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

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

Offset: 0

Author

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

Keywords

• hard
• nonn
• more
• fini

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. Chess: A006494, A048987, A079485.

Cf. Go: A007565, A048289.

Cf. Checkers: A133046, A133047.

Programs

Python

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

Extensions

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