A133047 -id:A133047 - cp's OEIS frontend

A133046 Starting from the standard 12 against 12 starting position in checkers, the sequence gives the number of distinct move sequences after n moves.

1, 7, 49, 302, 1469, 7361, 36768, 179740, 845931, 3963680, 18391564, 85242128, 388623673, 1766623630, 7978439499, 36263167175, 165629569428, 758818810990, 3493881706141, 16114043592799, 74545030871553, 345100524480819, 1602372721738102, 7437536860666213, 34651381875296000, 161067479882075800, 752172458688067137, 3499844183628002605, 16377718018836900735, 76309690522352444005

Offset: 0

Jonathan Schaeffer (jonathan(AT)cs.ualberta.ca), Dec 27 2007

Duplicate captures (viz. the situation where a king can capture the same pieces in different directions) are counted separately.

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 512.

A. Bik, Aart's Computer Checkers Page
M. Fierz, CheckerBoard
I. Korshunov, Standard correctness check for a move generator [in Russian]
Jonathan Schaeffer et al., Checkers is solved, Science, Vol. 317. no. 5844, pp. 1518-1522, Sep 14 2007.

Cf. A133047, A055213.

a(12)-a(20) computed by Aart Bik and sent by Richard Bean, Sep 18 2009
a(21)-a(26) computed by Aart Bik, with last two completed Sep 18 2012. Rein Halbersma was first to compute a(22). Murray Cash confirmed Aart's a(23) and a(24) results.
a(27)-a(28) first computed by Aart Bik, Sep 2012. Paul Byrne confirmed Aart's a(23)-a(28).
a(29) from Murray Cash, Nov 20 2020

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

cp's OEIS Frontend

A133046 Starting from the standard 12 against 12 starting position in checkers, the sequence gives the number of distinct move sequences after n moves.

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