Chronology of Knight's Tours

© 2002 — compiled by George Jelliss.

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PART 2. 1800 – 1899

A long dash (—) indicates missing information. A tilde (~) preceding a date indicates that it is merely notional. ms = manuscript. Locations of copies of the works, including shelf-marks or catalogue references, are indicated, where known, for the assistance of researchers. Abbreviations used for libraries: [C] = Cleveland Library, Ohio (Contains the J.G.White chess collection), [K] = Koninklijke Bibliotheek, the Hague (contains the van der Linde and Niemeier chess collections); [R] = Radcliffe Science Library, Oxford. Where I have not located a copy myself, the source of my information is indicated within square brackets. Abbreviations used for important bibliographic sources: [A] = Ahrens 1901, [L] = van der Linde 1874, [M] = Murray 1930.

« 1800s

(~1800 – ~1900). Two French manuscripts with long titles in the J. G. White Collection (undated, probably 19th century): Observations et remarques ..., and receueil de differentes figures .... [C]

—Bossut & —Lalande (1800). Dizionario Enciclopedico delle Matematiche Padova 1800, vol.5, pp.107 and 111. Euler's work.

Charles Hutton (1803). (Professor of Mathematics, Royal Academy Woolwich 1773); Editor of English editions of Ozanam's Recreations 1803, 1814.

—Pruen (1804). Introduction ... (to chess) Cheltenham. The de Moivre tour.

Joseph Dollinger (1806). Ein Hundert zehen ganz neu zusammengesetzte Schach-End-Spiele. Dnn vier und zwanzig verschiedene Arten den Springer durch alle Felder hin und zuruck zu spielen, ohne ein Feld doppelt zu beruhren. {110 quite new chess endgame compositions, then 24 dfferent ways for the knight to play through all the squares, there and back, without touching any square twice} Anton Pichler, Vienna 1806. [Bodleian library, shelfmark; Murray e.81]. The tours are presented on pages 61-84 as a series of square coordinates, e.g. a1-b3-a5-c4-d6-e4, etc. They are all very similar. No.5 differs in only two moves from the first closed tour given by Euler (1759). None are symmetric. Six have a complete border braid from a1 to h1 to h8. Although the tours are all reentrant the initial squares are taken to be all different, following the sequence: abcdefgh1, abcdefgh8, cf3, cf6, de4, de5. [also C]

« 1810s

Prof. Vicenzo Brunacci (1811). Compendio di Calcolo Sublime Milan, pp.74-78. [C]

(1815). "Promenade des Springers", Thee und Caffee-Zeitvertreib, 2 January 1815. [M]

Karl Brandan Mollweide (or Carolus Brandanus) (1816). Dissertationis Mathematicae de Quadratis Magicis Leipzig. See also 1823.

Charles Babbage (1817). "An Account of Euler's Method of Solving a Problem Relative to the Move of the Knight at the Game of Chess" Journal of Science and the Arts, published by John Murray for the Royal Institution, London, 1817-18; vol.3, art.5, pp.72-77 plus plate 2 at end of volume. Summary of Euler (1759), quoting six complete and three partial tours in numerical form. The article is signed "from a Correspondent", but Prof. D. E. Knuth has been able to identify the author, since Babbage included this item on his list of publications in later years. [R shelfmark RSL:1816-19 Per.1996 e.276] also [C].

Edmé Gilles Guyot (1817). Giuoci Fisici e Matematici i Piu Belli Finora Immaginati collagiunta della loro spiegazione e construzione, e del modo di eseguirli, Opera dilettevolissima ed istrutiva, l'Erede Pazzoni, Mantova, 2 vols, 1817–18, vol.2, pp. 98–115, Giuoco 31. Italian edition of Guyot (1769), edited by Teofrasto Cerchi (?). [C]

« 1820s

— Bingham (1820). Editor of an English edition of Ozanam's Recreations 1820. See 1725.

Pierce Egan (1820). Sporting Anecdotes ... delineation of the sporting world, London, p.156. [C]

— Grandin (Prof. de Collège de Navarre) (1820). Editor of editions of Ozanam's Recreations 1820, 1825, 1835. See 1725.

Robert Willis (Professor of Applied Mechanics, Cambridge 1837) (1821). An Attempt to Analyse the Automaton Chess Player of Mr. De Kempelen. To which is added a copious collection of The Knight's Moves over the chessboard. London 1821. The diagrams are numbered 1–18 (open tours of all boards smaller than 8×8, and closed tours 6×8 and 7×8) and 20–39 (tours 8×8, mainly from Euler and Ozanam) on plates 6–10 which are misbound amongst unrelated text. No author's name is stated, but Tomlinson (1845) identifies Willis as the author of the section on the automaton. [British Library shelfmark 1040.d.26(1); catalogued under Kempelen]. The automaton was built by Farkas Kempelen (also known as Wolfgang von K- or de K-) (b. 1734 – d. 1804) and was later exhibited by Johann Nepomuk Maelzel.

(1821). Edinburgh Philosophical Journal, vol.4, no.8, art.30, pp. 393–398, and plate 10, facing p.393. Review of Willis (1821); 3×8 and 7×8 tours. [R shelfmark RSL:1819–26 Per.1996 e.273.]

(1823). "On the knight's moves over the chessboard" Edinburgh Philosophical Journal, vol.9, no.18, art.5, pp.236–237, and plate 6, facing p.237. Further review of Willis (1821), reproducing the 20 tours 8×8. [R shelfmark RSL: 1819–26 Per.1996 e.273.]

Karl Brandan Mollweide (1823). "Uber das Problem des Rosselsprungs" in Klugel's Mathemat. Wörterbuch Leipzig 1823; part 4, pp.458-467, art: Springer. See also 1816.

(1823). The Youth's Miscellany of Knowledge and Entertainment, Sherwood, Jones & Co (publishers), London, vol.1, pp.261–283. [C]

(1823). Nouvelle Notation, 1823, pp. 439–449, 9 tours, one by Monneron. [L]

H. C. von Warnsdorf (1823). Des Rösselsprunges einfachste und allgeneinste Lösung Schmalkalden, 68pp + 16 figures, 32 tours. Warnsdorf's Rule: 'Play the knight to a square where it commands the fewest cells not yet used'. Strictly applied this falls far short of producing a completely determined tour. The best it gives, before it reaches a position where it is undecided on the next move, is 18 moves, starting b3–c1. Nevertheless, much has been written about this rule, and recently it has been the subject of computer studies. Warnsdorf claimed that when there is a choice of moves under the rule any of them can be taken, but examples can be given where this fails. [K 4069] also [C]

— von Müllner (1824). Literatur-Blatt no.39, 14 May 1824 and no.40, 18 May 1824; Allgemeiner Lit. Zeitung 1825, nos. 183 and 184, 32 tours. Reviews of Warnsdorf (1823). [C]

(1824). Easy Introduction ... (to chess), Philadelphia. The de Moivre tour.

F— P— H— (1825). Appendix to Studies of Chess (editor P. Pratt, based on Philidor), London, 6th edition 1825, pp.533-536 (not in 5th edition 1817). Quotes an Euler tour and gives three originals, with diametrally opposite numbers differing by 32, 16 and 8 respectively. The last two are the earliest examples of tours by the method of 'squares and diamonds' and one of them sums to 260 in every file, but not in the ranks. [Potocopy kindly supplied by M. Sheehan]

Clemens Rudolph Ritter von Schinnern (1826). Ein Dutzend mathematischer Betrachtungen {A dozen mathematical contemplations}, Geistinger, Vienna, 36 pages. Contains the first full account of the squares and diamonds method. 'Die Formeln für den geometrisch-aritmetischen Rösselsprung', pp.16–29. There are 8 tour diagrams but only 5 distinct tours, the others being reflections or reversals. All are semimagic, i.e. their files all add to 260 but not the ranks. The final tour has all 8 files and 6 ranks magic — the nearest approach to a fully magic tour before Beverley (1847). [Phoocopy kindly supplied by D. Singmaster]

(1827). The Franklin Journal and American Mechanics' Magazine, Franklin Institute, Philadelphia, vol.3, pp.132–134. [C]

— Mauvillon (1827), table 10, Figs 1–5. [L]

Giuseppe Ciccolini (1827). Il Nuovo Tesoro degli Scacchi, Rome, 4 tours (?) [L].

Domenico Lorenzo Ponziani (1829). Il Giuoco Incomparabile degli Scacchi (1st edition Modena 1769) Rome edition 1829, pp.29-30, quotes Ozanam's Recreations.

H. Silberschmidt (1829). Das Gambit: Gives a compartmental tour [M].

« 1830s

Adrien Marie Legendre (1830); Théorie des Nombres Paris (2nd edition), vol.2, pp.151 and 165. Concerning the number of knight's tours. [C]

Eduard Billig (1831). Der Rösselsprung mit Variationen Mittweyda. One tour 64 times according to Murray. [K 4070]

Alexandre Magne de Castilho (1831), Récueil de Souvenirs du Cours de Mnémotechnie, Saint-Malo. [C]

George Walker (1832). New Treatise on Chess. Warnsdorf's method. [Hoffmann 1893] See also 1840, 1841, 1844.

Ignaz Plödtiels (1834). Versuch einer allgemeinen Lösung des Rösserlsprunges ... (ms, was in collection of R. Franz, Berlin) [C].

(1834). "Das Schachspiel" Der Pfennig-magazin der Gessellschaft, F.A.Brockhaus (publisher). Leipzig, vol.2, no.70, pp.551–2, 559–60. [C]

Teodoro Ciccolini (Marchese di Guardiagrele) (1836). Del Cavallo degli Scacchi, Paris, Bachelier, 120 pages. Text followed by 25 'Tabula' (lists of tour formulas and some diagrams). One closed tour formed by the squares and diamonds system is given, followed by a large number derived from it by Euler's method (which destroys the squares and diamonds structure), I think with the aim of providing examples of tours from any cell to any other of opposite colour. Tabula 21 shows the original 8×8 tour presented on a circular (cylindrical) board formed by identifying the a and h sides of the board. Tabula 22 sketches a 10×10 tour formed by joining an 8×8 tour to a border braid along two edges. Tabulae 24–25 show a 10×10 symmetric tour formed by joining four 5×5 tours (like Euler's but not quatersymmetric). [Bodleian Library, Oxford:], [K 4071] and [C].

—Gasbarri (1836). Raccolta di Venticinque Nuovi Problemi di Scacchi ..., Firenze, pp.17—20. The tour from this source, quoted to me by F. Pratesi, is very similar to Euler's first closed tour and to one by Dollinger (1806), but incorporates an internal 3×4 tour. [C].

C. F. de Jaenisch (1837). Découvertes sur le Cavalier aux Echecs, Petersbourg. Not sure if this includes tours - may only deal with chessplay. [K 4072].

George Augustus Addison (1837). "General Solution to the Knight's Trick at Chess", Indian Reminiscences (or The Bengal Moefussal Miscellany), Edward Bull, London, pp.19-27. [British Library] and [C]

Christoph Wilhelm Zuckermandel (1838). Regeln nach denen alle Zauberquadrate ... , Nürnberg (1836/8?) pp.62–66. [C]

J. E. Thomas de Lavernède (1840). Mémoires (1838-39) de l'Academie Royale du Gard de Nimes published 1840; pp.151-179. Squares and diamonds method. [C]

« 1840s

(1840). The Knight Crusader's Games ... Houlston and Hughes (publishers). [C]

Edward Riddle (1840). Editor of English editions of Ozanam's Recreations 1840, 1844.

George Walker (1840). "Chess without a Chessboard by a Chess Player" Frazer's Magazine March 1840 p.316. Syllabic coordinates method of memorising a tour as a nonsense verse. Authorship revealed in G. Walker Phil. Mag. 1840. See also 1832, 1840, 1841, 1844.

Peter Mark Roget (Secretary of the Royal Society for 20 years) (1840). "Description of a Method of moving the Knight over every square of the Chessboard without going twice over any one; commencing at any given square and ending at any other given square of a different colour." London and Edinburgh Philosophical Magazine and Journal of Science 1840, series 3, vol.16 (January-June 1840), April, pp.305-309, plate facing p.305. Contrary to the impression given by many writers, Roget's method is distinct from the method of squares and diamonds. He divides the board into "four separate systems of 16 squares each" which he letters L, e, a, P. The moves L-L, e-e, a-a, P-P are those later classified by Murray (1949) as 'straights'. He advocates traversing each system separately, as far as possible, and gives three actual tours, none of which are of squares and diamonds type. [R Gen.Per.27], [K 4073] and [C].

George Walker (1840). "On Moving the Knight over every Square of the Chessboard alternately" London and Edinburgh Philosophical Magazine and Journal June 1840, series 3, vol.16 (January–June 1840) pp.498-501.See also 1832, 1840, 1841, 1844. (a) Reveals his authorship of the article in Fraser's Magazine (March 1840). (b) "Those who have not gone deeply into chess are hardly aware that a whole library has been written upon the knight's move, and ten thousand modes are printed in which the feat may be performed." (c) Warnsdorf's method. (d) Bibliography of 11 titles. [R Gen.Per.27], [K 4074] and [C].

George Walker (1841). Polytechnical Journal May 1841 p.243; September 1841 p.141. See also 1832, 1840 (two), 1844.

Prof. Christian Flamin Heinrich August Glaszer (1841). Abhandlung über dem Gang eines Springers auf dem Schachbrette, Jahresbericht von der königl. Studienanstalt zu Erlangen in Mittelfranken, 28 August 1841. Gives one tour, effectively by Euler's method, attributed to his late colleague Prof. Rothe. [M]

Charles Tomlinson (1841). Articles in Saturday Magazine 1841-44. See also 1845.

Jules de Poilly (?) (1842). Le Palamède 15 June 1842, series 2, vol.1, p.322. Knight's tour presented as verse cryptotour. Seems to be the earliest modern example of this type of composition. See also 1844, 1846. [Lucas 1882]

E. Troupenas (1842). "Problème du Cavalier. Parcourant toutes les cases de l'echiquier" Le Palamède 1842; vol.2, 15 October, pp.166-171, 15 November pp.221-226, 15 December pp.268-277. Six line diagrams. Contents: (a) pp.166-7; Tour of squares and diamonds type, and method of giving a tour by lettering the rows A-H, I-P, Q-Z, A-H, I-P, Q-Z, A-H, I-P (omitting W, Y), but method not subsequently used. (b) pp.167-272: Euler's methods: (c) pp.272-3: Vandermonde's method. (d) pp.273-6: Method of squares and 'lozenges'. Begins with a tour sent from England by a certain Mr — Anderson. A complex figure is given, presumably showing ways in which the squares and lozenges can be connected. (e) pp.276-7: Euler's 10×10 tour and another, with Euler's 12-cell cross tour as a centrepiece. [Photocopy kindly supplied by K. Whyld]

Victor Käfer (1842). Vollständige Anweisung zum Schachspiele, Graz, vol.4, pp.191–193. "24 symmetric designs of great merit" [Murray 1955]. Some of these tours appear to anticipate Bergholt's methods of construction by 'terminal loops'. [C]

— Perenyi (1842). pp.30-34. [Haldemann 1864]

Rev. R. Moon (1843). "On the Knight's Move at Chess" Cambridge Mathematical Journal; series 1, vol.3, no.17, February, pp.233–236. Method of annuli applied to square boards of any size. Index diagram for 8×8 board with outer borders lettered a, b, c, d and central 4×4 lettered A, B, C, D (as in Collini 1773). Suggests the sequences aDbCdAcB or aDcBdAbC. No actual tours. "If the number of squares in the board be odd, the same principle of division obtains. We shall still have a central square, which will have either five or seven places to a side, as the case may be, and a series of annuli divisible into circuits. It will be found, however, that in this case each annulus will consist of only two circuits, by reason of which the process is much simplified". [R shelfmark: RSL:1838-45 Per.1875 e.62] and [C].

Paul Heinrich von Fuss (1843). Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siecle, Académie Impériale des Sciences de Saint Pétersbourg (Johnson reprint, New York, 1968, vol.1, pp.654-5). Includes Euler-Goldbach letter (1757). [C]

Lieut. Paul Rudolph von Bilguer (1843); Handbuch des Schachspiels. 1st edition, pp.4–5, completed after Bilguer's death by von der Lasa. Three tours in numerical form by the methods of Euler, Collini and Warnsdorf. [Photocopy kindly supplied by P. Wood]

George Walker (1844). Chess Player's Chronicle London, 1844, p.180 and 1845, p.240. Describes a modern Persian ms, at that time in possession of Mr Rimington Wilson, translated by George Swinton, consisting of 29 chess problems, and a knight's tour; identical to the Rudrata tour reflected left to right and joined to a copy of itself on the lower half of the board. See also 1832, 1840, 1841.

Ferdinand Julius Brede (1844). Almanach für Freunde vom Schachspiel Altona. 24 tours. Some of these tours anticipate Bergholt's methods of construction. [M]

Edward Riddle (1844). Recreations in Science and Natural Philosophy. See also 1840. Version of Ozanam's Recreations. [C]

Jean Baptiste Alexandre Charpentier (1844). Du Jeu des Echecs a la Portee des Jeunes Gens ... Paris, pp.106-7. See also 1849. [L]

Jules de Poilly (1844). "Une des Marches du Cavalier aux Echecs" Le Palamède, vol.4, October p.466, November p.517. Knight's tour (de Moivre's) presented as 64-word verse cryptotour. See also 1842, 1846. [Photocopy kindly supplied by K. Whyld]

Charles Tomlinson (1845). Amusements in Chess John W. Parker, London. (a) Preface pp.v-ix; notes that the book is based on articles that appeared in the Saturday Magazine 1841-44. (b) Chapter 7, "The Knight's Move" pp.114-128: Discusses tours of small boards, giving square examples of sizes 5, 6 and 7, then three of size 8. Then expounds Roget's method (confusing it with the squares and diamonds). Then gives a tour constructed by the rule: 'keep as far from the centre as possible'. Then de Moivre's, said to follow the rule: 'Play the knight to that square where he has the least power'. Then a symmetric tour. Then one with diametrally opposite numbers differing by 16. Then a semi-magic tour. (c) Lesson 2, "The Moves": Has another open tour on p.155. [My own copy, now in BCPS Library.]

Jules de Poilly (1846). "Une des Marches du Cavalier aux Echecs" Le Palamède 1846, vol.6, September p.418, October p.476. Knight's tour presented as 64-word verse cryptotour. See also 1842, 1844. [Photocopy kindly supplied by K. Whyld]

Aaron Alexandre (1846). Collection des plus beaux Problèmes d'Echecs, Paris. Published simultaneously in English and German translations: Beauties of Chess, London. Praktische Sammlung bester Schachspiel-Probleme, Leipzig. Olms reprint, Zurich 1979, Problems p.341, Solutions p.63. Two tours of squares and diamonds type taken from Troupenas (1842). [Photocopes kindly supplied by J.D.Beasley and S.Pantazis]

Baron Tassilo von Heydebrand und von der Lasa (1846). "Losung des Rosselsprunges" Schachzeitung Berlin, vol. 1, 1846, pp. 188–191, vol. 2, 1847, pp. 79–86, 97–103. Collini, method of quartes etc. See also 1862. [Lucas 1895] and [A].

R. Franz (1847). "Rosselsprung" Schachzeitung 1847, vol.2, pp. 341–343. Closed, semi-magic, knight's tours showing differences of 16 or 8 in diametrally opposite squares. [A], [M]

Ferdinand Minding (of Dorpat) (1847). Bulletin de l'Academie des Sciences de St Petersburg, vol.6, number 14, 22 January. See also 1852.

William Beverley (1848). "On the Magic Square of the Knight's March" London and Edinburgh Philosophical Magazine and Journal of Science, Taylor & Francis, London, series 3, no. 220, August 1848, pp.101-105. Preface by Henry Perigal introduces letter by Beverley dated 5 June 1847. Gives magic tour and its reverse in numerical form. Notes that the four quarters are themselves magic (adding to 130), and that the numbers in the 16 blocks 2x2 also add up to 130. Notes other related numerical properties. Footnote p.247 by S. M. Drach, notes that the 4x4 squaes with diagonals 31-1 and 25-23 are also magic. [R] [C]

Dr. Carl Rädell (1848). "Uber die mathematische Behandlung des Schachspiels" Schachzeitung vol.3, pp.101-120. See also 1863-64. [L], [A].

Carle Adam le jeune (1848). Ms, Des Mouvements du Cavalier. [C]

(1848). "De oplossing van den paardensprung" Sissa, 1848, pp.150, 190, 213 and 1849, p.330. See also 1850, 53. [L]

Paul Heinrich von Fuss and N. von Fuss (1849). Leonhardi Euleri Commentationes Arithmeticae Collectae, Academiai Imperialis Scientarium Petropolitanae, (St Petersburg), vol.1, pp.337–355. Includes Euler's paper of 1759. [C]

W. Hanstein (?) (1849). William Beverley's Rösselsprung, Schachzeitung, 1849, vol.4, pp.21–24 [A]

Carl Wenzelides (1849). "Bemerkungen uber den Rosselsprung" (dated 1 October 1848) Schachzeitung 1849, vol. 4, pp. 41–93, 72 figures. [Photocopy kindly supplied by M. Stere]. See also 1850, 1851, 1858.

Carl Wenzelides and W. Hanstein (1849). "Der Rosselsprung in hochster Kunstvollendung" {The Knight's Tour in its Utmost Perfection} Schachzeitung 1849, vol.4, pp.94-97. Wenzelides' first symmetric magic knight's tour, with verse encomium, signed 'Hn' (= Hanstein) [Photocopy, M. Stere]

Q. Poirson-Prugneaux (1849). Introduction Pratique du Jeu des Echecs; apparently the source of the error that ascribes the squares and diamonds method to Ciccolini (1836). [Lucas 1895]

Jean Baptiste Alexandre Charpentier (1849). Anciens et Nouveaux Jeux Géometriques ... combinaisons curieuse sur la marche simple et double du cavalier aux echecs, Paris. See also 1844. [A], [C], [K 4076].

A. Herma (1849). Rösselsprunge aus deutschen Dichtern, Frankfurt, 1849. Poems in German presented in the form of knight's tours. [C]

« 1850s

T. von der Lasa (~1850?). Forschungen, p.164. Considered a Florentine tour 4×8 (end of 15th century), combined with a copy of itself on an 8×8 board, to be the oldest instance of a reentrant tour in existence. He was not aware of the earlier Arabic instances. [Murray 1902]

Turkish (~1850). A modern Turkish manuscript of chess problems 'which may be as old as the middle of the 19th century' in the J. G. White Collection, Cleveland Library. Four tours: Knight-Alfil tour, plus 3×8, 4×8 and 8×8 knight's tours. [Murray 1930]

W. Manning, (~1850?). Ms, The Knight's Tour, in the von der Lasa library, no date. [L]

Elias Stein (~1850?). Manuel del'Amateur du Jeu des Echecs, pp. 132–138, Delarme, Paris. [C]

(1850). De oplossing van den paardensprung, Sissa, 1850, p.144. See also 1848, 49, 53. [L]

T. Scheidius (1850). Het problema van den paardensprung, Sissa, 1850, p. 209. [L]

Carl Wenzelides (1850). Bemerkungen uber den Rösselsprung, Schachzeitung 1850, vol. 5, pp.212–221, 230–248. See also 1848, 1849, 1851, 1858 and von Oppen 1858. [A], [M]

Paolo Volpicelli (1850). Solution d'une Problème de Situation Relatif au Cavalier des Echecs, (séance du 2 September 1850) Comptes Rendus de l'Acad. Sci., Paris, 1850; vol.31 (10), pp.314–322. [A], [C], [K 4077]. See also 1872, 1874, 1875.

Carl Wenzelides (1851). Uber symmetrische Rösselsprünge, Schachzeitung 1851, vol. 6, pp. 286–297. [A] also pp.45, 129, 380. See also 1848, 1850, 1858.

Ferdinand Minding (1852). "On the Knight's Move at Chess" Cambridge and Dublin Mathematical Journal, vol.7, pp.147-156. Cites Euler, Legendre, Vandermonde, Collini, Mollweide. Gives some rather elaborate algebraic theory directed at the number of tours, but producing no definite results. Similar: "Uber den Umlauf des Springers auf dem Schachbrette (den sogenannten Rösselsprung)" Crelle's Journal (fur Reine und Angewandte Mathematik), 1852, vol. 44, pp. 73–83, Berlin. See also 1847. [C]

— Lamarle (1852). Solution d'un coup singulier, Mémoires de l'Academie Royale Belgique, vol.27. [M]

(1852). Sur le Problème du Cavalier au Jeu des Echecs, Nouvelles Annales de Mathématiques, Journal Redige par Terquem et Gerono, vol.11, Paris. [M] See also 1854.

Maharajah Krishnaraj Wodayer (1852). A square of silk, dated 31 July 1852, bearing a magic 8×8 knight's tour. Present whereabouts unknown (there may be more than one). The tour is also in Harikrishna (1871), Naidu (1922) and Iyer 1982. Sightings of the silk were reported at Margate Easter Chess Congress 1938 by Murray (1951) and in a London shop, A. Hammond, Emil, Burlington Gardens, 1947 by Akenhead (1947). See also Harikrishna 1871.

— Basterot, (1853). Source? Tour with fixed difference of 16. [L]

Dr T. Clausen (1853). Direkte Auflösung des Rösselsprung, Archiv der Mathematik und Physik ..., vol.21, pp.91–92. [A] also (same author and date?) series 3, vol. 3, pp. 136–151. [C]

Anselmo T. Quignano (1853). Saltes del caballo; Problema de Ajedres. [C]

(1853). De oplossing van den paardensprung, Sissa, 1853, pp.39, 69, 89, 102, 138. See also 1848, 49, 50. [L]

(1854). Sur le Problème du Cavalier au Jeu des Echecs, Nouvelles Annales de Mathématiques, Journal Redige par Terquem et Gerono, vol.13, May, pp.181–6, Paris. See also 1852. [C]

Arnold Pongracz (Graf zu Balassa Gyarmath) (1855). Rosselsprunge, Schachzeitung, p.238. [M]

Max Lange, (1856), Die Lehre vom Rosselsprung; Systematische Theorie des Rosselsprungs, Lehrbuch,, pp.1930–199; 199–202. [L]

l'Abbé Durand (1856). Géneralisation Complète du Problème d'Euler, La Régence, p.366. [M] See also 1862.

'Slyvons, Edmond' (E. S. A. L. Solvyns) (1856). Application de l`Analyse aux Sauts du Cavalier du Jeu des Echecs, Brussels. Includes a tour with a star in each corner and monogram tours. Proposed formulae for number of tours on n×n board in the form: S(closed) = n(n–1)(n–2)(n–3)(n–4)(n–5) (A+Bn+Cn²+...) and S(open) = (n–1)S(closed). [C], [K 4078/9]

Sir William Rowan Hamilton (1856). Letter to J. T. Graves, 17 October 1856, in The Mathematical Papers of Sir William Rowan Hamilton (editors H. Halberstam and R. E. Ingram), Cambridge, 1931, vol.3, pp.612–625. Concerns a game on the edges of a dodecahedron which gave rise to the term 'Hamiltonian tour'. [Bondy and Murty 1976]

Sir William Rowan Hamilton (1857). On the Icosian Calculus, British Association Report, Notices and Abstracts, 1857, p.3. Formal presentation of the ideas in the above letter. [A]

N. F. Leonhart (1857). Handbuch für angehende Schachspieler ..., pp.149–155, Altona. [C]

— von Oppen (1858). Rösselsprung, Schachzeitung, May 1858, vol.13, pp.174–175. Prompted by Jaenisch (see 1859) the editor, von Oppen, gives three magic tours (00m, 12e, 12m) that Carl Wenzelides had sent to Schachzeitung in 1849 but which were not published at the time owing to the death of the previous editor, Hanstein. [Murray 1930], [A] Also: Zur Theorie des Rösselsprungs, Schachzeitung, December, vol.13, pp.489–492. [L], [A] There is mention in one of these articles of work on magic knight's tours by Prof. A.F.Svanberg, with similar results to Wenzelides, but not published.

H. C. von Warnsdorf and von Mullner? (1858). In Schachzeitung 1858. See 1823.

Carl Friedrich Andreyevich Jaenisch, (also von J or de J or Yanich) (1859). Chess Monthly, pp.110–115, 146–151, 176–179. De la solution la plus parfaite du problème du cavalier; and; Solution of the problem of the knight's tour. The same article in French and English on facing pages, dated November 1858. Contains three new magic knight tours. A preview of his forthcoming treatise. See 1862. [Photocopy kindly supplied by K. Whyld]

G. Mann Jr (1859). 120 neue Rösselsprunge ohne nebst einer kurzen Einleitung, Nürnberg. [C]

(1859) English Cyclopaedia, Arts and Sciences, vol.2, London. Tour with difference of 16. [L]

« 1860s

(1860). Moving the Knight over all the Squares Alternately, Boy's Own Conjuring Book, Art.45, pp.246–251. After G. Walker (1832). [Singmaster, 1991]

Duncan Forbes (1860). History of Chess. Contains a translation of the Ala'addin (~1400) quarterboard tour problem. [quoted in Falkener 1892]

Eugéne de Chambure (1861). Du Problème Relatif a la Marche du Cavalier au Jeu des Echecs, Mémoires présenté a l'Institut Egyptien, Cairo, 1861, vol.1, pp.67–86 (published in Paris 1862). Lucas 1895 wrote: "Cet auteur se sert de quatre coulours; les développements de la méthode sont agréables à l'oeil, mais peu aisés placer dans la mémoire; les circuits qu'il donne sont des chefs-d'oeuvre de difficulté vaincue." [C; has three differing copies (manuscript, journal and offprint?)], [K 4080]

C. de Polignac (1861). Sur la Course du Cavalier au Jeu des Echecs, 19 April 1861, Comptes Rendus de l'Académie des Sciences, 1861, vol.52 (17), pp.840–842, Paris. [A], [C] See also 1881.

(1861). Cosmos: Revue Encyclopédique Hebdomadaire des Progrès des Sciences et leurs Applications aux Arts et l`Industrie, A.Tramblay, Paris 1861, vol.18, p.489; and 1862, vol.20, pp.282, 358-360. [C]

l'Abbé Durand (1862). Géneralisation Complète du Problème d'Euler, La Nouvelle Régence, 1862, pp.83, 117, 152. Continuation of his 1856 article. [L], [M]

C. F. de Jaenisch (1862). Traité des Applications de l'Analyse Mathématique au Jeu des Echecs, Saint-Pétersbourg., 3 vols (1862, 1862, 1863) bound as one. Vol.1 (items 1-79) deals with moves and powers of various pieces; p.229 gives charts of the least number of moves by knight from a1 to each other square, and of the number of shortest routes; Fig.6 is a tour with constant diametral diference 16; Vol.2 (items 80-176) Problème du Cavalier; p.46 the theorem about tour on 4×8 board being in two parts; pp.56-61 and 273-289 Warnsdorf's method; p.62 Collini's method; p.268 Number of tours, upper limit 168C63. Vol.3 (items 177-284) De la Réaction des pièces de l'echiquier. [Cambridge University Library], [British Library copy destroyed in war D-7915.e4], [K 4081]

(1862). Comptes rendus de l'Academie des Sciences, Paris 1862, vol.54, p.464. [L]

(1862). LaRégence, 1862, vol.3, pp.193-195. [L]

(1862). Bulletin de l`Académie Imperiale des Sciences de Saint-Pétersbourg, 1862, vol.6, pp.473-477. [L]

(1862). Mélanges Mathématiques ... 1862, (extracts from the previous reference) vol.3, pp.488-492. [L]

Baron Tassilo von Heydebrand und von der Lasa (1862). Schachzeitung 1862, vol.17, pp.65-69. Review of Jaenisch 1862. [A] See also 1846, 1847. Also: Bull. de la classe physico-mathém. de l'Acad. Impér. des sc. de St-Petersbourg, 1863, vol.6, pp.473-7. [A]

Christian Friedrich Gottfried Thon (1863); C. F. G. Thon's Meister im Schachspiel ..., Weimar, 1863, pp.194-195 (edited by Max Lange). [C] See also 1856, 1871.

Dr. Carl Rädell (1863). Mathematische Schachfragen 1863, pp.97, 172, 203, 363, and 1864 pp.48, 78. See also 1848. [L]

Prof. — Mercklein, (1863). Marche du Cavalier du jeux des echecs, Mémoires de la Société Impériale d'Agriculture, de Science et d'Arts séant a Douai, 1863, series 2, vol.7, pp.151-155. Divides the board into 16 blocks 2×2. [C] See also Le Sphinx 1866.

Samuel Stehman Haldeman, (1864). Prodromus: Bibliography of the Chess Knight's Tour, E. H. Butler & Co, Philadelphia, 1864. [C] And: Tours of a Chess Knight, E. H. Butler & Co, Phladelphia, 1864. "Describes methods of Warnsdorf, Demoivre, Bertrand (= Euler), Collini, Collini–Kempelen (= Roget), Ciccolini, magic square, 108 tours on 8×8." [Murray 1955], [C], [K 4082]

(1864). Nuova Enciclopedia Populare Italiana, 5th edn, Torino, vol.20, p.629. (= Euler). [L]

— Geynet (1865). Mémoire Relatif au Problème du Cavalier, Comptes Rendus de l'Académie des Sciences, Paris, vol.60. [M]

C. Tychsen (1865). Tidskrift för Mathematik, series 2, vol.1, pp.17-20. [C]

Rev. Serge de Stchoulepnikoff (1865). Ms, Twenty Solutions of the Problem of the Knight's Tour, Buffalo, 1865. [C] See also 1885.

A. C. Crétaine (1865). Etudes sur le Problème de la Marche du Cavalier au Jeu des Echecs ..., Paris [British Library, copy destroyed in war D-7915cc24], [C], [K 4085]

B. H. Braadbaart (1865). Two ms De Paardensprong in het Schaakspel ..., Zaandam, 1865, 54 pages, and 1866, 70 pages. [K 4083 and 4084], [C, copies]

Prof. — Mercklein (1866). Marche du Cavalier du jeux des echecs, Le Sphinx 1866, pp.67-70. A reprint of Mercklein (1863) but with the matter on the plate described in words instead. [K. Whyld, e-mail]

(1866). Merkwaardigheden betreffende het schaakspel, De Globe; Album van Buitenlandsche Lettervruchten, 1866, De gebroeders Abrahams, Middelburg. [C]

(1867). Il Passatempo, 1867, 2 vols, Bologna, 1867-69. [C]

Carle Adam, le jeune (1867). Ms, Le Kaleidoscope Echiquiéen. Traité Complet de la Marche du Cavalier sur les Echiquiers de 25, 26, 64, 100, 144 et 256 Cases, Rouen, 1867. With 2492 tours in 18 categories. [C] See also 1848.

A. Canel (1867). Recherches sur les Jeux d'Esprit, les Singularités et les Bizarries Littéraires, Principalement en France, 2 vols, Evreux. Contains a cryptotour. [Quoted in H. Haughton, The Chatto Book of Nonsense Poetry, Chatto & Windus, London, 1988, pp.274-5 and 511.]

Sam(uel) Loyd, (1867); The Queen's Tour, Le Sphinx, March 1867. [A. C. White 1913]

E. B. Cook et al (1868). American Chess Nuts, New York, p.396, gives the Loyd (1867) queen tour problem. [British Chess Problem Society Library]

Ernest Nivernais (1868). La Bibliothèque de M. Adam, Strategie, pp.246-251. Review of Adam (1867). [L]

(1869). The Tour of the Knight, Westminster Papers, 1869, p.18. [L]

« 1870s

(1870). An Arm-Chair in the Smoking Room ... from various pens, S. Rivers & Co, London, title page and pp.78-81. [C]

William Hand Browne (1870). Dictation Tours of the Chess Knight, New Eclectic Magazine, Baltimore, April, vol.6, pp.481-7. [C]

Charles van Tenac (or Tomas?) (1870). Traites Illustres des Jeux des Echecs ..., Passard, Paris, vol.1, pp.34-39. [C]

Howard Staunton (1870). Chess column in Illustrated London News. The column began 25 June 1842 and was edited by Staunton from 1845. In the 31 December 1870 issue he began a series of 16 cryptotours (numbered I–XVII, omitting XII) which attracted much attention, judging by the long lists of solvers. I vol.57 (1870) p.691 (problem), vol.58 (1871) p.67 (solution), II pp.139, 187, III pp.267, 322, IV vol.59 (1871) pp.139, 235, V pp.259, 339, VI pp.363, 435, VII p.575, vol.60 (1872) p.187 VIII pp.251, 315, IX pp.411, 555, X p.587, vol.61 (1872) p.22, XI pp.71, 143, XIII pp.239, 335, XIV p.575 vol.62 (1873) p.67 XV pp.187, 259, XVI p.92, vol.63 (1873) p.183, XVII vol.64 (1874) pp.115, 163 (verbal solution), 307 (geometrical solution). This last diagram is very attractively drawn, the tour being shown by white lines, with white dotted ends, on a board of black and grey squares. The 27 June 1874 issue, p.619, announces the death of Mr Staunton "who for many years had charge of this column". [Hastings Public Library], [British Library]

Pandit Harikrishna Sharma Jyotishacharya (1871). Kridakausalyam. The following notes appear in S.R.Iyer (1982) which is an edited version of this Sanskrit work, with notes in English. "The author of the present work is one Harikrishna, son of Venkataraman, who lived in Aurangabad in the Maharastra State. He wrote it in Saka 1793 (1871AD). These particulars are mentioned by the author himself in the last 5 verses of Kridakausalyam, which contains 700 verses in all. Harikrishna compiled a voluminous (encyclopedic) work [title in Indian script] which consists of six parts called Skandhas. Each part consists of several chapters, Kridakausalyam being the 20th chapter in the 6th part of that work. This chapter treats in 12 main sections of many indoor and outdoor games prevalent in his part of the country. The extract printed here [on chess and knight's tours] forms the 8th section. This book was printed in Devanagari script by Venkateshwar Steam Press, Bombay, in Saka 1822 (1900AD). Being long out of print, it has been reprinted with the financial assistance from the Sanskrit Division, Ministry of Education and Culture, Govt. of India, by Nag Publishers, Delhi, 1982, on the occasion of the 9th Asian games, Delhi-1982." The text relating to knight's tours is section 9 (pp.29-47) with 82 diagrams in an appendix of 61 pages. Tours 1-76 are said to have been "mentioned by the King of Karnataka, H. H. Shri Krishna Udayar" i.e. the Rajah of Mysore. The Rajah's 8×8 magic tour (see 1852) appears in this collection as number 37. The collection also contains, as tour 3, the earliest example of a magic tour on the 12×12 board. Tours 77-79 are the three versions of the one tour from Nilakant-ha's work (see 1767). Tours 80-82 are Harikrishna's own. [Iyer]

Victor Gorgias (1871). The Knight's Tour, Gentleman's Journal (An Illustrated Magazine of Literature, Information and Amusement), London, August. Open tours of 8×12 board in near-biaxial symmetry, also king's tour, pp.124-5, 196. [M] Also in Dubuque Chess Journal, several issues. [M]

(1871). De Leer van den Paardensprung, Sissa, pp.7-13, 35-37. After Lange (1856). [L]

Edmé Simonot (1872). Le Polygraphile décrit par son inventeur; la Polygraphile du Cavalier des Echecs, Paris. Contains a bibliography of over 50 authors, according to de Hijo (1882). [A], [C, also another edn 1882?]. Also: La Strategie, p.113. Summary of the above. [L]. Also: Un Merveille de Patience et d'Habilité, La Strategie, vol.5, pp.147-154. Review of Adam (1867) [Information from D.Pritchard. BL shelfmark PP(Paris) 1831g.]

Paolo Volpicelli (1872). Soluzione Completa e Generale, mediante la Geometria di Situazione, del Problema Relativo alle Corse del Cavallo sopra qualunque Scacchiere (session 4 Feb 1872) Atti della Reale Academia de Lincei, Rome, vol.25, pp.87-160, 364-454; vol.26 (1873) pp.49-187, 241-325. Also in French: Solution Complète du Problème Relatif au Cavalier des Echecs. Seconde note. Comptes Rendus de l'Académie des Sciences, Paris, vol.74, no.17, pp.1099-1102. A version of this work was also issued in book form (389 pages). This is scathingly reviewed by Bergholt (1916) saying "There is not a scintilla of merit or interest (as regards the professedly original portion of this work:– Part II, pp.70-389) in the book from beginning to end. ... Starting from any given cell, he proposes to tabulate every possible sequence of knight's moves, following out every path until it comes to an obligatory stop. This, he says, when completely carried out, will be a complete and general solution of the problem, free from all trial. ... The alleged 'solution', as will be seen at once, far from being free from all trial, is trial pure and simple. ... instead of seeking to abridge his trials (the true aim of all methods of solution of this and kindred problems), he takes a foolish pride in setting down every possible false step that can be taken." Pages 1-69 form a historical introduction. It includes 48 tours on the 8×8 "all of a very commonplace and uninteresting type". [A], [C: Italian and French versions], [K 4087 (Italian pp.87-160 only); 4088 (Italian complete)] See also 1850, 1874, 1875.

Monsieur H. Tarry (1872). Solution du Problème du Cavalier ..., Les Mondes, vol.28, no.2, pp.60-64. [L], [A]

J—. B—. D—. (1873). The Knight's Tour, The Leisure Hour (A Family Journal of Instruction and Recreation), vol.22, no.1133, pp.587-590, 752. (a) A cryptotour encoding "O County Guy" by Walter Scott, using three letters to a cell. (b) "Many a leisure hour, no doubt, has been spent, especially of late, in constructing and unravelling those tangles of letters, which, inscribed in the squares of an imaginary chessboard at intervals of a knight's move from each other, make up the elaborate puzzles which go by the name of Knight's Tours ... exercises of patience." (c) An account of Roget's method with seven examples. "The fullest statement on the subject in any modern book is probably to be found in Dr Roget's article ... but the solutions here suggested, though necessarily founded on common principles, are not given by the learned doctor, and have not, to the writer's knowledge, appeared before." (d) Erroneous counts of numbers of types of tour classified by end cells. p.752 tries to correct the errors but is still wrong. See next reference. [Leicester University Library: PER 05C.L4244.]

Heinrich Meyer (1873). The Knight's Tour, The Leisure Hour, vol. 22, pp. 813– 815. (a) Five tours showing a high degree of axial symmetry; the first is Vandermonde (1771) three are by V. Gorgias, the fifth is a symmetric magic tour. (b) Correction of J.B.D.'s account of the number of types classified by end cells: 115 non-reentrant + 21 reentrant. (c) References back to Euler. [Leicester University Library: PER 05C.L424], [K 4089]

A. Weber (1873). Monatsbericht der Königlichen Akademie der Wissenschaften zu Berlin, Berlin, pp.705-735. Sanskrit text of Nilakant-ha, German translation. [M]

J—. R—. (1873). Eine Rösselsprungs-Frage, Deutsche Schachzeitung, vol. 28, p.283. [A]

A. Régnier (1873). Problème de la Marche du Cavalier, Les Mondes, vol. 32, pp. 507-514. [A]

Paolo Volpicelli (1874). Réponse aux Critiques faites a ma Solution du Problème du Cavalier et des Echecs, Les Mondes, vol. 33, pp.126-130. [A] See also 1850, 1872, 1875.

Adolf Friedrich Stenzler (1874). Uber Nilakant-ha's Rösselsprung, Monatsbericht der Königlichen Akademie der Wissenschaften zu Berlin, Berlin, pp.21-26. [C]

T. von der Lasa (ed) (1874). Bilguer`s Handbuch des Schachspiels, 4th edn, p.12. [L]

Count Ligondès ('Palamède') (1874). Polygraphie du Cavalier, Orleans. [M] See also Feisthamel 1880. Also 1884, 1906, 1910, 1911.

Dr Antonius van der Linde (1874). Geschichte und Litteratur des Schachspiels, vol.1, p.245 (diagrams), pp.292-5 (history), vol.2, pp.101-111 (bibliography), also pp.337-8?. [Reprint by Olms, Zurich, 1981]. [British Library or 7913.ff.23] See also 1881.

(1875). The City of London Chess Magazine Vol.2, Nr.7, p.197. Gives details of the Italian book edition of P.Volpicelli (1872), 400 pages quarto, published in Rome. [Photocopy from K.Whyld]

Charles Bouvier ('Adsum') (1876). Tours in Le Siècle . See also 1882, 1883.

Dr — Exner (1876). Der Rösselsprung als Zauberquadrat, Progress, des Königlichen Gymnasium zu Hirschberg, W. Pfund. Contains 15 magic tours of which three are new and another gives a fifth arithmetical version of Jaenisch's 00a. [Murray 1951]. [A], [C], [K 4091]

— Mansion (1876). Sur les Carrés Magiques, Nouv. Corr. Mathém. vol. 2, pp. 161, 193. [A]

Percival Frost (1876). Quarterly Journal of Mathematics. Introduction to the following article by his brother.

Andrew Hollingworth Frost (1876). On the Knight's Path, Quarterly Journal of Pure and Applied Mathematics, London, vol. 14, no.54, pp. 123-125 (and plate 3 at end of volume). Construction of tours on square boards of any size by compartmental and bordering methods. Also applied to tours in cubes and on the four-handed chessboard. [Radclife science Library, Oxford, RSL: Math.Per.3], [C]

Andrew Hollingworth Frost (1877). A Simple Method of Tracing Paths of a Knight over the Squares of 5, 6, 7, 8 and their extension to higher squares, Quarterly Journal of Pure and Applied Mathematics, London, vol. 14, no.56, pp.354-359 (and plate 5 at end of volume). Improved examples of tours by bordering methods. Includes one 8×8 with diametrally opposite cells differing by 8. [Radcliffe Science Library, Oxford: RSL: Math.Per.3], [C]

Andrew Hollingworth Frost (1877). On Nasik Squares, Quarterly Journal of Pure and Applied Mathematics, London, vol. 14, no.57. 'Anglican missionary A. H. Frost designated Pamagic squares Nasik after the town in India where he was working about 1850-60'. [P. Bidev 1986]

C. Flye Sainte-Marie (1877). Note sur un problème relatif à la marche du cavalier sur l'echiquier (Séance du 18 Avril 1877) Bulletin de la Société Mathématique de France, Année 1876-77, vol.5, pp.144-150. Analysis of the 4×8 board, applicable to 4-rank boards of any length. He proves that any tour must be open, must start and end on the first or fourth ranks, must be in two halves joined by a single move on the two inner ranks, the two halves being on two fixed families of squares (the outer white and inner black, and the outer black and inner white) and shows how the number of tours can be calculated from the numbers of half-tours. For the 4×8 board he correctly counts the half-tours 118 a2, 32 c3, 54 e2, 42 g3 and the full tours 118×32 + 32×54 + 54×42 = 7772. Does not diagram any actual tours. [C], [British Library]. See also 1904.

Sam Loyd (1878). Rook tour problem, Chess Strategy, pp.175, 337. [A.C.White (1913)]

G. Lauber (1878). Aufklärung von Schachgeheimnissen ..., F. Schmeer & Sohne, Ratibor. [C], [K 4092]

T. P. Stowell (1879). Problem Relative to the Move of the Knight at the Game of Chess, The Analyst, Des Moines, Iowa, vol. 6, no.6, November. [C]

E. C. Caldwell (1879). The Knight's Tour on the Chessboard as a Magic Square, English Mechanic and World of Science, vol. 29, p.317. Gives two magic knight's tours, one quoted from 'a modern French Cyclopedia' Larousse's Dictionnaire Universel, (this is Jaenisch 12o in our catalogue) the other 'is original and has never before been published' (05b). Both are of squares and diamonds type. Notes '...the errors in excess in one section are made to compensate the errors in defect in another section, and vice versa.' [Leicester University Library]

« 1880s

M. A. Feisthamel (editor) (1880). Chess column in Le Siècle {The Age}, 1876-1885 published all then known magic knight's tours, and many new ones; 2 in 1880, 6 in 1881, 14 in 1882, 39 in 1883, 2 in 1884. The contributors of new tours were Count Ligondès (22), Edouard Francony, 'Célina' (11), C.E.Reuss (7), Charles Bouvier 'Adsum' (5), Jolivald (4), Béligne (2), Feisthamel (1), M.Wihnyk (1). [Murray 1951]

(1880). Heures de Leisir, Moniteur des Jeux d'Esprit et de Combinaissons ..., Paris 1880-81. [C]

Emmanuel M. Laquière (1880). Géometrie de l'Echiquier, Solutions Regulières du Problème d'Euler sur la Marche du Cavalier ..., Bulletin de la Société Mathématique de France, Paris, vol.8, pp.82-102, 132-158. Roget's method restricted to closed tours of the nets; such tours are 16-move paths of H or C shape. Laquière seems to have been the first to recognise the role of the linkage polygon formed by the alternate deleted and inserted moves. Applies squares and diamonds to the 4×8 board. [A], [C], [K 4093]

Emmanuel M. Laquière (1881). Note sur le Nombre des Marches Rentrentes du Cavalier ..., Bulletin de la Société Mathématique de France, Paris, vol.9, pp.11-17. Extends the figures of Sainte-Marie (1877) to find the number of 8×8 reentrant tours formed of two 4×8 tours. Gives total 31,054,144, which is the number of tours presented in numerical form, i.e. 16 times the generic number which is 1,940,884. Murray (1918) claimed this total was wrong, but Jelliss (1997), by an independent method, finds it correct. [A]

C. de Polignac (1881). Note sur la marche du Cavalier dans un Echiquier, Bulletin de la Société Mathématique de France, Paris, vol.9, pp.17-24. [A] See also 1861.

H. J. Kesson (= 'Ursus') (1881). Caissan Magic Squares, The Queen, The Lady's Newspaper, London, viii, ix, x. [P. Bidev 1986]

A. van der Linde (1881). Quellenstudien zur Geschichte des Schachspiels. pp.196-198. Tours in mediaeval manuscripts. [Photocopy from K. Whyld]. See also 1874.

G. E. Carpenter & S. Hertzsprung (1881). Brentano's Chess Monthly, New York, vol.1 (a) issue 1, May, p.36: G. E. Carpenter (problem editor for issues 1 and 2): "Prize Knight's Tour. To construct a complete knight's tour of the board, that, the squares being numbered consecutively from 1 to 64, the square numbers 1, 4, 9, 16, 25, 36, 49, and 64 will occupy one band or column of squares." (b) issue 5, September, pp.248-249: S. Hertzsprung, letter from Copenhagen dated 17 July 1881, gives four solutions of Carpenter's problem, with the square numbers in order of magnitude along the first, second, third and fourth ranks. He also calculates the number of ways of arranging the square numbers along the ranks. [British Library: PP1831.l]

Abbé Philippe Jolivald ('Paul de Hijo') (1882). Le Problème du Cavalier des échecs d'après les méthodes qui donnent la symétrie par rapport au centre, ouvrage contenant plus de quatre cent treize mille parcours du cavalier. Metz, Chez l'Auteur, 9 Rue Marchant. [K] This includes a comprehensive study of pseudotours with quaternary symmetry on the 8×8 board. Pages 23-51, 63-85, 87-97, 99-110, 129-135, 140-153, 158-159, 165-167 are almost entirely taken up tedious by lists of moves, but in between are to be found some interesting results.
… Chapter II 'Les méthodes de Vandermonde'pages 124–155, enumerates all possible ways of covering the 8×8 chessboard with four 16-move paths in quaternary symmetry. On page 154 de Hijo gives the total of 16-move closed paths in direct symmetry as 301, a figure quoted by later authorities, such as Edouard Lucas, Recreations Mathématiques (1895). The correct figure was shown to be 368 by T. W. Marlow (1985). An examination of de Hijo's list however shows that he did not in fact miss out any but made an error in stating the total.
… On pages 128 and 139 are diagrams of the 14 patterns formed at the four central squares in the cases of oblique and direct quaternary symmetry. These are lettered a-k, there being two each lettered c, f, i where two patterns use the same squares (the second is distinguished by using an italic letter). The formations j and k are the same in each case, having octonary symmetry. He also gave the formations descriptive names. The correspondence between his lettering scheme and our numbering is as follows: Octonary: 303 [j] Grande étoile (great star), 101 [k] Petite étoile (little star). Oblique quaternary: 304 [a] carrefour (crossroad), 304 [b] Four angles of 53 degrees, 204 [c] Roue à rochets (catherine wheel), 204 [c] Tourniquet à petites branches, 104 [d] Barres croisées, 104 [e] Tourniquet à grosses branches, 203 [f] Carré (square), 203 [f] Four angles of 37 degrees, 103 [g] Moulin à vent (windmill), 103 [h] Croix grecque à pans coupés, 102 [i] Croix grecque (greek cross), 102 [i] Croix de pas Dieu. Direct quaternary: 304 [a] Dents de scie (sawtooth), 304 [b] Quatre angles droits, 204 [c] X coupé, 204 [c] Double noeud (double knot), 104 [d] Cross of St Andrew, 104 [e] Grand sablier (large hourglass), 203 [f] Losange, 203 [f] Deux demi-étoiles (two half-stars), 103 [g] Double losange, 103 [h] Casse noisettes (nut crackers), 102 [i] Vis (screw), 102 [i] Petit sablier (small hourglass).
… On page 126 the special case of four circuits in oblique quaternary symmetry with no move cutting another (the solution to 'Aladdin's conundrum') is mentioned but not diagrammed. And on the next page 127 there is a diversion in which his methods are adapted to the 6×6 board and the five quatersymmetric solutions are listed, though only one is actually diagrammed; the other tours are in coded numerical form. Many authors have subsequently rediscovered this attractive result independently (including Bergholt 1918, Papa 1921, Cozens 1940).

John Augustus Miles (1882). Poems and Problems. See also 1884.

François Edouard Anatole Lucas (1882). Le Saut du Cavalier au Jeu des Echecs, Revue Scientifique, 22 ix 1882, pp.370–375. Contents: (a) Number of undirected knight moves on p×q board is (2p–3)(2q–3) – 1, i.e. 168 on 8×8 board. (b) The four-knights problem on 3×3 attributed to Guarini (now known to date back to al-Adli). (c) The three tours of a 3×4 board given by Euler. (d) Euler's two cross-shaped tours with diagonal axes of symmetry. (e) The de Montmort tour (he is mentioned as author of l'Essai d'analyse sur les jeux de hasard, Paris 1708 and 1714, but it is not clear if the tour appears in either of these sources) the de Moivre and de Mairan tours of 1722 are also mentioned but not diagrammed. (f) Mentions that according to the Encyclopédie of d'Alembert and Diderot the problem was known long ago ('tres anciennement') in India. (g) The Guarini 4×8 tour. (h) Symmetric tour by Euler formed by joining two 4×8 tours. (He seems to imply that this half-board tour is identical to the one by Gianutio, but though similar they are not in fact identical.) He claims the number of tours of this type is 3872. (i) Quotes a 64-word cryptotour poem from Le Palamède 1842, series 2, vol.1, p.322. (j) Notes on Vandermonde, Ballière de Laisement, Simonot, Crétaine, Solvyns, de Hijo, and symmetry. (k) More general but inconclusive notes on mathematical topics including Hamiltonian tours on networks. [C] There is a footnote on p.374 citing a domino problem in the 1883 vol.2 of Récréations Mathématiques by Lucas, but presumably he would have had prior knowledge of this; or else this copy is a later reprint with added notes. This entire article is reproduced in different format as Note I, pp.205–223 in vol.4 of Récréations Mathématiques published posthumously in 1894. See also 1889, 1891, 1894, 1895.

Count Ligondès (1884). Polygraphie ... Appliquée a ... Carrés Magiques, 1884. (See also 1874.) First of a series of privately printed works containing his results on magic knight's tours. Continued in 1906, 1910, 1911

— Biddle (1884). (Wahrscheinlichkeitsbetrachtungen uber den Rösselsprung) Educational Times Reprints, vol.41, pp.70–72 [A] or p.93–98 [?]

Grove Karl Gilbert (1884). The Problem of the Knight's Tour, Bulletin of the Philosophical Society of Washington, 10th meeting, 30 i 1884. Abstract only. Discusses conditions under which various symmetries are possible. 'It is determined empirically that the smallest square field on which the [closed] tour can be executed is that with 36 spots. Upon this field the number of possible tours with biradial symmetry is 21, of which 5 have also quadriradial symmetry.' The correct total is 22

John Augustus Miles (1884). British Chess Magazine, vol.4, p.72. The Beverley magic tour. [Murray 1930] See also 1882.

Thomas B. Rowland & Frideswide F Rowland (1884). Chess Fruits. Includes four cryptotours, one apparently sent to a British Chess Magazine knight's tour prize tourney, others from Sussex Chess Magazine and Brighton Guardian. [BCPS Library]

Auguste Héraud (1884); Jeux et Récréations Scientifiques ... 1884. See also 1893.

Mikhail Frolow (1884). Le Problème d'Euler et les Carrés Magiques, Nouvelle étude, suivie de Notes par Monsieurs H. Delannoy et Ed. Lucas. Paris [A]. See also 1886.

Ambikadatta Vyasa (1884). Chaturanga Chaturi. Twelve tours: (1) Euler's first open tour modified as a1 to h1. (2) Symmetric tour joining two half-board tours, similar to Euler. (3) Euler's 12-cell cross tour repeated four times in 8×8 board and joined up to make an open tour, omitting 16 cells in a pattern often seen on Indian boards. (4) Closed tour 7×8 in 3×8 and 4×8 compartments. (5) Open tour omitting 20 cells of 8×8, central 2×2 and 2×2 at middle of each edge. (6–7) Open and closed tours 5×8. (8) Symmetric tour 6×8 in two 3×8 compartments. (9-10) Near-symmetric open tours 4×8 and 3×8. (11–12) Open tours 6×6, one omitting four cells, b25e25. [Murray 1930]

— Syamakisora (1885). Risala i Shatranj. Tour both Rhombic and Rogetian. [Murray 1930]

Maxwell Wihnyk (1885). Deutsche Schachzeitung.

Mikhail Frolow (1886). Les Carrés Magiques, Paris. Contains: p.21 Magic knight's tours by Feisthamel (14d) and 'Palamède' (= Wenzelides 12e). pp.33–4 Border braid method to expand 5×5 to 9×9, 6×6 to 10×10 etc, attributed to H. Delannoy. Plate VII has 10×10 tours of (1,4) and (2,3) movers by A. H. Frost. [British Library: shelfmark 8535 3.18]. See also 1884.

Rev. Serge de Stchoulepnikoff (1885). ms Numerate Tables of all Symmetric Tours Arranged with Quarts... 1885. See also 1865.

Fritz Hofmann (1886). Sur la Marche du Cavalier, Nouvelles Annales de Mathématiques (Journal des Candidats aux Ecoles Polytechnique et Normale), 3rd series, vol.15, pp.224–6. Longwinded 'proof' that closed tour requires even number of moves. [Leicester University Library: shelfmark MA.PER.510.N9080]

Michael Haberlandt (1887). Der altindische geist ... 1887.

C. Jordan (1888). Palermo Rendiconti 1888.

C. Böcklin (1889). Archiev der Math. 1889.

C. Planck (1888). Article describing 4D magic 3^4, The English Mechanic. [Planck 1905]

F. A. P. Barnard (1888). Memoirs of the Ameruican National Academy of Sciences, vol.4. 'The general theory of magic squares, especially with respect to entry of terms by coordinates, is expounded.' [Cashmore 1908]

A. Herma (1889). Rösselsprünge aus deutsche Dichtern ... 1889. See also 1849.

E. Lucas (1889). Nouveaux Jeux Scientifiques..., La Fasioulette, p.302 and Fig. p.301. This is an 8×8 board with 64 links of length root 5 to form knight's tours. [D.Singmaster]

« 1890s

General Théodore Parmentier (1891). Articles in publications of congresses of Association Française pour l'Avancement des Sciences, Marseilles 1891, Pau 1892, Caen 1894. Catalogues all known magic tours to date. [Kraitchik 1927 p.3, Coxeter 1956 p.174]

T. B. Sprague (1891). Edinburgh Math Soc. 1891.

E. Lucas (1891). Théorie des Nombres 1891.

Edward Falkener (1892). Games Ancient and Oriental and How to Play Them, Longmans, Green and Co. [Reprint, Dover Publications, New York, 1961]. Pages 267–356 deal with magic squares and knight's tours, including magic knight's tours pp.319–336. He relates the story of Roget having issued a card with the caption 'Key to the Knight's Tour as a Magic Square', for which however there is no corroborating evidence. The semi-magic tour shown is from Tomlinson (1845). The magic tours quoted are Beverley's, one by Wenzelides, seven by Jaenisch (three of which he claims as his own), eight by Palamède (Ligondès), and the one by Caldwell. He gives a collection of 8×8 tours and tours of the four-handed chessboard with striking patterns. [My own copy, now in BCVS Lbrary]

Walter William Rouse Ball (1892). Mathematical Recreations and Problems, Macmillan and Co, London. I'm not sure if this first edition contained knight's tours. Subsequent editions were issued as follows: 2nd 1892. 3rd 1896 [This has subtitle ... of Past and Present Times, Ch.7, has Hamiltonian game pp.162–165, Knight's path pp.165–177, use of 'cells' for the squares of the board p.165; tour cited as 'Roget's Solution' is of squares and diamonds type and not semimagic.] 4th 1905, 5th 1911, 6th 1914, 7th 1917, 8th 1919, 9th 1920, 10th 1922 (reprinted 1926, 1928, 1931, 1937). For 11th and subsequent editions see Coxeter 1939. [Leicester University Library]

Dr Hermann Caesar Hannibal Schubert (1892). Monist 1892. See also 1898.

General Théodore Parmentier (1892). French Association meeting at Pau. See also 1891, 1894.

Vinayaka Rajarama Tope (1893). Buddhibalakrida, (Marathi). Ten tours. Six open tours 4×8 and four 8×8 tours: (1) Euler type open tour identical to Vyasa (1884) but rotated a half-turn. (2–3) Tomlinson/Roget tour, closed, and slightly modified version of same. (4) Closed tour formed of two of the six half-board tours linked asymmetrically. [Murray 1930]

Rev. Angelo John Lewis ('Professor Hoffmann') (1893). Puzzles Old and New, Frederick Warne & Co. London. [Reprint 1988 by Martin Breese Ltd] Knight's Tour Letter and Word Puzzles pp.248–50 with solutions pp.259–63, partial tours and one near axially symmetric. The Knight's Tour p.335–6 with 8 solutions pp.367–73, 2 by Euler, 1 'Du Malabar', 1 Monneron (= Nilakant-ha). 3 with near direct quaternary symmetry and 1 closed tour using Warnsdorf's method.

Auguste Héraud (1893); Jeux et Récréations Scientifiques ... 1893. See also 1884.

Georges Edouard Auguste Brunet (1894). Analysis Situs, Recherches sur Réseaux 1894.

Rudolph Wolf (1894). Studie über den Rösselsprung 1894.

General Théodore Parmentier (1894). French Association meeting at Caen, 11 viii 1894. See also 1891, 1892.

E. Lucas (1894). Récréations Mathématiques, Gauthier-Villars et Fils, Paris (4 vols, 1882, 83, 93, 94). Vols 3 and 4 edited by H. Delannoy, C. A. Laisant and E. Lemoine from papers left by Lucas at his death in 1891. Vol.4 , Part 6: 'Le Géometrie des Réseaux et le Problème des Dominos' includes pp.130-131 an extension of his results on the number of knight's moves on a p×q board to the case of generalised moves. Note I pp.205–223 reproduces his 1882 article on the knight's leap. [Leicester University Library].

E. Lucas (1895). L'Arithmétique Amusante, Gauthier-Villars et Fils, Paris. This was also edited by H. Delannoy, C. A. Laisant and E. Lemoine from papers left by Lucas at his death in 1891. Note 4, Part 6, Le Saut du Cavalier covers: Collini, Squares and diamonds (misattributed to Ciccolini), Jaenisch, Vandermonde and de Hijo, Warnsdorf, Sainte-Marie, Laquière, symmetries, and border method for extending tours to larger boards. This is based in part on Lucas 1889 and Frolow 1886.

(1895). La Strategie, vol.32, p.213, has mention of a brochure on knight tours by General Parmentier, presented at Caen. [D.Pritchard]

W. M. Cubison (1896). The Knight's Tour, Chess Monthly, vii-viii 1896. Closed tour 64×64. [M]

(1896). Chess Monthly, vol.17, p.340, knight tour supplement. [Information from D. Pritchard; British Library shelfmark PP1831i – also K.Whyld who states that the date is 1896 not 1831 as previously given here.]

H.Eschwege (1896). The Knight's Tour. In a continuous and uninterrupted Ride over 48 Boards or 3072 Squares. Adapted from Byron's 'Mazeppa'. Silsbury Brothers, Shanklin, Isle of Wight. Dedicated to Sir George Newnes, Bart, President of the British Chess Club. December 31st. Byron's poem is presented on a series of chessboards, one word to a square, commencing at f8 and proceeding to e2, whence to leap to f8 on the next board, and so on. The same tour, supplied by H. E. Dudeney, is used throughout, except on the 48th board where it is modified to end at h1. [British Library shelfmark 11647 ee37, catalogued under Byron, G.G.N.]

— Grossetaite (1896). Figaro, Magic tour (01e). [M]

H. Jacobi (1896). Uber zwei ältere Erwähnungen des Schachspiels in der Sanskrit-Litteratur, Zeitschrift der Morgenländischen Gesellschaft, Leipzig, vol.1, pp.227–233. Rudrata ~900. [Murray 1913 pp.21–22, 53–55]

Sam Loyd (1897). Puzzle in Tit Bits 1897.

E. McClintock (1897). On the most perfect forms of magic squares, American Journal of Mathematics, vol.19, p.99. 'The term Pandiagonal Squares seem to have been invented by EMcC'. [Bidev 1986].

Dr Hermann Schubert (1898). Mathematische Mussestunden. Eine Sammlung von Geduldspielen, Kunststücken und Unterhaltungsaufgaben mathematischer Natur. {Mathematical Pastimes (?). A collection of solitaire games, feats and entertaining problems of a mathematical nature.} Leipzig. Ch.23, pp.187–222, Rösselsprünge. A. Introduction. Gives the Lucas (1882) formula for number of moves on p×q board in the form 2(2pq – 3p – 3q + 4). B. History. C. Euler and Vandermonde. D. Collini. E. Polignac and Laquière. F. Sainte-Marie 4×8 and other small boards. G. Magic tours, three examples (00m, 12e, 12m all by Wenzelides). H. 3D tours 4×4×4 and 3×4×6. Ch.25, pp.269–286, Hamiltonian problem. [Photocopy kindly supplied by D. Singmaster]. Second edition. 1904 Rösselsprünge pp.201–238. [Leicester University Library] Third edition 1909, in 3 vols. Rösselsprünge, vol 3, pp.1–36. [Photocopy from D. Singmaster]. These editions have the same content regarding knight's tours as the first edition. Fourth edition: see 1924. Also: Zwölf Geduldspiel 1899.

Mlle. Stella Goulieux (1899). La Strategie, vol.32, p.114, 'M' (Magic?) Knight Tours [D.Pritchard].

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