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A157851 Number of possible Fischer Random Chess games at the end of the n-th ply. 0
960, 18882, 371766, 8224968, 181106056, 4433048830, 107637760217, 2854198413886, 75006431287937 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

LINKS

Table of n, a(n) for n=0..8.

Chessgames.com, Fischerandom Chess Generator

Chessvariants.com, Fischer Random Chess

Richard Pijl, The Baron

Wikipedia, Chess960

EXAMPLE

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.

CROSSREFS

Cf. Chess: A006494, A048987, A079485.

Cf. Go: A007565, A048289.

Cf. Checkers: A133046, A133047.

Sequence in context: A046788 A206079 A203732 * A278011 A282012 A147883

Adjacent sequences:  A157848 A157849 A157850 * A157852 A157853 A157854

KEYWORD

hard,nonn,more,fini

AUTHOR

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

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

STATUS

approved

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Last modified February 23 11:36 EST 2018. Contains 299579 sequences. (Running on oeis4.)