login
Number of rotationally symmetric solutions for queens on n X n board.
7

%I #35 Aug 06 2024 08:57:05

%S 1,0,0,2,2,0,0,0,0,0,0,8,8,0,0,64,128,0,0,480,704,0,0,3328,3264,0,0,

%T 32896,43776,0,0,406784,667904,0,0,5845504,8650752,0,0,77184000,

%U 101492736,0,0,1261588480,1795233792,0,0,21517426688,35028172800,0,0,406875119616,652044443648,0,0,8613581094912,12530550128640,0,0,194409626533888,291826098503680,0,0

%N Number of rotationally symmetric solutions for queens on n X n board.

%C From _Don Knuth_, Jul 17 2015: (Start)

%C Ahrens proved that a(n)=0 unless n=4k or 4k+1. He also proved that in the latter case, a(n) is a multiple of 2^k. He found all solutions when n was less than 20.

%C Kraitchik carried the calculations further (for n less than 28). In his book he tabulated only the values a(n)/2^k. He had correct entries for n=21 and n=25, but his values for n=20 and n=24 were 1 too small -- of course he had calculated everything by hand! (End)

%D W. Ahrens, Mathematische Unterhaltungen und Spiele, 2nd edition, volume 1, Teubner, 1910, pages 249-258.

%D Maurice Kraitchik, Le problème des reines, Bruxelles: L'Échiquier, 1926, page 18.

%H Tricia M. Brown, <a href="http://dx.doi.org/10.5642/jhummath.201601.08">Kaleidoscopes, Chessboards, and Symmetry</a>, Journal of Humanistic Mathematics, Volume 6 Issue 1 ( January 2016), pages 110-126.

%H P. Capstick and K. McCann, <a href="/A000170/a000170_1.pdf">The problem of the n queens</a>, apparently unpublished, no date (circa 1990?) [Scanned copy]

%H Gheorghe Coserea, <a href="/A033148/a033148.txt">Solutions for n=20</a>.

%H Gheorghe Coserea, <a href="/A033148/a033148_1.txt">Solutions for n=24</a>.

%H Gheorghe Coserea, <a href="/A033148/a033148.mzn.txt">MiniZinc model for generating solutions</a>.

%H YuhPyng Shieh, <a href="https://citeseerx.ist.psu.edu/pdf/e377e984a5788068793c7bb11f31a51f10ddf753">Cyclic Complete Mappings Counting Problems</a>

%H M. Szabo, <a href="https://web.archive.org/web/20031018133806/http://www.nexus.hu/mikk/queen/index.html">Non-attacking Queens Problem Page</a>

%Y Cf. A002562, A032522, A037223, A037224, A260189.

%K nonn,hard

%O 1,4

%A Miklos SZABO (mike(AT)ludens.elte.hu)

%E More terms from Jieh Hsiang and YuhPyng Shieh (arping(AT)turing.csie.ntu.edu.tw), May 20 2002