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%I #29 Dec 08 2021 05:35:39
%S 1,20,437,10461,270726,7456194,215666696,6485151199,201183083017,
%T 6401210746834,207969967925893,6875935591529309
%N Number of possible plays on the n-th move in Mirror Chess in which Black's play is always the mirror image of White (White must either mate or play such that Black can mirror the move).
%C 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.
%C If white cannot play a legal move under the rules of mirror chess then the game is considered to be a draw.
%C 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
%H Jeremy Gardiner, <a href="http://www.woomerang.com/mchess/">Mirror Chess</a>
%H Jeremy Gardiner, <a href="/A136257/a136257.txt">Mirror Chess 3 moves supplied by Francois Labelle</a>
%e 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
%e 1.c4 c5 2.Qa4 Qa5 3.Qc6 Qc3 4.Qxc8#, corresponding to the following diagrams:
%e r n Q . k b n r r n Q . k b n r r n Q . k b n r
%e p p p . p p p p p p p . p p p p p p . p p p p p
%e . . . . . . . . . . . . . . . q . . . . . . . .
%e . . . p . . . . . . . p . . . . . . p . . . . .
%e . . . P . q . . . . . P . . . . . . P . . . . .
%e . . . . . . . . . . . . . . . . . . q . . . . .
%e P P P . P P P P P P P . P P P P P P . P P P P P
%e R N B . K B N R R N B . K B N R R N B . K B N R
%e where upper/lowercase letters represent white/black pieces, and dots stand for empty squares. - _M. F. Hasler_, Dec 08 2021
%o (Python)
%o import chess
%o def A136257(n, B=chess.Board()):
%o if n == 0: return 1
%o count = 0
%o for m in B.legal_moves:
%o B.push(m)
%o if B.is_checkmate():
%o if n == 1: count += 1
%o else:
%o m.from_square ^= 56
%o m.to_square ^= 56 # reverse ranks through XOR with 7
%o if B.is_legal(m):
%o if n == 1: count += 1
%o else:
%o B.push(m)
%o count += A136257(n - 1, B)
%o B.pop()
%o B.pop()
%o return count # M. F. Hasler, Dec 08 2021
%Y Cf. A048987.
%K nonn,hard,more,fini
%O 0,2
%A _Jeremy Gardiner_, Apr 18 2008
%E a(2) corrected and a(3) from _Jeremy Gardiner_, Mar 03 2013
%E a(3) corrected and a(4)-a(11) from _François Labelle_, Apr 12 2015
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