OFFSET
0,2
COMMENTS
By the number of possible plays on the n-th move is meant the total number of legal lines of play for white under the rules of mirror chess at a depth of n moves from the standard initial position.
If white cannot play a legal move under the rules of mirror chess then the game is considered to be a draw.
Among the 270726 possibilities up to move 4, only 3 correspond to games ending in checkmate, all at move 4: see examples. - M. F. Hasler, Dec 08 2021
LINKS
Jeremy Gardiner, Mirror Chess
Jeremy Gardiner, Mirror Chess 3 moves supplied by Francois Labelle
EXAMPLE
A checkmate cannot occur earlier than at move 4, where we have the following possibilities: 1.d4 d5 2.Qd3 Qd6 3.Qf5 Qf4 4.Qxc8# or 3.Qh3 Qh6 4.Qxc8#, and
1.c4 c5 2.Qa4 Qa5 3.Qc6 Qc3 4.Qxc8#, corresponding to the following diagrams:
r n Q . k b n r r n Q . k b n r r n Q . k b n r
p p p . p p p p p p p . p p p p p p . p p p p p
. . . . . . . . . . . . . . . q . . . . . . . .
. . . p . . . . . . . p . . . . . . p . . . . .
. . . P . q . . . . . P . . . . . . P . . . . .
. . . . . . . . . . . . . . . . . . q . . . . .
P P P . P P P P P P P . P P P P P P . P P P P P
R N B . K B N R R N B . K B N R R N B . K B N R
where upper/lowercase letters represent white/black pieces, and dots stand for empty squares. - M. F. Hasler, Dec 08 2021
PROG
(Python)
import chess
def A136257(n, B=chess.Board()):
if n == 0: return 1
count = 0
for m in B.legal_moves:
B.push(m)
if B.is_checkmate():
if n == 1: count += 1
else:
m.from_square ^= 56
m.to_square ^= 56 # reverse ranks through XOR with 7
if B.is_legal(m):
if n == 1: count += 1
else:
B.push(m)
count += A136257(n - 1, B)
B.pop()
B.pop()
return count # M. F. Hasler, Dec 08 2021
CROSSREFS
KEYWORD
nonn,hard,more,fini
AUTHOR
Jeremy Gardiner, Apr 18 2008
EXTENSIONS
a(2) corrected and a(3) from Jeremy Gardiner, Mar 03 2013
a(3) corrected and a(4)-a(11) from François Labelle, Apr 12 2015
STATUS
approved