A225551 -id:A225551 - cp's OEIS frontend

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

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.

A225557 Longest checkmate in king and two bishops versus king endgame on an n X n chessboard.

Original entry on oeis.org

1, 9, 11, 13, 16, 19, 21, 24, 26, 29, 32, 35, 38, 40, 44, 46, 49, 52, 55, 58

Offset: 3

Vaclav Kotesovec, May 10 2013

			Longest win on an 8x8 chessboard: Ka1 Bd1 Bb4 - Kc4, 1.Bb4-d6 Kc4-d5! 2.Bd6-g3 Kd5-c6 3.Ka1-b1 Kc6-d5 4.Kb1-c1 Kd5-c6 5.Kc1-c2 Kc6-d5 6.Kc2-d3! Kd5-e6 7.Kd3-e4! Ke6-e7 8.Ke4-e5 Ke7-d7! 9.Ke5-d5 Kd7-e8 10.Bd1-h5 Ke8-e7 11.Bg3-e5! Ke7-d8 12.Kd5-d6! Kd8-c8! 13.Kd6-c6! Kc8-d8! 14.Be5-f6 Kd8-c8! 15.Bh5-g6 Kc8-b8! 16.Kc6-b6! Kb8-a8 17.Bg6-f5 Ka8-b8! 18.Bf6-e5! Kb8-a8! 19.Bf5-e4#, therefore a(8) = 19.

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.225 and 236 (2013), p. 350 and 368 (second edition, 2017).

Cf. A225551, A225552, A225553, A225554, A225555, A225556, A227437.

Conjecture: a(n) ~ 11*n/4.

a(21)-a(22) from Vaclav Kotesovec, Jan 11 2017

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

A274684 White to move: King and Queen vs. King: Number of positions with mate in n.

Original entry on oeis.org

2448, 5012, 9064, 19964, 26164, 32064, 32104, 15000, 2680, 8

Offset: 1

David A. Corneth, Jul 02 2016

After n = 10, all terms are 0 so the sequence ends at n = 10.
GBR code is 1000. Longest checkmate under perfect play is in 10 (FEN 8/8/8/5k2/8/8/1Q6/K7 w - - 0 1). For path of moves, see k4it-link. Data is found using Kryukov link, "Endgame Tablebases online". FEN is found in Kryukov link, "Longest checkmates in chess".
The listed examples, or more generally up to 6 man positions can be examined using the k4it link.

Hans Zellner, International Computer Chess Association Journal, 1987, Vol. 10, No. 2, p. 95.

Steven J. Edwards, Endgame tablebase position count summaries, 1994.
EG 121, GBR class 1000.00, 1996, p. 873.
k4it, All 6-men tables online (except 5 vs. 1).
K. Kryukov, Endgame Tablebases Online
K. Kryukov, Longest checkmates in chess
Wikipedia, GBR code

Cf. A225551, A274458, A274683.

Data and comments changed by Vaclav Kotesovec, Aug 01 2016

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A225557 Longest checkmate in king and two bishops versus king endgame on an n X n chessboard.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A274684 White to move: King and Queen vs. King: Number of positions with mate in n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions