A225557 -id:A225557 - cp's OEIS frontend

A225551 Longest checkmate in king and queen versus king endgame on an n X n chessboard.

1, 4, 6, 7, 8, 10, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57

Offset: 3

Vaclav Kotesovec, May 10 2013

nonn

			Longest win on an 8x8 chessboard: Ka1 Qb2 - Kf5, 1.Ka1-b1 Kf5-e6! 2.Kb1-c2 Ke6-f5! 3.Kc2-c3 Kf5-e6 4.Kc3-d4! Ke6-f5 5.Qb2-b6! Kf5-g4 6.Kd4-e4 Kg4-g5! 7.Qb6-c6 Kg5-h4 8.Ke4-f4 Kh4-h5! 9.Qc6-d6 Kh5-h4! 10.Qd6-h6#, therefore a(8) = 10.

V. Kotesovec, King and Two Generalised Knights against King, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001)
V. Kotesovec, Fairy chess endings on an n x n chessboard, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, p.359 (2013), p. 539 (second edition, 2017).

Cf. A225552, A225553, A225554, A225555, A225556, A225557, A227437.

Conjecture: For n > 8, a(n) = 2*n-3-floor(n/2).
a(n) ~ 3*n/2.
Empirical g.f.: x^3*(x^8-x^7+x^5-x^4-2*x^3+x^2+3*x+1) / ((x-1)^2*(x+1)). - Colin Barker, May 11 2013

A225552 Longest checkmate in king and rook versus king endgame on an n X n chessboard.

Original entry on oeis.org

3, 7, 10, 12, 14, 16, 18, 21, 23, 25, 28, 30, 33, 35, 38, 39, 42, 44, 47, 49, 52, 54, 56, 58, 61, 63, 66, 67, 70, 72, 75, 77, 80, 81, 84, 86, 89, 91, 94, 95, 98, 100, 103, 105, 108, 109, 112, 114, 117, 119, 122, 123

Offset: 3

Vaclav Kotesovec, May 10 2013

nonn

			Longest win on an 8 X 8 chessboard: Ka1 Rb2 - Kc3, 1.Ka1-b1 Kc3-d4! 2.Kb1-c2! Kd4-c4 3.Rb2-b8 Kc4-d5 4.Kc2-d3! Kd5-e6 5.Kd3-d4 Ke6-f5 6.Rb8-e8! Kf5-f4 7.Re8-f8 Kf4-g5 8.Kd4-e4! Kg5-g6! 9.Ke4-f4 Kg6-g7! 10.Rf8-f5! Kg7-h6 11.Rf5-f6 Kh6-g7 12.Kf4-f5 Kg7-h7! 13.Rf6-f7 Kh7-h8! 14.Kf5-g5 Kh8-g8! 15.Kg5-g6! Kg8-h8! 16.Rf7-f8#, therefore a(8) = 16.

V. Kotesovec, King and Two Generalised Knights against King, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001).
V. Kotesovec, Fairy chess endings on an n x n chessboard, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, pp. 360-361 (2013), pp. 540-541 (second edition, 2017).

Cf. A225551, A225553, A225554, A225555, A225556, A225557, A227437.

Conjecture: For n > 24, a(n) = 2*n - 3 + floor((n+1)/2) - floor(n/6).

a(n) ~ 7*n/3.

Terms a(52)-a(54) from Vaclav Kotesovec, May 17 2013

A225555 Longest checkmate in king and princess versus king endgame on an n X n chessboard.

Original entry on oeis.org

3, 8, 9, 11, 14, 17, 21, 24, 26, 30, 33, 37, 41, 44, 48, 51, 55, 59, 63, 67, 70, 75, 79, 83, 87, 91, 96, 101, 104, 109, 114, 118, 123, 127, 132, 136, 141, 145, 151, 154

Offset: 3

Vaclav Kotesovec, May 10 2013

nonn

A princess moves like a bishop and a knight.

			Longest win on an 8x8 chessboard: Ka1 PRc1 - Kc2, 1.PRc1-f4 Kc2-b3 2.Ka1-b1 Kb3-c3! 3.Kb1-c1! Kc3-d4 4.Kc1-d2! Kd4-e4! 5.PRf4-e3! Ke4-e5! 6.Kd2-d3! Ke5-d6 7.Kd3-e4! Kd6-c7 8.Ke4-d5 Kc7-d7! 9.PRe3-f4 Kd7-e7! 10.PRf4-e5! Ke7-e8! 11.Kd5-d6 Ke8-d8 12.PRe5-f6 Kd8-c8! 13.Kd6-c6! Kc8-b8! 14.PRf6-d4! Kb8-a8 15.Kc6-b6! Ka8-b8! 16.PRd4-e6! Kb8-a8! 17.PRe6-c7#, therefore a(8) = 17.

V. Kotesovec, King and Two Generalised Knights against King, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001)
V. Kotesovec, Fairy chess endings on an n x n chessboard, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, p.362 (2013), p. 542 (second edition, 2017).

Cf. A225551, A225552, A225553, A225554, A225556, A225557, A227437.

Terms a(41)-a(42) from Vaclav Kotesovec, Jul 19 2013

A225556 Longest checkmate in king, bishop and knight versus king endgame on an n X n chessboard.

Original entry on oeis.org

7, 14, 13, 22, 21, 33, 29, 47, 39, 64, 50, 78, 63, 93, 76, 112, 90, 131, 105, 151, 121, 171

Offset: 3

Vaclav Kotesovec, May 10 2013

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

			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.

Julius Telesin, Can B + S give checkmate on 1000 x 1000 chessboard?, EG 73, 1983, p. 190.
Vaclav Kotesovec, King and Two Generalised Knights against King, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001).
Vaclav Kotesovec, Fairy chess endings on an n x n chessboard, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, p. 224 and 231 (2013), p. 349 and 362 (second edition, 2017).
Chess Stackexchange, Mate with Bishop and Knight, 2018.
Johan Wästlund, The bishop and knight checkmate on a large chessboard, arXiv:2405.04421 [math.CO], 2024. See p. 13.

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

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

a(21) from Vaclav Kotesovec, Jan 11 2017
a(22) from Vaclav Kotesovec, Jan 15 2017
a(23) from Vaclav Kotesovec, Jun 22 2017
a(24) from Vaclav Kotesovec, Jun 30 2017

A225553 Longest checkmate in king and amazon versus king endgame on an n X n chessboard.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21

Offset: 3

Vaclav Kotesovec, May 10 2013

nonn

An amazon (superqueen) moves like a queen and a knight.

			Longest win on an 8x8 chessboard: Ka1 AMb1 - Kd4, 1.AMb1-f5! Kd4-c4! 2.Ka1-b1 Kc4-b4! 3.Kb1-b2 Kb4-a4 4.AMf5-c5#, therefore a(8) = 4.

V. Kotesovec, King and Two Generalised Knights against King, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001)
V. Kotesovec, Fairy chess endings on an n x n chessboard, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, p.364 (2013), p. 544 (second edition, 2017).

Cf. A172200, A051223, A178967, A225551, A225552, A225554, A225555, A225556, A225557, A227437.

Conjecture: for n > 10, a(n) = floor((n+2)/2).

Empirical g.f.: -x^4*(x^9-x^8+x^4-x-1) / ((x-1)^2*(x+1)). - Colin Barker, May 11 2013

A225554 Longest checkmate in king and empress versus king endgame on an n X n chessboard.

Original entry on oeis.org

3, 5, 7, 8, 10, 11, 13, 14, 16, 18, 19, 21, 23, 25, 26, 28, 30, 32, 34, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 53, 55, 57, 59, 61, 63, 64, 66, 68

Offset: 3

Vaclav Kotesovec, May 10 2013

nonn

An empress moves like a rook and a knight.

			Longest win on an 8x8 chessboard: Ka1 EMb1 - Kd4, 1.Ka1-a2 Kd4-e5 2.Ka2-b3 Ke5-f4 3.Kb3-c3 Kf4-e5 4.EMb1-b6! Ke5-f4 5.Kc3-d4 Kf4-g5 6.Kd4-e4 Kg5-g4! 7.EMb6-e6 Kg4-g3! 8.EMe6-f4! Kg3-h2! 9.Ke4-f3! Kh2-g1! 10.Kf3-g3 Kg1-h1! 11.EMf4-f1#, therefore a(8) = 11.

V. Kotesovec, King and Two Generalised Knights against King, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001)
V. Kotesovec, Fairy chess endings on an n x n chessboard, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, p.363 (2013), p. 543 (second edition, 2017).

Cf. A137774, A218244, A225551, A225552, A225553, A225555, A225556, A225557, A227437.

Conjecture: a(n) ~ 7*n/4.

A227437 Longest checkmate in king and 3 knights versus king endgame on an n X n chessboard.

Original entry on oeis.org

10, 10, 16, 18, 21, 25, 29, 33, 36, 41, 45, 50, 55, 60, 66

Offset: 4

Vaclav Kotesovec, Jul 11 2013

With 16 GB of memory it was possible to explore this ending on boards up to 15 X 15. On these boards the ending is a general win. I think that from some greater n this ending is drawn (and the sequence is finite), but this is only conjecture. - Vaclav Kotesovec, Jul 19 2013

			Longest win on an 8 X 8 chessboard: Ka1 Sa2 Sb1 Sg1 - Kf2, 1.Sg1-h3! Kf2-g3! 2.Sh3-g5! Kg3-f4! 3.Sg5-f7! Kf4-g3 4.Ka1-b2 Kg3-f4 5.Kb2-c2 Kf4-g3 6.Kc2-d2 Kg3-f4 7.Sb1-a3 Kf4-f3 8.Kd2-d3 Kf3-f4! 9.Sa3-c4 Kf4-f5! 10.Sc4-e5 Kf5-f4 11.Sa2-b4 Kf4-f5! 12.Sb4-c6 Kf5-e6 13.Kd3-e4! Ke6-f6! 14.Sc6-d4! Kf6-e7 15.Ke4-f5! Ke7-f8! 16.Kf5-e6! Kf8-g7 17.Sd4-f5! Kg7-f8 18.Se5-g6! Kf8-g8! 19.Ke6-f6! Kg8-h7! 20.Sf7-g5! Kh7-g8! 21.Sf5-e7#, therefore a(8) = 21.
(In the above, 'S' (for "Springer" in German?) stands for knight moves.) - _M. F. Hasler_, Apr 22 2022

John Beasley, How many knights does a king require?, British Endgame Study News, Special Number 8, p. 8, December 1997.
V. Kotesovec, King and Two Generalised Knights against King, ICGA Journal, Vol. 24, No. 2, pp. 105-107 (2001)
V. Kotesovec, Fairy chess endgames - new results 2008, Electronic edition of chess booklets by Vaclav Kotesovec, Vol. 2, (2008)
V. Kotesovec, Fairy chess endings on an n x n chessboard, Electronic edition of chess booklets by Vaclav Kotesovec, vol. 8, pp. 52-64 (2013), pp. 59-82 (second edition, 2017).
Wikipedia, Endgame tablebase

Cf. A225551, A225552, A225553, A225554, A225555, A225556, A225557.

a(16) from Vaclav Kotesovec, Jan 07 2017
a(17) from Vaclav Kotesovec, Sep 05 2017
a(18) from Vaclav Kotesovec, Jan 24 2018

cp's OEIS Frontend

A225551 Longest checkmate in king and queen versus king endgame on an n X n chessboard.

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A225552 Longest checkmate in king and rook versus king endgame on an n X n chessboard.

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A225555 Longest checkmate in king and princess versus king endgame on an n X n chessboard.

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A225556 Longest checkmate in king, bishop and knight versus king endgame on an n X n chessboard.

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A225553 Longest checkmate in king and amazon versus king endgame on an n X n chessboard.

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A225554 Longest checkmate in king and empress versus king endgame on an n X n chessboard.

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A227437 Longest checkmate in king and 3 knights versus king endgame on an n X n chessboard.

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