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Longest checkmate in king, bishop and knight versus king endgame on an n X n chessboard.
8

%I #45 May 15 2024 11:09:29

%S 7,14,13,22,21,33,29,47,39,64,50,78,63,93,76,112,90,131,105,151,121,

%T 171

%N Longest checkmate in king, bishop and knight versus king endgame on an n X n chessboard.

%C If n is odd then the bishop must be on a black square.

%H Julius Telesin, <a href="http://www.arves.org/arves/images/PDF/EG_PDF/eg73.pdf">Can B + S give checkmate on 1000 x 1000 chessboard?</a>, EG 73, 1983, p. 190.

%H Vaclav Kotesovec, <a href="http://problem64.beda.cz/silo/kotesovec_icga_2001.pdf">King and Two Generalised Knights against King</a>, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001).

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Fairy chess endings on an n x n chessboard</a>, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, p. 224 and 231 (2013).

%H Chess Stackexchange, <a href="https://chess.stackexchange.com/questions/1432/kbn-vs-k-checkmate-on-nonstandard-boards">Mate with Bishop and Knight</a>, 2018.

%H Johan Wästlund, <a href="https://arxiv.org/abs/2405.04421">The bishop and knight checkmate on a large chessboard</a>, arXiv:2405.04421 [math.CO], 2024. See p. 13.

%F Conjecture: a(n) ~ n^2/6 + 3*n if n is even and a(n) ~ n^2/6 + 2*n if n is odd.

%e Longest win on an 8 X 8 chessboard: Ka1 Bc1 Sb1 - Kc2, 1.Bc1-d2 Kc2-b3! 2.Bd2-f4 Kb3-c2! 3.Ka1-a2! Kc2-d3! 4.Ka2-b3! Kd3-e4 5.Bf4-h2 Ke4-d5 6.Kb3-c3 Kd5-e4 7.Sb1-d2! Ke4-d5! 8.Kc3-d3! Kd5-c5! 9.Sd2-c4! Kc5-b5 10.Kd3-d4 Kb5-c6! 11.Bh2-g3 Kc6-b7! 12.Kd4-d5 Kb7-a7! 13.Kd5-c6! Ka7-a6 14.Bg3-h2 Ka6-a7! 15.Sc4-b6! Ka7-a6! 16.Bh2-b8! Ka6-a5! 17.Sb6-d5! Ka5-a4! 18.Kc6-c5 Ka4-b3! 19.Sd5-b4! Kb3-b2! 20.Bb8-f4! Kb2-c3! 21.Bf4-g5 Kc3-b3! 22.Bg5-f6 Kb3-a4 23.Kc5-c4! Ka4-a5! 24.Bf6-d8! Ka5-a4! 25.Sb4-d3! Ka4-a3! 26.Bd8-e7 Ka3-a4! 27.Sd3-c5! Ka4-a3! 28.Kc4-c3! Ka3-a2! 29.Sc5-d3! Ka2-b1! 30.Kc3-b3! Kb1-a1! 31.Kb3-c2! Ka1-a2! 32.Sd3-c1! Ka2-a1! 33.Be7-f6#, therefore a(8) = 33.

%Y Cf. A225551, A225552, A225553, A225554, A225555, A225557, A227437.

%K nonn,hard,more

%O 3,1

%A _Vaclav Kotesovec_, May 10 2013

%E a(21) from _Vaclav Kotesovec_, Jan 11 2017

%E a(22) from _Vaclav Kotesovec_, Jan 15 2017

%E a(23) from _Vaclav Kotesovec_, Jun 22 2017

%E a(24) from _Vaclav Kotesovec_, Jun 30 2017