Chronology of Knight's Tours

© 2002 — compiled by George Jelliss.

On this page: scroll down or click on the required period:
Ancient (Before 500)Mediaeval (500 – 1500)1500s1600s1700s … or Onward to: Part 2: 1800s.

PART 1. (Before 1800)

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. For much fuller details of the mediaeval manuscripts the reader should consult H.J.R.Murray, A History of Chess (1913).

« Ancient (Before 500)

Chinese (~−2200). The 3×3 diagonal magic square is said by many sources to have been known in China about −2200, though what evidence there is for this date I'm not sure. [C.Berge Principles of Combinatorics Academic Press 1971, cites F.M.Müller Sacred Books of the East vol.XVI (The Yi-King) Oxford University Press 1882] Moves between succesive entries include the knight's move.

Minoan (~−1600). The classic labyrinth design that appears on Minoan coins, is attributed to the legendary engineer Daidalos (Latinised: Daedalus) and is associated with the Minotaur myth.

« Mediaeval (500 – 1500)

Ravenna (~530). Labyrinth elaborates on the Minoan design. [Described by Pennick 1990]

Celtic Art (~700–800). Includes interlacing designs, some of which can be interpreted as king tours. These are found in books and artefacts produced in monasteries of Ireland, Scotland and Northern England. In particular (in order of increasing complexity, not necessarily chronologically): panels of the Soicel Molaise book-shrine, horizontal borders of the eagle page of the Book of Dimma, a panel of the St Madoes Stone, vertical borders of the lion page of the Book of Durrow and the knot design in the Book of Kells. The latter appears to illustrate a saying of Alcuin. [See: Bain 1951, Pennick 1990, Laing & Laing 1992]

Al Adli ar Rumi (~840). Kitab ash-Shatranj {Book of Chess} from which one 8×8 closed tour survives, quoted in al-Hakim.

Rudrata (~900). Kashmirian poet, ascribed to the reign of Sankaravarman, 884-903, gave in his Kavyalankara rook, elephant and knight tours of the 4×8 board. See also Nami (1069), Jacobi (1896) [Murray 1913, pp.53–55].

Abu-Bakr Muhammad ben Yahya As Suli (~910). Kitab ash-Shatranj {Book of Chess – he wrote two} from which four 8×8 closed tours survive in several later manuscripts, including examples using alternating knight-fers and knight-alfil moves. [Murray 1913, pp.169–85, 335–8].

Ben Ishaq An-Nadim (988). Kitab al-fihrist. A bibliography which has a section on chess in which the Adli and Suli books are listed.

Basra (989). An Arabic encyclopaedia Raza'il edited in Basra by members of the philosophical association 'Brothers of Sincerity' describes the method of constructing the 3×3 magic square by moves of knight, fers and pawn. [Bidev 1986, part 2, p.34]

Nami of Guzerat (1069). Commentary on Rudrata's Kavyalankara.

Abu Ishaq Ibrahim ben al-Mubarak ben Ali al-Mudhahhab al-Baghdadi (1141). Manuscript #560 in Istanbul Library, title Kitab ash-shatranj mimma's-lafahu'l-Adli was-Suli wa ghair-huma {Book of chess: extracts from the works of al-Adli, as-Suli and others}. This has the four as-Suli tours. [Murray 1913, pp.171-2].

Somesvara III (~1150). Manasollasa {Delight of the Intelligent} contains an edge-hugging tour.

al-Buni (1200). Various manuscripts, include many 4×4 magic squares, using letters as numbers, with the top row spelling a word. [Singmaster 1991, pp.218–9].

Chartres (1220). Pavement labyrinth at the Cathedral of Notre Dame, Chartres [Dudeney 1917, Pennick 1990].

Muhammad ben Hawa ben Othman (1221). Manuscript, has the second as-Suli tour. [Murray 1913, pp.174-5].

'Bonus Socius'{Good Companion} (~1250 – ~1350). The name given to the compiler of the first large collection of mediaeval chess problems of which numerous manuscript copies survive, written over a long period. One of these is scribed by Nicolas de Nicolai, a scholar from Picardy who studied and lectured at the Lombard Universities. The knight's tour problem is presented on a 4×8 board in which a knight is to capture the other 31 chessmen in a given sequence. The sequence of capture of the pawns is not fully determined, allowing six different tours. [L].

Arabic (1257). An Arabic ms in the British Library contains two problems involving capture of pawns by a knight in the minimum number of moves [Murray 1902].

AngloFrench (~1275). The King's Library (now in British Library) has a manuscript in Anglo-French, containing two tours, one 4×8 identical to one of the Bonus Socius solutions, and one 8×8 formed of two 4×8 tours joined together. [BM shelf mark 13A xviii], [Murray 1913, pp.581, 589].

Abu Zakariya Yahya ben Ibrahim al-Hakim (~1350). Ms, Nuzhat al-arbab al-'aqulfi'sh-shatranj al-manqul {The delight of the intelligent, a description of chess}. This has two 8×8 tours, one attributed to al-Adli and the other to Ali ibn Mani. It survives in two mss, one scribed about 1350 has both tours, a later copy has only one. [Murray 1913, pp.175–6, 336].

Muhammad b. Mahmud al-Amuli (1352). Nafa'is al-funun fi'ara'is al-'uyun {Treasury of the Sciences}, Persian, survives in numerous mss of variable quality. It concludes with chapters on chess, including one 8×8 tour. [Murray 1913 (description) and 1930 (tour)].

Cairo (~1370). Ms in former Khedival Library, Cairo, ref: Mustafa Pasha #8201, with three of the tours given by al-Baghdadi 1141. [Murray 1913].

Ala'addin Tabrizi (also known as Ali ash-Shatranji, i.e. Ali the chessplayer) (~1400). Leading player at the Samarkand court of Timur. Known to have written a work on chess. Murray gives reasons to believe that a copy of this work may survive in a 16th century ms in the Royal Asiatic Society Library, London, ms Persian #211 (formerly #260). The preface promises tours on the whole, half and quarter boards, but the pages with the tours are missing. See Forbes (1860) for translation. [Murray 1913, pp.171, 177, 335 footnote].

Dresden (~1400). Ms, which gives the half tour problem without solution and sets as a wager game a tour over a board of 4×4 squares (a complete tour is impossible). [M].

'Civis Bononiae'{Citizen of Bologna} (~1450 – ~1500). The name given to the compiler of the second main collection of mediaeval chess problems. Like the earlier Bonus Socius collection this larger compilation also exists in numerous ms copies. The 4×8 tour is identical to one of the six Bonus Socius solutions. [See van der Linde 1874, pp.245, 292–5 and Murray 1913, pp. 643-8, 674].

Florence (~1490). Italian ms, end of 15th century, has a 4×8 tour. [L].

« 1500s

Johannes Chachi, of Terni (1511). Ms #791 Casanatense Library, Rome, 4×8 tour. [Murray 1913, pp.727, 730].

Paulo Guarini di Forli (or Paulus Guarinus) (1512). Ms in Cleveland Library, contains the Civis Bononiae tour, and the al-Adli problem of interchanging the two black and two white knights on a 3×3 board. [C]

Denis Janot, (printer) (~1530 – 40). Sensuit Jeux Partis des Eschez: Composez nouvellement Pour recréer tous nobles cueurs et pour eviter oysiveté a ceulx qui ont voulenté: desir et affection de le scavoir et aprendre et est appelé ce Livre, le jeu des Princes et Damoisellles. Paris. Contains the first printed tour; the same tour as Civis Bononiae [Vienna Library MUS II #195], [A], [M].

Persian (~1550). Ms in Royal Asiatic Society Library, London, Persian #211 (formerly #260). See entry for Ala'addin ~1400. [M]

Orazio Gianutio della Mantia (1597). Libro nel quale si tratta della maniera di giuocar a Scacchi, con alcuni sottilissimi partiti, nuovamente composto, Antonio de Bianchi, Torino. Has one 4×8 tour. [D. Singmaster notes the tour is not in the 1817 English translation by Sarratt.]

« 1600s

Bhatta Nilakant-ha, Bhagavantabhaskara (~1600 / ~1700 dating doubtful). An encyclopedia of ritual, law and politics, which has a section on chess at the end of the fifth book. Gives one tour, presented in three different ways. See Monneron 1776, Weber 1873, Stenzler 1874.

B. Frénicle de Bessy (1693). "Des Carrés Magiques" Divers Ouvrages de Mathem. et de Physique, Par Messieurs de l'Academie Royale des Sciences de Paris, 1693, pp. 423-507. Also: "Des Quarrés ou Tables Magiques ", including: Table Générale des Quarrés Magiques de Quatre, Mem. de l'Acad. Roy. des Sc. depuis 1666 jusqu`a 1699, vol. 5, pp. 209-354 (Paris 1729). Enumerates and lists all 880 diagonally magic squares of size 4×4. [Benson and Jacoby 1976 give a computer print-out of all the squares. They also cite Frénicle`s 880 Basic Magic Squares of 4×4 Cells, Normalized, Indexed and Inventoried by K. H. de Haas, no date given. Among these magic squares #100 is a knight + wazir tour, and two others are queen tours, all three axially symmetric; though attention was not drawn to these cases until recently (Jelliss, 1986).]

« 1700s

Jacques Ozanam (1725). Recreations Mathematiques et Physiques first published 1694. The edition of 1725 (published after Ozanam's death, editor not named) is the first to contain the three knight's tours by the mathematicians Jean-Jacques d'Ortous de Mairan, (Directeur de l'Academie Royale des Sciences), Abraham de Moivre and Pierre Rémond de Montmort; according to a marginal note, the tours were contributed by de Mairan in 1722. [K 4066]

This work has appeared in numerous revised and translated editions, under various editors and publishers: French: Montucla 1750. 1778, 1790; Grandin 1820, 1825, 1835; English: Hutton 1803, 1814, 1840; Riddle 1844. A history is given by Rouse Ball (1939, pp. 2–3).

Jean Etienne Montucla (1750). Editor of editions of Ozanam's Recreations, 1750, 1778, 1790.

Jean le Rond d'Alembert and Denis Diderot (1751). Compilers of the Encyclopédie 28 vols (inc 11 of plates), 1751 – 72. Lucas (1882) cites this source as saying that a method of solution of the knight's problem on the usual board was known long ago ('tres anciennement') in India; and Kraitchik (1927, p.1) says more specifically that they trace it to the Brahmins (Hindu priests) 2000 years ago. However, Lucas and Kraitchik give no page reference and I have been unable to locate the comment; they may have confused it with Monneron 1776. [K], [Bodleian Library, Oxford]

Leonhard Euler (1757). Letter to C. Goldbach 1757 contained a symmetric tour. See Fuss 1843.

Leonhard Euler (1759). "Solution d'une Question Curieuse qui ne Paroit Soumise a Aucune Analyse" {Solution of a curious question which does not seem to have been subject to any analysis} Mémoires de l'Academie Royale des Sciences et Belles Lettres, Année 1759, vol.15, pp.310-337, Berlin 1766. The first mathematical paper on knight's tours, much quoted in later sources. This paper, although presented in 1759 did not appear in print until 1766 and the first review of it, quoting his first two tours, appeared in the Journal Encyclopédique in 1767. See also Fuss 1849 and Euler 1923.

Lelio dalla Volpe (printer) (1766). Corsa del Cavallo per tutti scacchi della scacchiere, Bologna. First to show tours by printed geometrical diagrams. [C], [K 4064]

Monsieur B— (1767). In a letter to the editors of the Journal Encyclopédique enquired about knight's tours and the editors responded with a review of Euler's paper, quoting his first two tours, vol.1, part 3, February 1767, pp. 124–128. [Photocopy kindly supplied by D. E. Knuth]

Edmé Gilles Guyot (1769). Nouvelles Récréations Physiques et Mathématiques Chez Gueffier, Paris, 1769-70 (4 vols); vol.2 pp.230-242 (recreation 60) and plate 2 (de Moivre's tour). Also Paris, 1786 (3 vols), 1772–75, 1786, 1798, 1799, 1800–01. (Translations: German 1772; Giuoci Fisici e matematici 1818). [A], [C]

Philip Stamma (1770). Essai sur le Jeu des Echecs 1737. Hamburg edition 1770 pp. 70-72. First use of modern 'algebraic' (letter and number) notation for chess. [C]

Alexandre-Théophile Vandermonde (1771). "Remarques sur les Problèmes de Situation", L'Histoire de l'Academie des Sciences avec les Mémoires, Année 1771, Paris 1774, vol.15, Memoires, pp.566-574 and two plates. One 8×8 tour formed in two stages from four equal circuits. Also a 4×4×4 tour. [Photocopy kindly supplied by D. E. Knuth]

Edmé Gilles Guyot (1772). Neue physikalische und mathematische Belustigungen, Augsburg 1772. German translation of Guyot (1769). [A]

Cosimo Alessandro Collini (1772). Monsieur Colini [sic], Secrétaire intime de S. A. E. Palatine, "Résponse à un problème sur le jeu des échecs", Journal Encyclopédique vol.6, part 3, September 1772, pp.453–462; vol.7, October 1772, part 1, pp. 112–118, part 2, 283–290. Describes Collini's method of forming tours from the pattern of eight concentric circuits, capable of providing tours beginning and ending at any given squares of opposite colour. Expanded into book form 1773. [Copy from D. E. Knuth]

A— C— B— (1773). Writing from Pont-à-Mousson, 17 January 1773, "Observation sur la solution d'un problème relatif au jeu des échecs", Journal Encyclopédique, vol.2, February 1773, pp.133–135. [Copy from D. E. Knth]

Le Chevalier W— (1773). Capitaine au régiment de Kinski, writing from Prague, 20 April 1773, "Lettre ... sur un problème de l`échiquier", Journal Encyclopédique, vol.6, August 1773, pp. 123–125. Gives a closed tour. This tour was quoted in the 1790 and later editions of Ozanam's Recreations, where his regiment is described as 'dragons, au service de l'Impératrice-Reine'. [Copies from D. E. Knuth and D. Singmaster]

Cosimo Alessandro Collini (1773). (At various times Private Secretary to Voltaire and to the Elector Palatine): Monsieur C—, Solution du Problème du Cavalier au Jeu des Echecs, à Mannheim, chez Tobie Loeffler, au Chandelier d'Or. Book of 60 pages. This work is a fuller account of his method published in the Journal Encyclopédique 1772. [K]

(1774). Magazino Toscano [Information from K. Whyld]

— Monneron (1776). Writing from the East Indies, supplied two tours to the Nouveau Dictionaire, Pancoucke, Paris. The first of these is the Nilakant-ha tour. The second is ascribed to a Malabarese (Malabar is in south-west India). Both tours are quoted in Hoffmann (1893) who gives the first the heading 'Du Malabar' as if that was the name of its author. This may be the source of the statement, quoted by Lucas (1882) and Kraitchik (1927), that tours were known in India 2000 years ago.

— Chapais (1777). Essais analytique ... . K. Whyld wrote 2002: Jean Mennerat of France now owns this manuscript. It was written shortly after 1777. Pages 485-496 relate to knight tours. [K. Whyld, e-mail 2002] This item was previously listed mistakenly at 1910.

L. Bertrand (1778). Developpement Nouveau de la Partie Elementaire des Mathematiques Geneva. Includes an account of his contribution to Euler's paper of 1759. [Jaenisch 1862]

Jean Etienne Montucla (1778). Editor of editions of Ozanam's Recreations, 1750, 1778, 1790. [C], [K 4067]

— Montferrier (1778). Dictionnaire des Sciences Mathématiques, vol.1, p.489, 1778 and Encyclopédie Méthodique, vol.1, section: 'Partie', Padua, p. 574. [L]

Denis Ballière de Laisement (1782). Essai sur les Problèmes de Situation, Rouen, chez Jean Racine, 74 pages. According to de Hijo (1882 p.4) he proposes forming tours using a 'planchette', which is a little chessboard with a pin in each square, the path being traced by a silk thread stretched from pin to pin. This hardly seems more practical than pencil and paper! D. E. Knuth reports that it gives 35 problems, including a tour 16×16. [C], [K 4068]

Karl Gottlieb von Windisch (1783). Letter dated 18 September 1783, in a book by him published that year (and in an English translation of 1819) gives an eye-witness account of exhibitions of Kempelen's Automaton, including making the knight's tour. [Information from K. Whyld]

Richard Twiss (1787). Chess, 1787, vol.1, three tours. Chess, 1789, vol.2, two tours, one on circular board. [M], [Information from K. Whyld]

Giorgio Fontana (1789). Opuscoli Matematici Supra il Teorema della Composizione delle Forze e Sopra il Calcolo Integrale delle Differenze Finite, Pavia, p.45. [L]

Jean Etienne Montucla (1790). Editor of editions of Ozanam's Recreations, 1750, 1778, 1790. The 1790 edition, problem 23, pp. 178–182, adds the Chevalier W— (1773) tour to the other three.

Fr. Wilhelm A— (1791). "Der Rosselsprung durch ale 64 Felder des Schachbrettes", ms Cleve, (was in the collection of von der Lasa). One tour repeated 64 times according to Murray (1930). [M]

Herzog Ernst II von Sachsen-Gotha (1797). Reichs-Anzeiger, 18 September 1797. Compartmental tour formed of two subboards 4×5 and two 4×3. [A]

Shir Muhammad-Khan (1798). Sardarnama manuscript. Described by Murray as Persian, but written in central India. Contains the Nilakant-ha tour. [Murray, 1913, p.65 footnote and p.181]

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