A136257 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).

1, 20, 437, 10461, 270726, 7456194, 215666696, 6485151199, 201183083017, 6401210746834, 207969967925893, 6875935591529309

Offset: 0

Author

Jeremy Gardiner, Apr 18 2008

Keywords

• nonn
• hard
• more
• fini

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

Examples

			From _M. F. Hasler_, Jul 05 2025: (Start)
a(1) = 20 because White can make any of the 20 possible starting moves, to each of which Black can reply with the "same" (mirror) move.
Then, depending on the starting move, there are between 19 (for 1. a3, f3, f4 and h3) and 30 (for 1. e3) possible moves, for a total of a(2) = 437 moves (for each of which Black can reply with the "same" move).
At move 3, there is a total of 48 moves that are legal but not mirror-legal:
- After 1.c4: 2.d3 (or d4) 3.Qa4+, 2.d3 3.Qa4+, 2.Qb3 3.Qb5 (or Qxb6), and 2.Qa4 3.Qxa5 (or Qxd7+).
- After 1.d4: 2.c4 3.Qa4+, 2.e3 (or e4) 3.Bb5+, and 2.Qd3 3.Qb5+.
- Similarly after 1.c3 and 1.d3, and 20 more with 1.e3 or 1.e4.
- After 1.f4: 2.e3 3.Qh5+ and 2.e4 3.Qh5+, and after 1.g3: 2.Bh3 3.Bxd7+.
- Finally, there are also 9 sequences of only knight moves that end in a check, like 1.Nc3 2.Ne4 3.Nd6+ (or Nf6+). (End)
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
		

Links

• Jeremy Gardiner, Mirror Chess
• Jeremy Gardiner, Mirror Chess 3 moves supplied by Francois Labelle

Crossrefs

Cf. A048987.

Programs

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 not B.is_checkmate():
            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()
        elif n == 1: count += 1
        B.pop()
    return count  # M. F. Hasler, Dec 08 2021

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