Sources page biographical material

5.I.1. EIGHT QUEENS PROBLEM
See MUS I 210-284. S&B 37 shows examples. See also 5.Z. See also 6.T for examples where no three are in a row.
Ahrens. Mathematische Spiele. Encyklopadie article, op. cit. in 3.B. 1904. Pp. 1082 1084 discusses history and results for the n queens problem, with many references.

Paul J. Campbell. Gauss and the eight queens problem. HM 4:4 (Nov 1977) 397 404. Detailed history. Demonstrates that Gauss did not obtain a complete solution and traces how this misconception originated and spread.

"Schachfreund" (Max Bezzel). Berliner Schachzeitung 3 (Sep 1848) 363. ??NYS

Solutions. Ibid. 4 (Jan 1849) 40. ??NYS (Ahrens says this only gives two solutions. A. C. White says two or three. Jaenisch says a total of 5 solutions were published here and in 1854.)

Franz Nauck. Eine in das Gebiet der Mathematik fallende Aufgabe von Herrn Dr. Nauck in Schleusingen. Illustrirte Zeitung (Leipzig) 14 (No. 361) (1 Jun 1850) 352. Reposes problem. [The papers do not give a first name or initial. The only Nauck in the first six volumes of Poggendorff is Ernst Friedrich (1819-1875), a geologist. Ahrens gives no initial. Campbell gives Franz.]

Franz Nauck. Briefwechseln mit Allen für Alle. Illustrirte Zeitung (Leipzig) 15 (No. 377) (21 Sep 1850) 182. Complete solution.

Editorial comments: Briefwechsel. Illustrirte Zeitung (Leipzig) 15 (No. 378) (28 Sep 1850) 207. Thanks 6 correspondents for the complete solution and says Nauck reports that a blind person has also found all 92 solutions.

Gauss read the Illustrirte Zeitung and worked on the problem, corresponding with his friend Schumacher starting on 1 Sep 1850. Campbell discusses the content of the letters, which were published in: C. A. F. Peters, ed; Briefwechsel zwischen C. F. Gauss und H. C. Schumacher; vol. 6, Altona, 1865, ??NYS. John Brillhart writes that there is some material in Gauss' Werke, vol. XII: Varia kleine Notizen verschiednen Inhalts ... 5, pp. 19-28, ??NYS -- not cited by Campbell.

F. J. E. Lionnet. Question 251. Nouvelles Annales de Mathématiques 11 (1852) 114 115. Reposes problem and gives an abstract version.

Giusto Bellavitis. Terza rivista di alcuni articoli dei Comptes Rendus dell'Accademia delle Scienze di Francia e di alcuni questioni des Nouvelles Annales des mathématiques. Atti dell'I. R. Istituto Veneto di Scienze, Lettere ed Arti (3) 6 [= vol. 19] (1860/61) 376-392 & 411 436 (as part of Adunanza del Giorno 17 Marzo 1861 on pp. 347 436). The material of interest is: Q. 251. Disposizione sullo scacchiere di otto regine, on pp. 434 435. Gives the 12 essentially different solutions. Lucas (1895) says Bellavitis was the first to find all solutions, but see above. However this may be the first appearance of the 12 essentially different solutions.

C. F. de Jaenisch. Op. cit. in 5.F.1. 1862. Vol. 1, pp. 122-135. Gives the 12 basic solutions and shows they produce 92. Notes that in every solution, 4 queens are on white squares and 4 are on black.

A. C. Cretaine. Études sur le Problème de la Marche du Cavalier au Jeu des Échecs et Solution du Problème des Huit Dames. A. Cretaine, Paris, 1865. ??NYS -- cited by Lucas (1895). Shows it is possible to solve the eight queens problem after placing one queen arbitrarily.

G. Bellavitis. Algebra N. 72 Lionnet. Atti dell'Istituto Veneto (3) 15 (1869/70) 844 845.

Siegmund Günther. Zur mathematische Theorie des Schachbretts. Grunert's Archiv der Mathematik und Physik 56 (1874) 281-292. ??NYS. Sketches history of the problem -- see Campbell. He gives a theoretical, but not very practical, approach via determinants which he carries out for 4 x 4 and 5 x 5.

J. W. L. Glaisher. On the problem of the eight queens. Philosophical Magazine (4) 48 (1874) 457-467. Gives a sketch of Günther's history which creates several errors, in particular attributing the solution to Gauss -- see Campbell, who suggests Glaisher could not read German well. (However, in 1921 & 1923, Glaisher published two long articles involving the history of 15-16C German mathematics, showing great familiarity with the language.) Simplifies and extends Günther's approach and does 6 x 6, 7 x 7, 8 x 8 boards.

Lucas. RM2, 1883. Note V: Additions du Tome premier. Pp. 238-240. Gives the solutions on the 9 x 9 board, due to P. H. Schoute, in a series of articles titled Wiskundige Verpoozingen in Eigen Haard. Gives the solutions on the 10 x 10 board, found by M. Delannoy.

S&B, p. 37, show an 1886 puzzle version of the six queens problem.

A. Pein. Aufstellung von n Königinnen auf einem Schachbrett von n² Feldern. Leipzig. ??NYS -- cited by Ball, MRE, 4th ed., 1905 as giving the 92 inequivalent solutions on the 10 x 10.

Ball. MRE, 1st ed., 1892. The eight queens problem, pp. 85-88. Cites Günther and Glaisher and repeats the historical errors. Sketches Günther's approach, but only cites Glaisher's extension of it. He gives the numbers of solutions and of inequivalent solutions up through 10 x 10 -- see Dudeney below for these numbers, but the two values in ( ) are not given by Dudeney. He states results for the 9 x 9 and 10 x 10, citing Lucas. Says that a 6 x 6 version "is sold in the streets of London for a penny".

Hoffmann. 1893.

Chap. VI, pp. 272 273 & 286 = Hoffmann-Hordern, pp. 187-189, with photo.

No. 24: No two in a row. Eight queens. Photo on p. 188 shows Jeu des Sentinelles, by Watilliaux, dated 1874-1895.

No. 25: The "Simple" Puzzle. Nine queens. Says a version was sold by Messrs. Feltham, with a notched board but the pieces were allowed to move over the gaps, so it was really a 9 x 9 board.

Chap. X, No. 18: The Treasure at Medinet, pp. 343 344 & 381 = Hoffmann-Hordern, pp. 237-239. This is a solution of the eight queens problem, cut into four quadrants and jumbled. The goal is to reconstruct the solution. Photo on p. 239 shows Jeu des Manifestants, with box.

Hordern Collection, p. 94, and S&B, p. 37, show a version of this with same box, but which divides the board into eight 2 x 4 rectangles.

Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 1: The famous Italian pin puzzle. 6 queens puzzle. No solution.

Lucas. L'Arithmétique Amusante. 1895. Note IV: Section I: Les huit dames, pp. 210-220. Asserts Bellavitis was the first to find all solutions. Discusses symmetries and shows the 12 basic solutions. Correctly describes Jaenisch as obscure. Gives an easy solution of Cretaine's problem which can be remembered as a trick. Shows there are six solutions which can be superimposed with no overlap, i.e. six solutions using disjoint sets of cells.

C. D. Locock, conductor. Chess Column. Knowledge 19 (Jan 1896) 23-24; (Feb 1896) 47 48; (May 1896) 119; (Jul 1896) 167-168. This series begins by saying most players know there is a solution, "but, possibly, some may be surprised to learn that there are ninety-two ways of performing the feat, ...." He then enumerates them. Second article studies various properties of the solutions, particularly looking for examples where one solution shifts to produce another one. Third article notes some readers' comments. Fourth article is a long communication from W. J. Ashdown about the number of distinct solutions, which he gets as 24 rather than the usual 12.

T. B. Sprague. Proc. Edinburgh Math. Soc. 17 (1898-9) 43-68. ??NYS -- cited by Ball, MRE, 4th ed., 1905, as giving the 341 inequivalent solutions on the 11 x 11.

Benson. 1904. Pins and dots puzzle, p. 253. 6 queens problem, one solution.

Ball. MRE, 4th ed., 1905. The eight queens problem, pp. 114-120. Corrects some history by citing MUS, 1st ed., 1901. Gives one instance of Glaisher's method -- going from 4 x 4 to 5 x 5 and its results going up to 8 x 8. Says the 92 inequivalent solutions on the 10 x 10 were given by Pein and the 341 inequivalent solutions on the 11 x 11 were given by Sprague. The 5th ed., pp. 113-119 calls it "One of the classical problems connected with a chess-board" and adds examples of solutions up to 21 x 21 due to Mr. Derington.

Pearson. 1907. Part III, no. 59: Stray dots, pp. 59 & 130. Same as Hoffmann's Treasure at Medinet.

Burren Loughlin & L. L. Flood. Bright-Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909. The eight provinces, pp. 14-15 & 65. Same as Hoffmann's Treasure at Medinet.

A. C. White. Sam Loyd and His Chess Problems. 1913. Op. cit. in 1. P. 101 says Loyd discovered that all solutions have a piece at d1 or equivalent.

Williams. Home Entertainments. 1914. A draughtboard puzzle, p. 115. "Arrange eight men on a draughtboard in such a way that no two are upon the same line in any direction." This is not well stated!! Gives one solution: 52468317 and says "Work out other solutions for yourself."

Dudeney. AM. 1917. The guarded chessboard, pp. 95 96. Gives the number of ways of placing n queens and the number of inequivalent ways. The values in ( ) are given by Ball, but not by Dudeney.

n 4 5 6 7 8 9 10 11 12 13

ways 2 10 4 40 92 (352) (724)   - -

inequivalent ways 1 2 1 6 12 46 92 341 (1766) (1346)
Ball. MRE, 9th ed., 1920. The eight queens problem, pp. 113-119. Omits references to Pein and Sprague and adds the number of inequivalent solutions for the 12 x 12 and 13 x 13.

Blyth. Match-Stick Magic. 1921. No pairs allowed, p. 74. 6 queens problem.

Hummerston. Fun, Mirth & Mystery. 1924. No two in a line, p. 48. Chessboard. Place 'so that no two are upon the same line in any direction along straight or diagonal lines?' Gives one solution: 47531682, 'but there are hundreds of other ways'. You can let someone place the first piece.

Rohrbough. Puzzle Craft. 1932. Houdini Puzzle, p. 17. 6 x 6 case.

Rohrbough. Brain Resters and Testers. c1935. Houdini Puzzle, p. 25. 6 x 6 problem. "-- From New York World some years ago, credited to Harry Houdini." I have never seen this attribution elsewhere.

Pál Révész. Mathematik auf dem Schachbrett. In: Endre Hódi, ed. Mathematisches Mosaik. (As: Matematikai Érdekességek; Gondolat, Budapest, 1969.) Translated by Günther Eisenreich. Urania Verlag, Leipzig, 1977. Pp. 20 27. On p. 24, he says that all solutions have 4 queens on white and 4 on black. He says that one can place at most 5 non attacking queens on one colour.

Doubleday - 2. 1971. Too easy?, pp. 97-98. The two solutions on the 4 x 4 board are disjoint.

Dean S. Clark & Oved Shisha. Proof without words: Inductive construction of an infinite chessboard with maximal placement of nonattacking queens. MM 61:2 (1988) 98. Consider a 5 x 5 board with queens in cells (1,1), (2,4), (3,2), (4,5), (5,3). 5 such boards can be similarly placed within a 25 x 25 board viewed as a 5 x 5 array of 5 x 5 boards and this has no queens attacking. Repeating the inflationary process gives a solution on the board of edge 5³, then the board of edge 5⁴, .... They cite their paper: Invulnerable queens on an infinite chessboard; Annals of the NY Acad. of Sci.: Third Intern. Conf. on Comb. Math.; to appear. ??NYS.

Liz Allen. Brain Sharpeners. Op. cit. in 5.B. 1991. Squares before your eyes, pp. 21 & 106. Asks for solutions of the eight queens problem with no piece on either main diagonal. Two of the 12 basic solutions have this, but one of these is the symmetric case, so there are 12 solutions of this problem.

Donald E. Knuth. Dancing links. 25pp preprint of a talk given at Oxford in Sep 1999, sent by the author. See the discussion in 6.F. He finds the following numbers of solutions for placing n queens, n = 1, 2, ..., 18.

1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 3 65596, 22 79184, 147 72512, 958 15104, 6660 90624.