A019319 Number of possible chess diagrams after n plies.

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

Offset: 0

Author

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

Keywords

• nonn
• hard
• nice
• fini

Comments

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

References

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

Links

• 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

Crossrefs

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

Extensions

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