFan Google Group, Oct 14 2024.

Cf. A006494, A048987.

cp's OEIS Frontend

A048987 Number of possible chess games at the end of the n-th ply.

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A019319 Number of possible chess diagrams after n plies.

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A079485 Number of chess games that end in checkmate after exactly n plies.

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A157851 Number of possible Fischer Random Chess games at the end of the n-th ply.

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A007545 Number of chess games with n plies (another version).

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A114037 Number of games of Gothic chess after n plies plus number of games that terminate (i.e., mate) in fewer than n plies.

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A366476 Number of possible chmess games at the end of the n-th ply.

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