Chronology of Knight's Tours

© 2003 — compiled by George Jelliss.

Back to: Part 2: 1800s … or on this page: scroll down or click on the required period: 1900s1910s1920s1930s1940s1950s1960s1970s1980s1990s … or Onward to: Part 4: 2000s.

PART 3. 1900 to 1999

This page updated August 2003. Work on all the pages is a continuing process. Let me know of missing references that you think should be included. For fuller names of authors look in the Biobibliography page. The word ‘and’ between authors names indicates joint authorship, while the word ‘with’ indicates written by the first author but including one or more items by the other. A long dash (—) indicates missing information. A tilde (~) preceding a date indicates that it is merely notional. ms = manuscript. PFCS = Problemist Fairy Chess Supplement, FCR = Fairy Chess Review. In references to these publications the problem numbers are cited at the date proposed, the solution (with diagram) will usually be found in a later issue. 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.

« 1900s

C. Planck (1900). The N Queens Problem, British Chess Magazine, 1 March 1900. Article includes a 'chess magic square', and attributes the term Caissan Squares to the late Henry Jones (Cavendish). [Bidev 1986]

F. Fitting (1900). Z. für Math. und Physik 1900.

W. Ahrens (1900). Mathematische Spiele, section IG1 in Encyk. Math. Wiss. (Teubner, Leipzig) vol.1, part 2, pp.1080-93, Bibliography, estimates on number of tours, results on semi-magic tours. [Singmaster 1987] See also 1901, 1910.

W. Ahrens (1901). Mathematische Unterhaltungen und Spiele (Teubner, Leipzig) Ch. 11, Der Rösselsprung, pp.165–208. §1 Knight's move. §2 Guarini's problem. §3 Euler's method. §4 Warnsdorf. §5 Small boards and Euler's crosses. §6 Minding algebraic approach. §7 Collini and borders. §8 Vandermonde. §9 Squares and Diamonds and Roget, misattributed to Ciccolini. §10 Frost's compartmental method. §11 Magic tours, one example: (12o) §12 Cubic tours 4×4×4 by H. Schubert. §13 Number of tours: Sainte-Marie's results on 4×8. §14 Rook non-intersecting paths. [Leicester University Library] See also 1900, 1910.

A. Boutin (1901). L'Intermédiaire des Mathématiciens, Paris, vol.8, pp.153–4. Question on the number of rook tours of a rectangular board. Partly answered by Sainte-Marie (1904). [Coxeter 1956, p.186].

L. R. Babu (1901). Mo'allim ul Shatranj. 27 tours. (1–10) The first one is de Moivre's tour with the middle three moves reflected. The next nine are similar, the eighth incorporating a central 3×4 tour, the last two closed. (11) Euler's first symmetric bipartite tour. (12-17) Three open and three closed tours omitting various squares. (18–21) Four tours with approximate symmetry. (22–26) Open tours 7×8, 6×8, 5×8, 4×8, 3×8. (27) Open 10×10 tour joining four 5×5 tours. [Murray 1930]

H. J. R. Murray (1902). The Knight's Tour, Ancient and Oriental, The British Chess Magazine, pp.1-7. Notes on early history of the knight's tour. 1. Considers Rudrata's 4×8 the earliest tour. 2. Diagrams two reentrant 8×8 knight's tours, two alternating-move tours (dated 10th to 12th centuries), and two knight puzzles (dated 1257) from Arabic manuscripts, noting the various methods of presentation of the tours. 3. Mentions the Persian ms in the Royal Asiatic Society. 4. Knight's tour in Anglo-Norman ms. 5. Five 4×8 tours in European chess manuscripts, including Guarini and Gianutio. 6. The Nilakant-ha tour. 7. Modern Indian tour omitting four corner cells of each quarter (consists in effect of four copies of Euler's small cross-shaped tour linked together). “Mr Blackburne has often exhibited his knowledge of the tour at the end of a blindfold sitting.” [BCPS Library]

F. Fitting (1902). Archiv der Math. und Physik 1902.

F. Fitting (1904). Das Rösselsprungproblem in Neuer Behandlung Lipsk 1904. [P Wood] See also 1924.

C. Flye Sainte-Marie (1904). L'Intermédiaire des Mathématiciens, Paris, vol.11, pp.86–88. Partial answer to Boutin (1901). Gives recurrence formula for number of oriented wazir tours on rectangular board 4×y as F(y) = 2F(y–1) + 2F(y–2) – 2F(y–3) + F(y–4) given F(1) = 0, F(2) = 1, F(3) = 2, F(4) = 6. This generates F(5) = 14, F(6) = 37, F(7) = 92, F(8) =236 etc. [Leicester University Library]

H. Schubert (1904). Mathematische Mussestunden Eine Sammlung von Geduldspielen, Kunststücken und Unterhaltungs-aufgaben mathematischer Natur. (Leipzig) p.195 4×8 tour, pp.201-238 Rosselsprünge, p.204 formula 2(2ab - 3a - 3b + 4), p.235-7 4×4×4 closed tours, p.238 3×4×6 closed tour. [Mathematical Association Library, Leicester University]

C. Planck (1905). The Theory of Path Nasiks (A. J. Lawrence, Rugby). Theory relating to magic squares constructed by A. H. Frost (1878).

A. Rilly (1905). Le Problème du Cavalier des Echecs (Troyes). Two 32-move circuits linked by a 3-step rook move to form a diagonally magic square. [Coxeter 1956, p.185]

H. Fritsche (1906). Brett oder Rösselsprungspiel ... {Board for knight's tour game}. Patentschrift 179770, 4 February 1906 approved 18 December 1906. [C]

Ligondès (1906). La Mode du petit Journal, Magic tour. [M]

H. E. Dudeney (1906). Tribune, 2, 3 October and 3 December. Queen paths of 5 and 15 moves and Bishop path of 17 moves. [Coxeter 1956, pp.186-7]

A. C. Pearson (1907). The Twentieth Century Standard Puzzle Book (London). Selected from a column in the London newspaper The Evening Standard. Tours on pages 83-90 of part 1 (visual patterns), p.86 of part 2 (6×6 tour) and pp.31, 33-34, 37 and 118-121 of part 3 (cryptotours).

C. Planck with T. H. Neal, T. R. Dawson, J. R. Mattey (1907-1910). Chessboard Puzzles series in Chess Amateur: vol 2 (October 1907 - September 1908) puzzles #1-17; vol.3 (October 1908 - September 1909) puzzles #18-44; vol.4 (October 1909 - September 1910) puzzles 45-55. #1: pp.52, 82, routes of Ke1 to e8 in 7 moves = 393. #12: pp.276, 370, king tours of first two ranks (by THN); formulae in terms of Fibonacci numbers. On 2×8 there are 8576 routes from a1. #25-6, pp.83 (December 08), 147 (February 09), Knight tours of 5×5, total 1728. #28, pp.115, 179, rook tour in 15 moves a1-a6, (posed by TRD after JRM). #31, p.179, Guarini's problem. #32, p.179, Euler symmetric knight tour in two half-boards. #50 (Feb 1910), p.83, Knight 6-move routes from a1. [BCPS Library]

W. S. Andrews (editor) (1908). Magic Squares and Cubes 1908.

M. Cashmore (1908). Chess Magic Squares. On magic squares constructed using chess moves. Published 'in a journal of Capetown'. Cites Planck 1900. Cites Encyclopaedia Britannica (no date given) for construction of Nasik squares. [Article reproduced in P.Bidev (1986) pp.24, 46-48, 82-85]

« 1910s

W. Ahrens (1910). Mathematische Unterhaltungen und Spiele I 325 (?). Use of knight's tours as a secret code. [Singmaster]

Ligondès (1910). La Mode du petit Journal, Two magic tours 1910 and 1911. [M]

H. E. Dudeney (1910). Queen, 12 November. Open knight's tour of 80-cell Greek Cross board. [Bergholt 1916]

Ligondès (1911). A work on magic two-knight tours in quaternary symmetry. [Murray 1951]

H. J. R. Murray (1911). The Knight's Tour on the Half-Chessboard, The British Chess Magazine, vol.31, pp.417-422, also p.466. “Strictly speaking the knight's tour has more to do with mathematics than chess...” Duplicates Sainte-Marie (1877) apparently without knowledge of it, but his counting of the tours is erroneous. Considers tours formed by connecting half-tours of the half-boards (a special case of Bergholt's domino method).

G. W. Cutler (1911). The British Chess Magazine, vol.31, December, pp.464-5. An account of the squares and diamonds method (misattributed to C. E. Orme in the 1860s). Editor cites Roget 1840 and Leisure Hour 1873.

A. Fraser (1911). The British Chess Magazine, vol.31, December, p.465. Monogram tours showing MD and ECC (for Edinburgh Chess Club).

H. J. R. Murray (1913). A History of Chess, Oxford (reprinted 1962, 1969) pp.53-5 Rudrata (~900), pp.64-5 Nilakant-ha (~1600 or 1700), pp.169-183 Ch.10, The Arabic and Persian Literature of Chess. pp.335-6 Tours from al-Adli and as-Suli etc. pp.579-582 and 589, King's Library ms and its tours. pp.643-8 and 674 the Civis Bononiae group of mss and tour. p.727 and 730 the Chachi ms and tour. [My own copy, now in BCPS Library].

A. C. White (1913). Sam Loyd and his Chess Problems 1913.

T. R. Dawson (1913). Caissa's Playthings, Cheltenham Examiner 1913.

A. C. Pearson (1915). Twentieth Century Standard Puzzle Book revised? 1915. See 1907.

E. Bergholt (1916). Some Original Knight's Tours, Queen, 25 December 1915: Two 8×8 symmetric tours called Arabesque (a symmetrisation of the de Montmort tour) and Hour-Glass, admittedly based on one in Pearson (1907) 1 January 1916: Two 8×8 symmetric tours called St George and St Andrew showing crosses, with discussion of symmetry terminology. 8 January 1916: Two tours based on Vandermonde and two others, all showing maximum direct quaternary symmetry. 22 January 1916: Two symmetric closed tours of 80-cell cross-shaped boards improving on Dudeney (1910) and one of 84 cells showing oblique quaternary symmetry. 29 January 1916: Four 10×10 tours two showing maximum direct quaternary symmetry and two oblique quaternary symmetry. [Offprint among Murray's papers.]

E. Bergholt (1916). Memoranda on the Knight's Tours, communicated to W. W. Rouse Ball Esq. [These manuscripts, in a retyped format, were preserved among H. J. R. Murray's papers under the heading 'Miscellaneous Articles and Notes by Ernest Bergholt'. The series continues as 'Memoirs' 1918.] First Memorandum: 24 February 1916, deals with terminology for symmetry in tours, direct and oblique, binary and quaternary, and methods of construction, with examples 6×8, 7×7, 8×8, and on 80-cell cross. Second Memorandum: 14 March 1916, gives a scathing review of Volpicelli (1872). Third Memorandum: 30 March 1916, analyses the 6×6 board for quaternary symmetry by his methods, confirming the five solutions found by de Hijo (1882), and uses one, repeated four times, to make a 12×12. Fourth Memorandum: missing, but there is a hand-written note on enumeration of 16-move circuits in quaternary symmetry, classified by pattern of central angles, which may have been omitted because duplicating the work of de Hijo (1882), which he cites in later memoirs. Fifth Memorandum: 1 April 1916, 'domino' method of marking alternate pairs of cells (similar to Sainte-Marie's method on the 4×8 board) and covering each set with a path, then joining the two paths; examples 8×10, 8×8, with methods of closure and symmetrisation. Sixth Memorandum: 22 June 1917. (1) Corrects mistake of Euler (1759) by giving closed tours 3×10 and 3×12. (2) Introduces new species of diametral symmetry, i.e. 'Bergholtian symmetry' (so named by Murray presumably, though possibly by himself), with examples 5×8, 6×7 etc. (3) Criticises Jaenisch. Elaborates his methods for symmetry on 8×8, with example by Käfer (1842). Results on piece-wise symmetric tours such as F.P.H (1825). (4) Supplementary note, dated 27 August 1917, on spelling of Col(l)ini.

H. J. R. Murray (1917). The Knight's Tour on the Half-Chessboard, The British Chess Magazine, vol.37, pp.305-9, 355 and 392 Revisits his article of 1911, but his totals are still incorrect. Gives four partially magic tours formed by joining half-board tours. Gives an interesting cryptotour however. p.355 corrects misprints. p.392 has letter from E. Bergholt citing Sainte-Marie and Laquière and reply from Murray. His total 7772 now agrees with that of Sainte-Marie. But disagrees (wrongly in my view) with Laquière.

J. F. Magee with A. C. White and W. Mason (1917). The Good Companions Chess Problem Club Folders. vol.4, nr.9, March. pp.134-5 and 140. (by ACW?) The first Euler tour is quoted as having been used by the operator of the Automaton (see Willis 1821 and Tomlinson 1845). Reproduces a photo of a template for use in the machine showing this tour (this was in the library of George Allen, sold to the Library Co of Philadelphia – information from Oxford Companion to Chess). [In 1999 I wrote to The Library Company (1314 Locust Street, Philadelphia, Pennsylvania 19107-5698 USA) about this item and received the reply: “Unfortunately, and very embarrassingly for us, we have been unable to put our hands on the chart itself. Two members of the staff remember handling the chart about a year ago in order to have it photographed, but that is the last time anyone saw it. In our very small experience with situations like this, the object has always turned up eventually. ... We do have a negative of it and could create reproductions (photographs, slides, or scans) from that.” Signed: Jessy Randall.] pp.136–9 (by ACW?) Reproduces photos of tours in the ms Abdul Hamid 1140 (= Baghdadi 1041) plus two tours by Euler and a magic tour (12a). pp.207–8. (by W. Mason) Describes a mnemonic for memorising a knight's tour. The ranks are named un, oo, ee, or, iv, ix, en, et and the files MLKHGFDB. This provides monosyllabic names for the 64 squares which can be memorised, in the order of a tour, as a piece of nonsensical doggerel. This appears to be the method described by George Walker in Fraser's Magazine 1840. [BCPS Library]

T. R. Dawson and C. D. Locock (1917). Some Puzzles 16: A Personally conducted tour Pittsburgh Gazette-Times 19 August 1917. Black knight g8, moving only to check, and barred from 20 squares, is forced by moves of the white king to make a tour of 44 squares. [Dawson ms in BCPS Library]

H. E. Dudeney (1917). Amusements in Mathematics (Thomas nelson and Sons Ltd, London). (a) Rook or wazir tours: #242 pp.69, 198, 3×4 unicursal in 19 moves. #244, pp.70, 198, 8×8 maximum length unicursal 16 moves, 70 units. #247, pp.71, 199, 4×5 unicursal in 36 or 35 moves. #250, pp.72–3, 200, 4×6 with trick of immediate return to visit the start square. #253, pp.73–4, 201, formula (m+n)!/m!n! for number of shortest wazir paths corner to corner of m×n rectangle. #320-324, pp.96–8, 223–5, 8×8, 16-move closed, 21 open, 57 longest, etc. (b) Bishop, Queen etc: #325, pp.98, 225, bishop 17 moves. #328–333, pp.98–101, 225–7, Queen 14 moves, citing Loyd 1868, two using Nightrider line, 12-move tour on 7×7, longest Q 5 moves from d1 without intersection: d1-h1-a8-h8-h2-c7 = 17 + 12 root 2 = 33.9705 whch is slightly longer than d1-h1-h8-a1-a8-g8 = 24 + 7 root 2 = 33.8995. (c) Knight: #243, pp.79, 198, and #339, pp.103, 229, these problems both use the unique quarterboard tour that solves Ala'addin's conundrum. #334–341, pp.101–3, 227–230, two 7×7 tours, 12 corner-to-corner 4×5 tours, two equal circuits on 4×8, compartmental tour, tour of faces of 8×8×8 cube, the move from face to face being made as if the two formed a single 8×16 board (see Eschwege 1896), and the Guarini problem.

W. Ahrens (1918). Mathematische Unterhaltungen und Spiele (Teubner, Leipzig). 'has dealt with various designations of Panmagic squares ... paragraph 3: Pandiagonale magische Quadrate pp.40-41, Note 2' 'He has condemned the term carrés diaboliques, preferred by Ed. Lucas 1882 and adopted by G. Arnoux, B. Portier and other authors, in our days by Martin Gardner in Mathematical Puzzles and Diversions.' [P. Bidev 1986]

E. Bergholt (1918). Knight's Tours, The British Chess Magazine. A series of letters pp.7–8, 48, 74, 104, 195, 262. (1) His idea of 'perfect' quaternary symmetry on the 8×8 board, formed of a mixture of sets of four moves in direct and oblique quaternary symmetry. Presented as problems to complete tours including given central moves. Five example solutions. (2) Example tour with two three-move lines. The problem of a symmetric tour with maximum three-move lines. Solution with eight such lines. (3) The existence of closed tours on 3-rank boards, contrary to a statement of Euler, with examples 3×10and 3×12 (as in his 1916 sixth memorandum). (4) Binary symmetry on 7×8 board (a further example to those given in the 6th memorandum).

E. Bergholt (1918). Memoirs on the Knight's Tours, communicated to Mr H. J. R. Murray. These continue the series of Memoranda previously sent to W. W. Rouse Ball. These memoirs develop Bergholt's idea of 'Complete' or 'Perfect' Quaternary Symmetry, (termed by Murray 'Mixed' QS) on the 8×8 and 12×12 boards, where true quaternary symmetry is not possible. Seventh Memoir: 16 i 1918 Method of Terminal Loops on 12×12 board and 8×8 with examples from J. Brede (1844). Addendum 22 i 1918, Further example from Brede, and on 88-cell shaped board. Eighth Memoir: 23 ii 1918. Method of Terminal Loops with Symmetrical Nodes. Generates 14 tours on 8×8. Ninth Memoir: 7 iii 1918. Coalescing Loops. Explained by 12×12 examples. Generates 78 tours 8×8. These memoirs were reproduced in Games and Puzzles Journal issue 18, see 2001.

G. L. Moore (1918). Demonstration that there are only two possible diametral knight's tours with eight three-move lines in each. 18 April 1918. [Five-page manuscript among Murray's papers.]

B.B., J. W. Brown and A. M. B. Irwin (1919-1920). The Knight's Tour Notes & Queries (series 12). Vol.5 (1919): Initial query (by BB) April pp.92-93. Reply (by JWB) May pp.136-7 and December p.325. Vol.6 (1920): Reply (by AMBI) April pp.91-93. [Richmond 1975]

« 1920s

G. L. Moore (1920). The Knight's Tour, [Thirty-page manuscript among Murray's papers.] (1) History and Euler's method. (2) Discussion of types of symmetry, including 'combined' examples by Bergholt from BCM. (3) Closed tours on 3-rank boards, including the four symmetric tours 3×10 and one example 3×14 with direct symmetry. (4) Impossibility of closed tours on 4-rank boards. (5) Tours 8×8 with three-move lines in given positions. (6) The five quatersymmetric tours 6×6. (7) Quatersymmetric tours on 8×8 omitting four squares in all nine possible positions. (8) Investigations of maximum symmetry in tours 8×8 and on 80-cell cross. Supplement dated 4 xii 1920 (21 pages, mainly diagrams) seeks to enumerate symmetric tours with six three-move lines.

U. Papa (1920). Il problema del cavallo degli scacchi La Scienza Per Tutti (fortnightly review) 15 August: 27(16) 241-244. [see Papa (1922)]

I. Ghersi (1921). Matematica Dilettevole e Curiosa (2nd edition, Ulrico Hoepli, Milan). pp.64-66, Puzzle of shortest path between given points, via certain intermediate points in any order. p.81 Diagram of 8×8 H-shaped pseudotour. pp.74-85 Knight's tours. p.319 The 3×3 magic square as a tour. p.320 (Figs 261 and 265) Magic king tour. p.321 Fig.262 step-sidestep magic square tour of (0,1), (1,1), (1,4) moves. [British Library]

S. R. Naidu (1922). Feats in Chess, published in India. Ch.2, with two plates, contains 100 diagrams of tours on the 8×8 or portions thereof, also the 12×12 and a T-shape of 80 cells. The magic tours diagrammed are 00b and 27b (in three versions). [Murray 1930 and 1955 §126] “It had a brief mention in the BCM at the time. There is a copy in the Edwin Gardiner collection (British Library). Purely knight's tours - very off-beat.” [D. Pritchard 1992] Many of the results are the same as in Harikrishna (1871).

U. Papa with H. R. Bigelow, T. R. Dawson, F. Douglas, H. A. Adamson and O. T. Blathy (1922-23). The Problem of the Knight's Tour, Chess Amateur, Half Hours section. Translated by H. R. Bigelow from Papa (1920). Edited and supplemented by T. R. Dawson. Part 1, July, pp.315–6. §1-2 Historical sketch. §3–5 Discusses counting of tours, taking account of symmetry, on square and rectangular boards of any size. §6 N-cell open tours with diametral symmetry have constant sum N + 1, with 5×7example. Part 2, August, pp.345–7. §7 Closed diametral tours have of N cells have constant difference N/2, example 6×6. §8 Closed tours with double diametral symmetry. Exactly five on 6×6, two diagrammed. §9 Closed tours with central diametral symmetry (= Bergholtian), add to N + 1, example 5×8. §10 Closed tours with axial symmetry, example 5×6. §11 Bipartite tours formed of two equal components. Open and closed examples 6×7. §12 Magic tours, example (12a). §13 Euler's 20-cell cross tour. §14 Mentions transforming the board into a cylinder or sphere, and Vandermonde's 3D tour 4×4×4. Part 3, September, pp.378–9. §15–16 Methods of enumerating tours without duplications, taking account of symmetry, in terms of primitive squares from which the tours must start. §17–19 Analysis of tours on 4-rank boards, but results are not accurate. Part 4, October, pp.22–23. §20–23 Tours on small boards continued, all rectangles with both dimensions less than 9 considered. The results are unfortunately not correct. §24 Dawson notes two estimates for the number of tours on the 8×8. (i) 168C63 attributed to Dudeney (actually in Jaenisch 1862) approximated as < 122×(10^45). (ii) F. Douglas estimate of (2^6)(3^20)(5^8)(7^8) approximated as < 210×(10^28). Part 5, June 1923, p.287. §24 continued. H. A. Adamson says Douglas count ought to be doubled, also further refines it to (2^22)(3^24)(5^8)(7^6)×2. §25 O. T. Blathy by a statistical argument arrives at the much lower estimate of 6×(10^11). [BCPS Library and K.Whyld]

C. T. Blanshard (1923). A Knight's Tour, Chess Amateur, August, p.349. Gives a tour he calls the 'four-loop tour' said to have been found by him before the war, i.e. in 1913. This is in effect formed by simple linking of the unique pseudotour formed of four non-intersecting circuits (Ala'addin 1400, de Hijo 1882). [BCPS Library]

L. Euler (editors?), Opera Omnia 1923, (1) 7, 26-56. Includes reprint of Euler (1759).

R. C. Temple (1923) and L. R. R. Iyer. (1923). The Indian Knight's Tour at Chess Indian Antiquary Letter by Temple no.52 December 1923 pp.351-354. Reply by Iyer no.53 December 1924 pp.262-264. [Richmond 1975]

H. Schubert and F. Fitting (1924). Mathematische Mussestunden, Fourth edition, edited by Fitting. §24 pp.201–214 Hamiltonsche Rundreisen. §25 pp.215–245. Rösselsprüunge. This has a new section quoting a figure of 10,298 [too high, see Duby 1964] for all tours on the 6×6 board, and describing the method of enumeration in relation to a board of 22 cells 4×5 with squares added at the middle of the odd sides. [D. Singmaster] Further editions: 6th 1940, 10th 1943, 11th 1953.

M. Hogrefe (1924). Weser Zeitung 13 July 1924. The first tour with ten three-unit lines (Parmentier had previously achieved nine). [Problemist Fairy Chess Supplement 1932]

(1924). Burmese Chess Guide. Contains tours in Burmese numeration, 129 tours 8×8, those I have examined being asymmetrical, two 6×6, two on 6×6 with corners removed, eight on 4×4 with corners removed, one on 3×8, and some other diagrams with non-numerical symbols. It is unclear if this is original work. It may be derived from Volpicelli 1872. [D. Pritchard 1990]

R. Inwards (1925?). Chess, A Knight's Tours Two albums with hand-drawn diagrams of symmetrical 8×8 knight tours, undated, accompanied by a letter from T. R. Dawson. [K 4063]

A. Sharp (1925). Linaludo: The Knight's Tours, A Fascinating Game of Patience (E. Marlborough & Co Ltd, London). In the 'Introductory' and in an advertisement at the end there is reference to a companion work 'Handbook of Linaludo: The Knight's Tours' said to contain over 250 diagrams (this may mean blanks). Murray (1930) wrote: “This is an attempt to popularise the drawing of knight's tours, in which the writer adopts a vocabulary of his own. He proceeds by the method of trial. In drawing tours on a basis of symmetry, he generally aims at filling his diagram by a number of complete reentrant chains, rather than by one complete tour with a maximum of the desired symmetry.” I find more of interest in this work than Murray, since Sharp's terminology of ‘degrees of symmetry’ (single symmetry = binary, double symmetry = quaternary, treble symmetry = octonary) coincides with my own use in Chessics 22 (1985); also his ‘canonical forms’ correspond to my ‘pseudotours’, and he expounds the principle of the ‘linkage polygon’ of deleted and inserted moves when linking up component paths. This is a procedure I have often found practical in constructing tours myself, or converting pseudotours to true tours. He also pioneers the covering of large areas by repetitive patterns of knight's moves. [BCPS Library] [Cleveland Library also has another item with the subtitle Sketch Book.]

E. Lancel (1925). L'Echiquier (Bruxelles), Chess magazine edited by Lancel. (a) #2 February p.44: Magic knight's tour attributed to Mansion (actually by Wenzelides). (b) #4 April pp.83-84 (solutions #7 July p.151): knight tour by M. A. Errera in two shaped halves, not 4×8. Euler's 10×10 tour. Rook tour. L-shaped cryptotour of three 5×5s, one letter per square. [J. Beasley]

M. Kraitchik (1926). Le Problème du Cavalier L'Echiquier (Bruxelles). A series of articles. The author is not stated but the work is identical to Kraitchik (1927). #15 March pp.57-59, §1 Statement of the problem. §2 History. Diagrams of the de Montmort (dated 1708) and Moivre (1722) tours. §3 On number of solutions. §4 Open and closed tours. §5 Symmetry. §6 Euler's cross-shaped tours. §7 Symmetry with respect to a median. Examples 8×8 approximately symmmetric, 8×8–2, 5×6, 9×10. §8 Quaternary symmetry. Examples 6×6, 7×7–1, 8×8–4. #16 April pp.78-80, Ch.2 Méthodes générales. §9-12 Euler's method. §13 Warnsdorf. #17 May pp.98-100, §14-16 Warnsdorf continued and tourability of shaped boards. #18 June pp.123-124, Ch.3 Methods applicable to 8×8. §17 Vandermonde. §18 Collini. §19 Squares and Diamonds and Roget. #19 July pp.147-148, Ch.4 Quelques parcours remarquables. §20 Compartmental tours. §21 Two half-boards. #20 August pp.166-168, §22 Graphical examples. §23 Half of tour covering a regular pattern of cells. Ch.5 Conditions graphiques. §24 Graphical representation. §25 Tours with two-move lines. §26 Tours with 3-move lines. §27 Tours with maximum moves in given directions. §28 Equipartition of directions. §29 Tours showing crosses, letters etc. §30 Four stars in corners. #21 September(?) Ch.6 Conditions numériques. §31 Relation of numbering and symmetry. §32-34 Magic conditions. §35 On board of side 4k + 2. Semimagic 6×6 example. §36 On board of side 4k. Two 8×8 examples. §37 Jaenisch. #22 October pp.212-215, Ch.7 Boards 4×N. §38-50. Extensive enumerations based on Sainte-Marie (1877) results. #23 November pp.237-240, Ch.8 Généralisations. §51-53 Some general rules. §54 Boards 3×N. §55-57 Board 5×5. #24 December pp.261-263. §58 Board 6×6, gives the 17 singly symmetric cases. §59 Symmetric open and compartmental tours 7×7. §60-62 Larger boards. [J. Beasley]

M. Kraitchik (1927). Le Problème du Cavalier (cover has: Bruxelles, Edité par l'Echiquier, but title page says: Paris, Gauthier-Villars et Cie). Much of this book is reproduced from l'Echiquier 1926 but in smaller format. The book adds a preface, three further chapters and some extra paragraphs. There are now 81 sections. Ch.9 Généralisation de la marche du Cavalier. §63 Definition of sauteur (a, b). §64 Tours by (3, 1), seven examples. §65 Numerical property of (3, 1) tours; all cells in same row are of same parity. §66 Two open tours and one closed on 10×10 by (3, 2). §67 Open tour 9×9 and closed quatersymmetric tour 10×10 by (4, 1). §68 Fixed distance leapers can have two types of move, example (7, 4) and (8,1). §69 Closed 8×8 tour by (3, 4) + (5, 0). §70 Closed tour by (7,1) + (5,5) on white cells of 12×12. Ch.10 Parcours doublement symétriques. §71 Definition = oblique quaternary symmetry. §72 Every such tour covers 4N cells, is closed, and cells corresponding by quarter-turn have constant difference of N. §73 Odd order squares. Tours on 7×7 minus centre, 23 diagrams. Examples of side 9, 11, 13, 15. §74 Side 4k + 2. Examples 10, 14. §75 Side 4k. Examples 8, 12, 16 minus four centre cells. Ch.11 Board 3×N. §76-81. Enumerations up to 3×14.

(1927?). Le Problème du Cavalier, Cahiers de l'Echiquier Francaise, pp.166-168, 256, 388-391. Historical discussion. Monogram tour CEF by T. R. Dawson. Cryptotour from Le Palamède 1842. 'Polygraphie du Cavalier' lettered tour problem, solved by a quotation from Barbey d'Aurevilly. Alternative CEF monogram tour by H. Rohr. Further historical notes and two magic tours from Jaenisch (1862).

J. Kürschak (1926); [Nash-Williams (1959)]

F. Rixey (1927); Jumblegrams. Cryptotours and paths on small boards. [D. E. Knuth]

T. R. Dawson and F. Douglas (1928); Chess Amateur problem 1259; path by Nightriderhopper over rook.

J. G. White (1928); bequeathed his Chess Collection to Cleveland Library, Ohio.

S. Vatriquant (1929); L'Echiquier various items including knight tours on cylinders.

« 1930s

M. Kraitchik (1930); La Mathématique des Jeux Brussels, pp.360, 402, on number of tours. [Ball and Coxeter 1939]

H. J. R. Murray (~1930); ms Early History of the Knight's Tour (the ms is undated but in a similar format to his 1942 ms). The ‘Early History’ section is short, interspersed with and followed by diagrams and reviews. In chronological order they are: Persian ms (~1800), Turkish ms (~1850), Journal of Science (1817), Ciccolini (1836), Glaszer (1841), Wenzelides in Schachzeitung (1858), Laquière (1880), Vyasa (1884), Miles (1884), Syamakisora (1885), Falkener (1892), Tope (1893), Eschwege (1896), Babu (1901), naidu 919220, Sharp (1925). the Ciccolini review also includes a note on Murray's own work on combination of 5×5 tours to form 10×10, dated 2 January 1917. The ms concludes with two versions of an extensive bibliography. [Bodleian Library, Oxford]

G. Kowalewski (1930); Alte und Neue math. Spiele .... [?]

W. Pauly (1930); L'Echiquier December. Longest nonintersecting closed knight path on 8×8 mentioned but not diagramed. [Diagram in Murray (1955)]

T. R. Dawson and many others; The Problemist Fairy Chess Supplement began publication in August 1930, changed its name in August 1936 to The Fairy Chess Review and continued to appear until April 1958. Each of its 9 volumes, except the last, consists of 18 issues (3 years' issues, appearing every other month: August, October, December, February, April, June). PFCS/FCR was edited by T. R. Dawson to Vol.8 #1 August 1951, then by C. E. Kemp for one issue February 1952, then by D. Nixon to Vol.9 #9 April 1956, and by C. E. Kemp again to Vol.9 #21 April 1958. Tours first appear in February 1932. It not only includes many original results but also cites references to other obscure sources that would have been lost. In many cases the tour was stated as a problem and the solution published in a later issue, consequently it is often necesary to see both entries to understand the tour. [BCPS Library].

M. Kraitchik (1931); Sphinx (editor 1931-39). A periodical on recreational mathematics. [Preface to Kraitchik (1953)].

F. Cassani (1931); Die Schwalbe August. Grasshopper over pawns maximumming tour. [Chessics 15 (1983)]

(1931); Zurcher Illustrierte, 27 May 1931. A 10×10 tour. [Murray (1955)]

S. Hurst (1931); The Governor's Secretary ... 1931. [?]

R. E. Ingram (1931); see Hamilton 1856.

T. R. Dawson and D. R. Dawson (1932); Evening News (London) 24 December. Figured tour with the double triangular numbers (2, 6, 12, 20, 30, 42) in symmetrical array. [Dawson ms, BCPS Library]

T. R. Dawson, H. A. Adamson, G. Fuhlendorf, P. C. Taylor, A. H. Haddy (1932) PFCS Knight's tours with square numbers in a knight chain. Problems 329 (TRD), 376 (HAA), 377 (TRD), 378 (GF), 379 (PCT), 380-381 (TRD, these also have the cubes in a chain), 448 (PCT), 449 (AHH), 450-451 (TRD), 490-491 (AHH), 492-493 (TRD), 564-565 (AHH), 566-567 (TRD), 591 (PCT), 605-606 (AHH), 607-608 (TRD).

T. R. Dawson and V. Onitiu (1932) PFCS, Knight's tours with ten 3-move lines. Problems 330 (TRD), 452-458 (VO), 459 (TRD).

F. Douglas, W. E. Lester, E. J. van den Berg, F. Hansson (1932) PFCS Tours by other movers: (3,4)-mover problem 403 (FD), root-85-leaper problem 404 (WEL) and (EJV), rook + bishop + knight problem 501 (FH), camel problem 535 (FH).

M. B. Lehmann (1932); Der Geometrische Aufbau Gleichsummiger Zahlenfiguren (Schellenberg, Wiesbaden), pp.301-362 Magische Rösselsprung: p.309 four pseudotours, pp.317-339 catalogue of all known magic knight tours. pp.340-348: 13 diagonally magic two-knight tours. pp.349-359 work of E. Lange: tours on boards of sides 12, 16, 24. pp.360-361 diagonally magic king tours. p.362 magic bishop tour (two links not diagonal).

E. Lange (1932); in Lehmann (1932) pp.349-359: (non-diagonal) magic tours boards of sides 12, 16, 24.

E. Lange (1932-3); Sphinx tours with maximum moves in one direction and in four directions.

T. R. Dawson, A. H. Haddy, P. C. Taylor, F. Dignal, A. Kniest (1933) PFCS, Knight's tours with square numbers in a knight or giraffe chain. Problems 658-659 (AHH), 660-661 (TRD), 716 (TRD), 717-719 (PCT), 907-908 (FD), 909 (AK), 910 (TRD), 980 (FD), 981 (TRD), 1051-52 (FD), 1053-54 (TRD).

F. Hansson and T. R. Dawson (1933) PFCS Camel tours on 8x8 normal and cylinder boards. problems 714-715 (FH), June p.125: representation of camel tours by knight tours by 45 degree turn (TRD).

T. R. Dawson (1933); Evening Standard (London) 10 August, shortest rook AP circuit [= PFCS 985a].

W. Jacobs, T. R. Dawson, D. H. Hersom, H. Perkins (1933) PFCS Tours with move-lengths in arithmetical progression. Problem 736 (WJ), 985-986 (TRD), 987-991 (DHH), 1061 (TRD), 1062-63 (WJ), 1064 (HP).

M. B. Lehmann (1933); Sphinx, a new 8×8 magic knight tour.

(1933); Zurcher Illustrierte, 7 July 1933. [Murray (1955)]

F. Dignal, T. R. Dawson (1934); PFCS, Knight's tours with square numbers in a knight chain. Problems 1132-35 (FD), 1303-06, 1449-52, 1525-28, 1593-96 (all TRD).

T. R. Dawson, G. Fuhlendorf, H. Perkins, W. H. Reilly, P. C. Taylor (1934); PFCS Tours with move-lengths in arithmetical progression. Problems 1200-05 (TRD), 1206 (GF, HP, WHR and PCT), 1207 (WHR).

M. Kraitchik (editor) (1935); Comptes Rendus du Premier Congres International de Récr7#233;ation Mathématique (Brussells, Libraire du Sphinx 1935, 131 pages). reviewed in PFCS 2(16)2 p.174 (February 1936): “... contains an account of the Congress, group photographs, and the full text of 19 papers read at the meetings ...”.

T. R. Dawson (1935) Comptes Rendus du Premier Congres International de Récr7#233;ation Mathématique (Brussells, Libraire du Sphinx) Knight's tours with numerical conditions. They show, in 8×8 knight tours: AP with cd = 9 along one rank, two APs with cd = 7 in a 3×3 area, multiples of 8 on diagonal, squares in two 8queens arrangements, squares and cubes in knight chains, six tours with triangular numbers in triangles.

E. Huber-Stockar (1935) Comptes Rendus du Premier Congres International de Récr7#233;ation Mathématique (Brussells, Libraire du Sphinx) Remarkable study on partially generalised knights and their tours. That is tours by compound pieces with moves in given directions.

L. Okunev (1935); Kombinatornye Zadachi ia Schachmatno ... 1935. [?]

T. R. Dawson, S. H. Hall (1935) PFCS Knight's tours with square numbers in a knight chain, or other formation. Problems 1674 -77 (TRD), 1704 (SHH, squares in a circle), 1705 (TRD, squares in a circle), 1813-1816, 1834-37, 1917-20 (all TRD).

T. R. Dawson and S. H. Hall (1936) PFCS Knight's tours with other numbers in formation. Problem 2178 (multiples of 7 in square), 2179 (odd primes in rectangle).

H. J. R. Murray (1936) Beverley's Magic Knight-Tour and its Plan PFCS/FCR February 1936, p.166. Problems 2107-09. April p.177 2239-2240, June p.187 2286-87. (Here PFCS changed to FCR) August p.3 2350-51, October p.18 2466-67, December p.29 2441-42. also: February 1937 p.41 Problem 2636.

T. R. Dawson (1936) PFCS/FCR Knight's tours with square numbers in a knight chain, Problems 2288-89. (PFCS changed to FCR August 1936) Problems 2352-53, 2468-69, 2544-45.

T. R. Dawson (1937) /FCR Knight's tours with square numbers in a knight chain, Problems 2637-39, 2702-2705, 2785-2788, 2869-2872.

G. E. McGuffey (1937) /FCR Cryptotour. October p.86 Problem 2932.

G. Fuhlendorf, S. H. Hall (1937) /FCR Knight's tours with square numbers in line. 2933 (GF), 2934-35, 3037-38 (SHH).

W. E. Lester, (1937) /FCR Rook + knight tours with square numbers in line without intersection. Problems 2930-31, 3035-36.

— Vazsonyi (1938); [cited by Nash-Williams (1959)].

S. H. Hall, T. R. Dawson (1938) FCR February p.110. Grasshopper over other piece tours. Problems 3106-3107, 3178 (SHH over knight), 3179 (TRD - over rook).

F. Hansson, T. R. Dawson (1938) FCR Knight's tour with squares in a knight's chain. Problem 3108, 3180, 3251 (FHH - all these also with cubes in chain), 3324, 3388, 3465 (TRD).

W. E. Lester (1938) FCR Bishop + knight tour with squares in diagonal. Problem 3109.

P. C. Taylor (1938) FCR Monogram tours. Problem 3181 (spells TRD), 3252 (spells CMF), 3325 (spells HAA), 3389 (spells FRA), 3466 (spells WHR).

S. H. Hall (1938) FCR Knight's tour with multiples of 7 in 3x3 array. Problem 3249.

W. E. Lester (1938) FCR Rook tour with 56 turns. Problem 3250.

F. Hansson (1938), FCR Camel tour determined by W of moves. Problem 3322.

W. E. Lester, O. E. Vinje, S. H. Hall (1938) FCR Fiveleaper tours. Example in text August p.140 (OEV), Problem 3323 (WEL), 3463 (SHH).

W. E. Lester (1938) FCR Camel + knight partial tours on 4 by 4. problem 3387.

E. Huber-Stockar (1938) FCR Knight + fers tour quaternary. Problem 3464.

W. W. Rouse Ball and H. S. M. Coxeter (1939); Mathematical Recreations and Essays (Macmillan and Co, London), edited by Coxeter (Rouse Ball died 1925): 11th edition 1939, (and 12th 1974, 13th 1987) [see 1892 for earlier editions]. pp.161-192 deal with Chessboard Recreations including knight's tours. pp.193-221 deal with Magic Squares.

D. R. Dawson (1939) FCR Lettered knight tour. Problem 3586 (spelling THANK YOU), 4131 (DDD).

S. H. Hall, E. Huber-Stockar F. Hansson (1939) FCR Knight + camel tours. Problems 3587 (SHH - on 6×6), 3588 (SHH - on 6×8), 3589 (EHS - on 7×5), 3762 (SHH - compartmental on 8×8), 3936 (FH) - 8×8 with 2 camel moves, vertical axis, 3937 (EHS - 8×8 quaternary).

P. C. Taylor, A. Lapierre (1939) FCR Monogram tours. Problem 3590 (PCT - spells HAR), 2591-92 (AL - spell FCR, TRD), 3763 (AL - spells AL), 3764 (PCT - spells ACW).

P. C. Taylor (1939) FCR 5x5x5 knight tour with squares in a knight chain and cubes in a row on the midle plane. Problem 3928.

E. Huber-Stockar (1939) FCR Giraffe tour. Problem 3938 (32-move journey on 8×8 that can be repeated to form 8×16 tour).

O. E. Vinje (1939) FCR Fiveleaper tour 8×8 axial symmetry. Problem 3939. Closed 8×8 tour of (2,3)-rider. Problem 3946.

H. J. R. Murray (1939) FCR Magic tours. Problems 4132-4134.

V. Onitiu (1939) FCR Knight's symmetric tour with squares in knight chain. Problem 4135 “VO notes that he examined 144 dispositions of the squares, all that are possible for diametral symmetry, and the above is the only case leading to a tour. Moreover every move of the tour is determined, so that the tour in UNIQUE in all the millons possible.”

« 1940s

W. H. Cozens (1940); Cyclically symmetric knight's tours, Mathematical Gazette, December pp.315-323. The 6×6 and 10×10 tours. [FCR] See also 1956.

W. H. Cozens (1940) FCR Closed tour of 204-cell board in shape TRD. Problem 4471.

E. Huber-Stockar (1940) FCR Monogram tour. Problem 4528 (spells TRD).

H. J. R. Murray (1940) FCR Magic tour, Problem 4556.

H. H. Cross (1941) FCR 10x10 zebra tour. Problem 4709.

E. Huber-Stockar, R. J. French, S. H. Hall (1941) FCR 5x5 and 10x10 knight + camel tours: Problem 4710 (EHS). Two centrosymmetric 4x4 tours by dabbaba + knight: Problem 4772 (EHS). Three axisymmetric 4x4 tours by dabbaba + knight: Problem 4842 (RJF, SHH). Centrosymmetric 4x4 tour by knight + 3-leaper: Problem 4892 (EHS).

O. E. Vinje (1941) FCR Knight tour of 50 squares unguarded by knights f2,g6: Problem 4773.

T. R. Dawson (1941); British Chess Magazine January. Figured tour with pentagonal numbers in pentagon. [BCPS Library]

E. Huber-Stockar (1942) FCR Centrosymmetric 6x8 Camel + Knight tour. Problem 5085.

H. J. R. Murray (1942) FCR Vol.5, Issue 1, August. Magic Knight's Tour 16x16 by a new extended squares and diamonds method. Problem 5226.

T. R. Dawson (1942); British Chess Magazine January. Figured tour with squares and cubes in knight chains. [BCPS Library]

M. Kraitchik (1942); Mathematical Recreations (W. W. Norton Co), first edition. See also 1949, 1953.

H. J. R. Murray (1942); ms The Knight's Problem Preface: “This sumarises the results of over 30 years' research on the knight's tours, in which I have been greatly assisted by John Keeble, Ernest Bergholt, G. L. Moore and especialy by T. R. Dawson who allowed me to use the valuable material he had collected and lent me many books which otherwise would have been inaccessible to me.” The chapters are: I Introductory, II Chains, III Symmetry, IV Construction (this includes an exposition of his Theory of Slants), V The Classification of Tours, VI Some Chessboard Problems, VII Chessboard tours described by two or more knights concurrently, VIII Enumeration of Chessboard Tours, IX Some Numerical Problems, X-XIV Magic Tours (Beverley, Method of Quartes, Tours with Chains, Larger Boards, Two-Chain Magic Tours), XV Compound Tours, XVI-XX Tours on Boards ... (Enumeration or construction of tours on narrow boards, then on boards with an odd side, then side 4n + 2, then side 4n, and finally boards with omission of cells), XXI Mathematical. Appendix: Magic Tours on the Chessboard (I have not seen this, it was presumably a catalogue, see Murray (1951)). [deposited 1991 at Bodleian Library, Oxford].

T. R. Dawson (1943); British Chess Magazine January. Figured tour with squares in eight-queen formation. [BCPS Library]

F. Schuh (1943); Wonderlijke Problemen; Leerzaam Tijdverdrijf Door Puzzle en Spel. (W. J. Thieme & Cie, Zutphen). See also 1968.

N. M. Gibbins with E. Huber-Stockar (1944); Chess in 3 and 4 dimensions, Mathematical Gazette, May 1944, pp.46-50. Gives the smallest lattice for a knight's tour in space chess as 3×3×4 and credits EHS of Geneva. [D. Pritchard, and cited by Kumar 2009]

J. C. McCoy (1946); Magic knight's tour Scripta Mathematica #12, March, pp.79-86. [Richmond 1975]

H. J. R. Murray (1949); The Classification of Knight's Tours, British Chess Magazine, December, pp.397-400. His 'straights' and 'slants' terminology.

« 1950s

G. Bain (1951); Celtic Art, The Methods of Construction.

H. J. R. Murray (1951); ms The Magic Knight's Tours, a Mathematical Recreation 1951; Preface: “My original intention in writing this book was to celebrate the centenary of the composition of the first magic knight's tour y William beverley and its publication in 1848. This however proved to be impossible although the book was completed in the early years of the world war.” Chapters: I Definitions, II History, III Construction - Beverley's use of contraparallel chains, IV Construction - Quartes, V Construction - Chains, VI Magic Knight's Tours on the Chessboard, VII Magic Tours on Larger Boards - Introductory, VIII Magic Tours on the 12x12 Board, IX Magic Tours on the 16x16 Board, X Magic Two-Knight's Tours. Chapter VI, much the longest, is subdivided: Matrices, Arithmetical tours, Geometrical tours, Arithmetical properties, Geometrical properties, Diagonal sums, Semi-diagonal sums, Index. [deposited 1991 at Bodleian Library, Oxford]

M. Kraitchik; Mathematical Recreations (Dover Publications, New York) 2nd edition (revised). See 1943. Chapter 11 "The Problem of the Knight", pp.257-266. Summarises results given in his 1927 work, with a few corrections. No references given.

H. J. R. Murray (1955); Following Murray's death in 1955, many of his unpublished manuscipts were deposited in the Bodleian Library, Oxford. These are mainly earlier drafts of is 1942 and 1951 mss, together with other notes and many diagrams of knight's tours. It may well repay further study. 100: bibliographical data. 101-104: Original Investigations into the Knight's Tour, begun in 1911. Half-boards, 5×5 board, straights and slants, etc. 105: Ch.18 Mixed Quaternary Symmetry. 106: Ch.19-34 Magic tours. 107: Notes on the Knight's Problem (1909-1943). 108:Noes on the Knight's Problem, Vol.2. Deals with 3×n, 4×n, 5×5 to 5×10, 6×6 to 6×10, 10×10. 109-125: Rough notes. 126: Bibliography and Critique. Reviews of Slyvons, Jaenisch, Mercklein, Haldeman, Volpicelli, Lapierre, de Hijo, Parmentier, Falkener, Cubison, Rilly, Bergholt, Papa, Naidu, Blathy, Hogrefe, Sharp. [Oxford shelfmark; MSS H.J.MURRAY 101-126].

(1955); A Catalogue of the Chess Collection in the Royal Library the Hague (Koninklijke Bibiotheek, Bibliotheca van der Linde - Niemeijeriana). Mathematics pp.168-171. Lists 56 titles. [K]

(1955); Catalogue of the Chess Collections (Cleveland Public Library, Ohio) knight's tours pp.333-340. [D. E. Knuth]

H. ApSimon (1956); Knight's rotationally symmetric reentrant tour. Mathematical Notes 2592, Mathematical Gazette p.52. Two quatersymetric solutions of the 6×6 problem.

W. H. Cozens (1956); Mathematical Notes 2761, Mathematical Gazette pp.124-5. In resoponse to ApSimon (1956) Gives diagrams of the five quatersymmetric 6×6 tours.

W. H. Cozens (1957?); Mathematical Notes 2884, Mathematical Gazette p.117. Gives four quatersymmetric 10×10 tours and estimates 200,000 of this type. (See T. W. Marlow 1999).

B. D. Price (1957?); Mathematical Notes 2884, Mathematical Gazette p.288?. Gives a pseudotour of two circuits in direct quaternary symmetry that may be linked symmetrically. says there is one other solution of this type. [L. J. Upton]

C. St.J. A. Nash-Williams (1959); Abelian groups, graphs and generalized knights. Proceedings of the Cambridge Philosophical Society 55, pp.232-238. [cited in Knuth 1994]

R. C. Read (1959); Constructing open knight's tours blindfold!, Eureka #22. Roget's method with refinements. [J. D. Beasley]

« 1960s

E. Albert (1962); Generalized symmetric random walks, Scripta Mathematica 27(2) 185-187. The probability of a piece being on a particular square after a long random walk is proportional to its mobility there. [Offprint from the author]

G. d'Hooghe (1962); Les Secrets du Cavalier, Le Problème d'Euler, Brussels. Attractively printed book. Warnsdorf, Euler and compartmental methods. Reproduces incorrect count of 3×8 tours from Kraitchik. Features a 16×16 magic tour by H. E. de Vasa. Some enumerations. [T. H. Willcocks]

T. H. Willcocks (1962); Magic Knight Tours on Square Boards, Recreational Mathematics Magazine December pp.9-13. History and theory, plus new examples 12×12 by THW and 16×16 diagonally magic by H. E. de Vasa, very similar to that in d'Hooghe (1962) but not quite the same.

D. Luke (1962); Recreational Mathematics Magazine December pp.13-14. Knight's tour on a Moebius board.

J. A. H. Hunter and J. S. Madachy (1963); Mathematical Diversions (Van Nostrand, Princeton) pp.86-87. [cited in Willcocks 1962!]

J. J. Duby (1964); Un algorithme graphique trouvant tous les circuits Hamiltoniene d'un graphe, Etude #8, IBM France, Paris, 22 October 1964. In English with French subtitle and summary. Finds there are 9862 closed knight's tours on the 6×6 board. [D. Singmaster 1987]

P. Bidev (1964-65); FIDE Magazine. #2 1964, #4 1964 and #2 1965. Material concerning magic squares and chess, included in his book of 1986. [W.Korn (1980)]

J. S. Madachy (1966); Mathematics on Vacation (Charles Scribner's Sons, New York). I have not seen this edition, but presume that, like Madachy (1979), it reproduces the 16×16 diagonally magic closed knight tour by H. E. de Vasa given in Willcocks (1962), though without acknowledgement of Wilcocks or de Vasa.

E. Bonsdorff, K. Fabel and O. Rihimaa (1966); Schach und Zahl (Walter Rau Verlag, Dusseldorf). pp.35-39: Die Beweglichkeit der Schachfiguren {Mobility of Chessmen} by K. Fabel. pp.40-50: Wanderungen von Schachfiguren {Travels of the Chessmen} by K. Fabel.

J. Erlebach (1967); Mathematische Mussestunden 13th edition of Schubert.

A. S. M. Dickins (1967); A Guide to Fairy Chess (Q Press, Kew). Generalisation of leaper.

M. Gardner (1967); Problems that are Built on the Knight's Move in Chess. Mathematical Games column. Scientific American 217(4) October pp.128-132, 1967. [cited by Richmond 1975, and by Balof and Watkins 1996]

F. le Lionnais and E. Maget (1967); Dictionnaire des Echecs Polygraphie du Cavalier pp.304-305. Tour given is Warnsdorf example from Bilguer (1843). [P. Wood]

Ira Pohl (1967): A Method for Finding Hamiltonian Paths and Knight's Tours, Stanford Linear Accelerator Center, SLAC-PUB-261, January 1967, 18pp. [PDF from A. Kumar] Warnsdorf's rule generalised using second level tie-breaking; this always yielded a knight’s tour from all squares of the 8 by 8 board. "The generalized rule (for method of order k) is: Consider all paths of k moves and count the remaining number of connections for each path. Select the first move of the path whose number is maximum, providing this path is not a dead end. Ties are broken by going to k + 1 moves. In our case k has been 1."

F. Schuh (1968); The Master Book of Mathematical Recreations (Dover Publications, New York). English edition Schuh (1943). Deals with pseudotours on small boards, and angles in closed knight tours.

S. W. Golomb with L. R. Welch (1968); Of Knights and Cooks and the Game of Checkers Journal of Recreational Mathematics 1 (3) 130-138. Magic tour (00a). Discusses 3×3, 3×4, 3×10, 4×n tours. His cook is our camel, a {1,3}-mover. Ascribes 45 degree knight - camel transform to colleague L. R. Welch (but known to T. R. Dawson 1930s). Gives one camel tour and equivalent knight tour.

W. J. Mannke (1968); A Magic Square Journal of Recreational Mathematics 1(3) 139. An 8x8 diagonally magic square composed of one 32-move and four 8-move king-paths joined by longer leaps.

L. D. Yarbrough (1968); Uncrossed Knight's Tours Journal of Recreational Mathematics 1(3) 140-142. Seeks maximum length nonintersecting paths on all rectangles up to 9×9. His 8×8 example is same as Dawson 1930.

T. H. Willcocks (1968); The Construction of Magic Knight Tours, Journal of Recreational Mathematics 1 (4) 225-233. Gives table of known 8×8 tours with diagonal sums. Describes methods of construction for larger tours and gives examples 12×12, 16×16 and 20×20.

R. E. Ruemmler, D. E. Knuth and M. Matsuda (1969); Letters to the Editor Journal of Recreational Mathematics 2 (3) 154-157. Improvements and extensions to the nonintersecting paths in Yarbrough (1968).

J. A. H. Hunter Construction of Odd-order Diabolic Magic Squares Journal of Recreational Mathematics 2 (3) 175-177. Gives a 7×7 diagonally magic tour by the step-side-step method using camel and knight moves.

D. J. W. Stone (1969); On the Knight's Tour Problem and its Solution by Graph-Theoretic Methods. MSc Thesis. Dept of Computing Science, University of Glasgow, January 1969. Confirms Duby (1964) count of 6×6 tours.

« 1970s

C. Trigg (1970); Knight's tours into non-magic squares Journal of Recreational Mathematics 3 (1) 3-8. [K. Whyld]

J. Stewart (1971); Solid Knight's Tours Journal of Recreational Mathematics 4(1) p.1, January 1971. Shows an 8×8×8 tour by stacking 8×8 tours. [T. W. Marlow]

R. Wieber (1972); Das Schachspiel in der arabischen Literatur von den Anfängen bis zur zweiten Hälfte des 16 Jahrhunderts, Verlag für Orientkunde Dr H. Vorndran, Walldorf-Hessen. [D. Singmaster 1987].

J. Fisher (1973); The Magic of Lewis Carroll (Penguin books) pp.89-91. Various highly symmetric 8×8 tours [L. J. Upton].

R. H. Merson (1973); Letter to editor Games and Puzzles On non-intersecting knight paths. [D. Pritchard]

W. W. Rouse Ball and H. S. M. Coxeter (1974); Mathematical Recreations and Essays (University of Toronto Press) 12th edition: tours section regrettably not revised.

D. A. Betts (1974); Chess, an Annotated Bibliography of Works Published in the English Language 1850 - 1968 (The Chess Player, Nottingham) pp.502-504. Section 41, problems: miscellaneous problems and puzzles. [P. Wood]

P. Richmond (1975); An Annotated International Bibliography of Chess Articles in Non-Chess English-Language Periodicals pp.42-43. Section 15: Knight's tour. Lists 22 titles. [P. Wood]

D. Singmaster (1975); Enumerating unlabelled Hamiltonian circuits. International Series on Numerical Mathematics, #29, Birkhauser, Basel, pp.117-130. Estimate of 10^(23 ± 3) tours of 8×8 board. [Offprint from author].

W. H. Benson and O. Jacoby (1976); New Recreations with Magic Squares. Includes print-out of all 880 magic squares 4×4.

E. Y. Gik (1976); Matematika na Shakhmatnoy Doske Moscow 1976. [K. Whyld]

G. P. Jelliss, T. H. Willcocks and W. H. Cozens (1976-78); The Five Free Leapers Chessics 1 (2) p.2. Conditions for leapers to be able to reach all cells on a board. Smallest square boards tourable by given leapers (by THW). Proof of impossibility of 8×8 giraffe closed tour. 1 (6) pp.4-5 Quaternary zebra and giraffe tours of 10×10 (by WHC). Quaternary (3,4) and (2,5) tours of 14×14 (by THW). Continued 1980.

G. P. Jelliss with J. J. Secker (1976-9); Angles in Knight's Paths Chessics. 1 (1) p.2 Proof that every closed knight tour contains a right angle. Paths of ‘interchessic missiles’. 1 (3) p.5 Proofs that every 8×8 closed knight tour contains (a) an orthogonal angle, (b) an obtuse angle, (c) at least four different angles. 1 (5) pp.4-5 Examples of tours with maximum and minimum nmbers of various angles (one by JJS). 1 (7) p.10 Five tours with angle maxima (four by JJS). 1(8) p.9 Note on two tours by V. Onitiu (1932).

G. P. Jelliss (1978) Tours and Paths Chessics. 1 (5) pp.7-8. Lettered tour spelling KNIGHT TOUR. Figured tour with octuples on diagonal. Symmetric closed fiveleaper tour.

C. M. B. Tylor (1978) Tours and Paths Chessics. 1 (5) pp.7-8. Moose-over-moose tour. Symetric Grasshopper-over-king tour.

A. S. M. Dickins (1978) Chessics. 1 (5) p.8. Figured tour with squares in sequence on fourth rank.

P. Cull and J. De Curtins (1978) Knight's tour revisited. Fibonacci Quarterly 16:3 (1978) 276-286.

G. P. Jelliss (1979); Honeycomb Leapers Chessics. 1 (7) pp.3-5. Analysis of moves on boards with hexagonal cells. Five tours of the Glinskian Hexagonal Chess board. 1 (9) p.4 Correction.

G. P. Jelliss (1979); Diffracting Pieces Chessics. 1 (7) p.7. Three tours by diffracting (= back-reflecting) fers, knight and alfil.

G. P. Jelliss (1979); Eccentric Knight Tours Chessics. 1 (8) pp.10-11. The 12 tours (of 32 cells) possible by a knight that moves only to or from centre or edge cells.

G. P. Jelliss (1979); Rook around the Rocks, A New Construction Task, The Problemist, vol.10, #22, November 1979. Wazir tours of 8×8 (and smaller) determined uniquely by blocks (minimum 4).

E. Y. Gik (1979); Shakhmatnye Dosugi 1979.

P. Berloquin (1979?); Jeux et Paradoxes, series (1971-79) in Science et Vie. See also 1982. [D. E. Knuth]

J. S. Madachy (1979); Madachy's Mathematical Recreations (Dover Publications, New York). This is a slightly revised edition of Madachy (1962) under a new title. p.88: Reproduces the 16×16 diagonally magic closed knight tour by H. E. de Vasa (1962), though still without acknowledgement.

« 1980s

W. Korn (1980); "Chess", Encyclopedia Britannica, 15th edition. Cites Bidev 1964.

G. P. Jelliss (1980); The Five Free Leapers (continued) Chessics 1 (9) pp.8-9. Proof of impossibility of 8×8 giraffe open tour. Maximum non-intersecting paths by giraffe, antelope and zebra. 1 (10) p.5 Three maximum length paths by A, G and Z.

G. P. Jelliss (1980-81); Clockwork Mouse Chessics 1 (10) p.7 (solution and diagram 1 (11) p.11). Uniquely determined clockwork mouse tour.

G. P. Jelliss and S. Ylikarjula (1980-82); Progressive Leapers Chessics 1 (10) p.7 (solution and diagram 1 (11) p.11). Progressive leaper autostalemates. 1 (12) p.16 (1981). Note on earlier progressive leaper result by T. R. Dawson (1930). 1 (14) p.16. Further results on closed paths (two by SY).

G. P. Jelliss (1981); Honeycomb Leapers (continued) Chessics. 1 (11) pp.1 and 8-9. Seven-leaper tours on boards of sides 5 to 8.

G. P. Jelliss (1981); Mobility of Leapers Chessics 1 (11) p.7.

G. P. Jelliss (1981); Missiles in Orbit Chessics 1 (12) pp.8-9. Interchessic missiles revisited (some results seem to relate to chaotic attractors).

G. P. Jelliss and T. W. Marlow (1981); Rook around the Rocks Chessics 1 (12) pp.7-9 and 12-13. Further results and proofs.

C. M. B. Tylor (1982) 2-by-2 Torus Chessics 1 (13) p.11 and 1 (14) p.12. Knight tours on a 2×2 torus related to wallpaper patterns.

G. P. Jelliss (1982-84); Chessics and the I Ching Chessics 1 (13) pp.12-13 and 1 (14) p.13. Includes chessboard tours by hyperwazir. 2 (19) pp.30-31. Relates hyperwazir tour to solution of the ring-of-rings toy.

G. P. Jelliss and C. Grimstone (1983) Chessics 1 (15) p.4. Lettered tours spelling CHESSAYS and CHESSICS (on RP board).

T. W. Marlow (1983) Chessics 1 (15) 7 and 1 (16) p.5. Grasshopper over rook paths.

P. Berloquin (1982) Manoeuvres de Cavalerie, Jeux at Strategie no.18, November-December 1982, pp.24-26. [D. Pritchard]

S. R. Iyer (1982); Indian Chess. An edition of the knight's tour section in Harikrishna (1871), with english commentary.

E. Y. Gik (1982); Zanimatel'nye Matematicheskie Igry.

E. Y. Gik (1983); Shakhmatny i Matematika 1983. [K. Whyld]

G. P. Jelliss (1984); Chessics 2 (18) p.22. Alternating tours by prince, king and emperor.

G. P. Jelliss (1984-5); Intersections in Knight Tours Chessics 2 (19) pp.25-27. Tours showing nonintersected and maximally intersected moves. 2 (20) pp.37-38 and 48, and 2 (21) pp.49-50 and 60. Crosspatch pseudotours and tours derived from them.

R. B. Eggleton and A. Eid (1984). Knight's Circuits and Tours. Ars Combinatoria, 17A (1978) 145-167. [MR 85i:05146] [reported to me by R. K. Guy and S. Pantazis, and cited in various other sources, but I have been unable to obtain a copy]

D. Hooper and K. Whyld (1984); Oxford Companion to Chess magic tour (00m) p.168.

G. P. Jelliss (1985); Wazir Tours with Squares in a Row Chessics 2 (21) p.56. Unique 6×6 solution and similar results 10×10, 14×14.

G. P. Jelliss (1985); in The Problemist, vol.12; #1 (January 1985), pp.3 and 16 (also on a New Year Card) a closed tour by simple-linking the C pseudotour.

G. P. Jelliss (1985); Special Issue: Notes on the Knight's Tour Chessics 2(22) pp.61-72. Figured tours. Origin of the knight's tour. Ala'addin's conundrum. Symmetry and counting of tours. Small board tours 3xn. Knight's tour as conjuring trick. Lettered tours. Tours 4xn. Tours 5xn. Tours 6x6. Synthetic tours. 10x10 tours. Square-symmetric (octonary) pseudotours.

J. Brügge (1985); Der Wanderer (Q, K, R, N) und sein Schatten (1, 2, 3, ..., 64) ; Magische Quadrate in Verbindung mit Figurenwanderung auf dem 8×8 Brett, Die Schwalbe #94, August 1985, pp.505-9. Diagonally magic tours by queen, king and rook.

S. Rabinowitz (1985-6); A Magic Rook's Tour Journal of Recreational Mathematics 18(3) 203-204. [Pickover 2002]

T. H. Willcocks (1985-6); Grasshopper-over-knight tours Chessics 2(23) p.84 and 2(25) p.104.

G. P. Jelliss with T. W. Marlow and T. H. Willcocks (1985-7); Special Issue: Theory of Leapers Chessics 2(24) pp.85-100. Wizard. Sleeper and Kraken. Mobility patterns. Journeys of 2, 3, 4, 6, 8 moves. 16-move symmetric paths (by TWM). Fxed-distance leapers (with fiveleaper tours by THW and TWM). Freedom of leapers. Amphibians. Houston's problem (shortest path between given points). Pterodactyl. Verse-tours. 2(25) pp.106-7. More on Figured tours, Square-symmetric pseudotours. Four-star tours. Early history of tours. 2(29/30) p.161. Improved solution (by THW).

G. P. Jelliss (1986); in The Problemist, vol.12; #9, (May 1986), p.159, salute to Halley's Comet; #10 (July 1986), A Unique Magic Tour, p.196, diagonally magic emperor tour with biaxial symmetry.

G. P. Jelliss with T. H. Willcocks (1986-7); Special Issue: Magic Tours Chessics 2(22) pp.113-128. de Vasa 48x48 example. The concept of magic. General principles. Magic tours on 2xeven, 3xodd, 4x4 boards. Diagonally magic king and queen tours. Structures of some magic tours. Catalogue of 8x8 magic knight tours. 12x12 magic tour (by THW). 2(29/30) p.163 solution and corrections.

P. Bidev (1986); Schach aus Indien oder China? / Did Chess Originate in India or China (Igalo, Yugoslavia, Part 1 German, Part 2 English). Expounds his theories about origin of chess related to 8×8 magic squares. [BCPS Library]

W. W. Rouse Ball and H. S. M. Coxeter (1939); Mathematical Recreations and Essays (Dover Publications, New York) 13th edition, reset: tours section regrettably not revised.

S. Collings (1987); Chessics 2(29/30) p.160. Note on 5-rank tours, answering a question in issue 22.

G. P. Jelliss and T. W. Marlow (1987); 3d tours Chessics 2(29/30) p.162. Sheherazade tour 7×11×13 (by GPJ). 4×4×4 tour (by TWM).

A. S. M. Dickins (1987); tours of side 6, 7, 8 Chessics 2(29/30) p.163.

W. L. Litmanowicz and J. Gizycki (1987) Szachy od A do Z Warsaw. Skoczka problem pp.1117-8. [P. Wood]

D. Singmaster (1987); Sources in Recreational Mathematics, a preliminary edition. [Part copy from D. Pritchard]. See 1991.

T. W. Marlow (1987); The Games and Puzzles Journal, vol.1 p.11 Magic Knight Tours (preview of his Problemist article 1988).

G. P. Jelliss (1987); The Games and Puzzles Journal, vol.1, p.27. Enumeration of Solitaire capture paths.

Mike Fox and Richard James (1987); The Complete Chess Addict Faber 1987. Magic tour 00m diagrammed.

T. W. Marlow (1988); Magic Knight Tours, The Problemist vol.12, #19, January 1988, p.379. Five new 8×8 magic knight's tours (two also appeared in The Games and Puzzles Journal 1987); reports computerised count of all such tours of quartes type, total 78.

'Alban' (= W. Scotland) (1988); in Crossword 1988. Uses a 10×10 knight's tour and the first verse of Lewis Carroll's Jabberwocky to make a crossword puzzle.

J. E. Hofmann (1988); Uber Rösselringe auf dem Brett von 6² Zellen (edited by C. J. Scriba), Elemente der Mathematik 43 (1) January 1988, pp.1-32. [S. Pantazis]

G. P. Jelliss (1988); The Games and Puzzles Journal, vol.1, no.3 p.45, no.4 p.61 Sopwith manoeuvres (directed wazirs on honeycomb).

J. Brügge (1988); Symmetriebruch als Phänomen semi-magischer {i.e. magic but not diagonally magic} Springerwanderungen, Die Schwalbe #112, August 1988, pp.449-454. Analysis of three 12×12 magic tours by T. H. Willcocks.

D. Eperson (1988). Patterns in Mathematics, Basil Blackwell Ltd, Oxford. Chapter 7. 'Knight's tours'. pp.53–58. Solutions pp.136–139. The introduction on p.53 gives two 8×8 closed tours (described as 'unicursal'), in numerical and graphic forms. The first is symmetric with oblique quaternary elements, the second has near axial symmetry, and its central area is formed of the symmetric (4×4 – 2) tour. These appear to be original and are said to be constructed using the methods of Euler and de Moivre. There follows a series of six questions and nine problems involving knight's paths forming symmetric patterns on a 5×5 board. These, as with the rest of the book, are designed for use in primary schools and in remedial courses for students with 'mathophobia'.

J. Labelle and Y-N. Yeh (1989); Dyck paths of knight moves Discrete Applied Maths no.24, pp.213-221. [S. Pantazis]

G. P. Jelliss (1989); The Games and Puzzles Journal, vol.1 no.11 p.178 Chess lettering (delineated by knight circuits) and Figured wazir tours.

T. H. Willcocks (1989); The Games and Puzzles Journal, vol.1 no.11 p.178 Rook around the rocks examples on boards of side 12 and 13.

« 1990s

G. P. Jelliss (1990); Knight's Tours and Other Chessic Paths: A Chronological Bibliography. A 32-page bibliography, basis of the present list, distributed to interested correspondents.

R. H. Merson (1990); personal communication with new results on nonintersecting paths: see Merson (1999).

N. Pennick (1990); Mazes and Labyrinths (Robert Hale, London). Ch.1 Ancient Rock and Stone Labyrinths. Ch.2 Ancient Turf Mazes. Ch.3 Pavement and Church Labyrinths. Ch.4 Puzzle Hedge Mazes. Ch.5 Modern Labyrinths. Bibliography pp.192-202. [Hastings Library]

Peter Wong (1990); Bouncer Tour. Variant Chess 1(3) July-September 1990 p.31 problem, 1(4) October-December 1990 p.47 solution.

A. J. Schwenk (1991); Which Rectangular Chessboards have a Knight's Tour? Mathematics Magazine 64(5), pp.325-332, December 1991. [info from L. J. Upton, also cited by Balof and Watkins 1996]

D. Singmaster (1991); Sources in Recreational Mathematics, An Annotated Bibliography (5th preliminary edition, November 1991, South Bank Polytechnic, London). Begun in 1982. Relevant sections are: 5E pp.66-70 Euler circuits and mazes. 5F pp.70-77 Hamiltonian circuits (this includes knight's tours). 5Y pp.100-101 Number of routes in a lattice. 6AK pp.142-144 Polygonal path covering lattice (includes queen and rook tours). 7N pp.214-228 Magic squares (this includes magic knight tours). [D. Singmaster]

G. P. Jelliss (1991); Knight's Tour News. Variant Chess 1(6) June 1991 p.75. GPJ: Magic Rook Tours, diagonal example. T.W.Marlow: Magic 5-Leaper Tours. Complementary tour question. R. H. Merson: Non-Intersecting Paths, tetraskelions, {1,2} on 7×7, {2,3} om 9×9, {3,4} on 11×11.

G. P. Jelliss (1992); Knight's Tour News. Variant Chess 1(7) March 1992 p.91. T. W. Marlow, diagonal magic knight tour on 8×8 torus. M. Dukic magic square, partly a king tour. Variant Chess 1(8) December 1992 p.105. GPJ: Magic 8×12 knight tour.

G. P. Jelliss (1992); Figured Tours Mathematical Spectrum (vol.25 (1) pp.16-20, Applied Probability Trust, University of Sheffield).

D. E. Knuth (1992); personal communications. Celtic tours, defined in letter to GPJ dated 23 December 1992.

L. and J. Laing (1992); Art of the Celts.

D. Wells (1992). The Penguin Book of Curious and Interesting Puzzles, Penguin Books, London. The chessboard knight's tour problem is mentioned on p.37 (#132) and two symmetric tours are diagrammed on p.218. Knight's tours of smaller boards are mentioned on p.151 (#484) where it is stated, wrongly, that "on a 4×4 board, wherever you start, either four or six squares will be omitted from your tour" (the minimum is 1 for an open tour, 2 for a closed tour). He asks (a) "What is the smallest rectangular board on which it is possible to do a complete tour?" and (b) for a tour with rotational symmetry unaltered by quarter turn. In the solutions on p.339 he answers (a) wrongly: "The smallest in area, and measured by the length of the shortest side, is 3×7" giving a symmetric example b2 to f2 (this overlooks the existence of tours on the 3×4 board) for (b) he diagrams one of the five 6×6 tours with 90° rotational symmetry.

I. Stewart (1992) Another Fine Math You've Got Me Into (Freeman, New York). [Pickover 2002]

B. Datta and A. N. Singh (revised by K. S. Shukla) (1992): Magic Squares in India, Indian Journal of History of Science 27(1), pp.51-120, 1992. [PDF from A. Kumar] "Magic squares have been known in India from very early times. It is believed that the subject of magic squares was first taught by Lord Shiva to the magician Manibhadra." /// "But the mathematics involved in the construction of magic squares and other magic figures was first systematically and elaborately discussed by the mathematician Narayana (AD 1356) in his Ganitakaumudi." /// "The earliest unequivocal occurrence of magic squares is found in a work called Kaksaputa composed by the celebrated alchemist and philosopher Nagarjuna who flourished about the 1st century AD. One of the squares in this work is called Nagarjuniya after him; so there can be no doubt that he really did construct some squares. The squares given by Nagarjuna are all 4×4 squares." The "Nagarjuniya" magic square is formed of an arithmetical progression of even numbers 6 to 44, but with 8, 12, 38 and 42 omitted, and adds to 100:

        30 16 18 36
        10 44 22 24
        32 14 20 34
        28 26 40  6 
"Another [4×4] square is found in a work of Varahamihira (d. 587 AD)." /// This is a "mystic square" formed of the numbers 1 to 8 each occurring twice, with magic sum 18.
         2  3  5  8
         5  8  2  3
         4  1  7  6
         7  6  4  1 
"A 4×4 square occurs in a Jaina inscription of the 11th century, found in the ancient town of Khajuraho." /// This is a magic square in the modern sense (it is #623 in Frenicle's list).
         7 12  1 14
         2 13  8 11
        16  3 10  5
         9  6 15  4 

D. E. Jackson (1993); in Journal of Recreational Mathematics.

S. W. Golomb (1993); in Journal of Recreational Mathematics. pp.309-310.

M. Reid (1993); in Journal of Recreational Mathematics.

M. Valtorta and M. I. Zahid (1993). Warnsdorff's tours of a knight. Journal of Recreational Mathematics 25:4 (1993) 263-275. Tries to show the weak property that a tour following Warnsdorf's rule is possible on all tourable boards.

M. Valtorta and M. I. Zahid (1993). Tie-breaking rules for 4×n Warnsdorff's tours. Congr. Numer. 95 (1993) 75-86. [cited by Cairns 2002]

S. P. Hurd and D. A. Trautman (1993). The knight's tour on the 15-puzzle. Mathematics Magazine 66:3 (1993) 159-166. [cited by Cairns 2002]

G. P. Jelliss (1994); Generalised Knight's and hamiltonian Tours Journal of Recreational Mathematics.

D. E. Knuth (1994); Leaper tours Mathematical Gazette.

Takefuji, Y. and K.-C. Lee (1994); "Finding Knight's Tours on an M×N Chessboard with O(MN) Hysteresis McCulloch-Pitts Neurons. IEEE Transactions on Systems, Man and Cybernetics 24(2): 300-306. [cited by Hingston and Kendall 2005]

A. Conrad, T. Hindrichs, H. Morsy and I. Wegener (1994) Solution of the knight's Hamiltonian path problem on chessboards. Discrete Appl. Math 50:2 (1994) 125-134. [cited by Cairns 2002]

G. H. J. van Rees (1996); Knight's tours and circuits on the 3×n chessboard. Bull. Inst. Combin. Appl. 16 (1996) 81-86. [cited by Grant Cairns 2002]

M. Loebbing and I. Wegener The number of knight's tours equals 33,439,123,484,294 - counting with binary decisioon diagrams. Electron. J. Combin. 3:1 (1996). Their number is wrong, as is pointed out in Comments in the same journal issue by B. McKay.

Barry A. Balof and John J. Watkins (1996); Knight's Tours and Magic Squares. Congressus Numerantium 120, pp.23-32. [PDF from A. Kumar]

G. P. Jelliss (1996); Figured Tours. A 22-page A4 booklet, containing more than 225 tours, including all 100 of T. R. Dawson's tours with square numbers in a knight chain.

G. P. Jelliss (1996); Puzzle Corner. Variant Chess 3(21) Autumn 1996 p.2, 22, 3(22) Winter 1996 p.28, 3(23) Spring 1997 p.47, knight's tours of shaped boards, with octonary symmetry.

G. P. Jelliss (1996); The Games and Puzzles Journal, vol.2 #13 May 1996: p.201 Figured tour 12×12 with Fibonacci numbers.

G. P. Jelliss (1996); The Games and Puzzles Journal, vol.2 #13 May 1996: pp.208-217 reproduces the articles by E. Bergholt from Queen 1915-16, and his first four unpublished Memoranda 1916 (also related to work by H. J. R. Murray and G. L. Moore p.214). #14 December 1996: pp.230-237, 244 reproduces his unpublished Memoranda 5 and 6 from 1916-17;

G. P. Jelliss (1996); The Games and Puzzles Journal, reproduces H. J. R. Murray's History of Magic Knight's Tours (1951 ms) vol.2 #14 (December 1996): p.225 the 12×12 magic tour by Krishnaraj Wadiar, Rajah of Mysore; pp.238-244, with commentary and new material. #15 pp.266-267 Continuation. #16 p.291 Conclusion.

T. H. Willcocks (1996); The Games and Puzzles Journal, vol.2 #14 (December 1996): p.230 research on biographical details of Bergholt (he was Ernest George Binckes, b. Worcester 1856, d. Letchworth 18 Nov. 1925).

G. P. Jelliss (1997); Variant Chess 3(24) Summer 1997 pp.77, 79. Clockwork mouse tour. 3(25) Autumn 1997 p.92. Tour on corridor board by E. Bergholt.

G. P. Jelliss (1997); The Games and Puzzles Journal, vol.2 #15 (December 1997): p.249 King-tour interpretation of the Boook of Kells knot design; p.250, review of Colin C. Adams' The Knot Book; p.252 results on 8×8 magic squares; pp.262-263, more on King tours in Celtic Art; p.265 Enumeration of closed knight's tours on smaller boards; p.270 Mystic rectangles.

T. W. Marlow (1997); The Games and Puzzles Journal, vol.2 #15 (December 1997): p.264 All known diagonally magic 8×8 King tours with biaxial symmetry.

John J. Watkins and Rebecca L. Hoenigman (1997); Knight's Tours on a Torus. Mathematics Magazine 70(3), pp.175-187 (page range cited varies), 1997. [cited by Watkins 1997, Hill and Tostedo 2004]

J. Gullberg (1997); Mathematics from the Birth of Numbers, W. W. Norton & Co, New York. Three magic knight's tours, and one magic two-knight's tour, are quoted on p.209 in section 5.6 (pp.205–214) on 'Magic Squares and Their Kin'. The first is Beverley's tour (27a, reflected in the principal diagonal) but misattributed to Euler. The two-magic tour is by Feisthamel (source not given) its middle link is a {0,3} move f4-c4. The other two magic knight tours are by Jaenisch (00a and 12o) though the second of these is misattributed to Wenzelides.

O. Kyek, I. Parberry and I. Wegener (1997) Bounds on the number of knight's tours. Discrete Appl. Math. 74:2 (1997) 171-181. [cited by Cairns 2002]

B. D. McKay (1997); Knight's Tours of an 8x8 Chessboard. Department of Computer Science, Australian National University. [cited by Hingston and Kendall 2005] "McKay calculated the number of closed tours on a standard 8×8 chessboard to be 13,267,364,410,532."

M. Petkovic (1997); Mathematics and Chess Dover Publications, New York, 1997, pp.53, 64. [cited by A. Kumar 2009]

John J. Watkins (1997); Knight's Tours on Triangular Honeycombs. Congressus Numerantium 124, p.81-87, 1997. [PDF from A. Kumar]

I. Parberry (1997); An Efficient Algorithm for the Knight's Tour Problem. Discrete and Applied Mathematics 73: 251-260. [cited in Gordon and Slocum 2004]

Ian Parberry (1997): Scalability of a Neural Network for the Knight’s Tour Problem. (1997) [cited by Hill and Tostado]

Ian Stewart (1997); Mathematical Recreations: Knight’s Tours. Scientific American, April 1997, pp.102-104. [cited by Hill and Tostedo]

Edward D. Collins (1997); A Knight’s Tour. In fact five knight's tours, the last being Beverley's, as usual misattributed to Euler.

G. P. Jelliss (1998); Variant Chess 3(28) Summer 1998. Figured emperor tour by W. E. Lester (1938).

P. C. B. Lam, W. C. Shiu and H. L. Cheng (1999). Knight's tour on hexagonal nets. Congr. Numer. 141 (1999). [cited by Cairns 2002 – see also Jelliss 1979]

J. D. Beasley (1999) Circular Chess in Lincoln Variant Chess 4(31) pp.33-34, 48, 4(32) p.55. Refers to the 4×16 circular chessboard tour in Twiss 1789 and gives two further examples of his own showing symmetry, which is impossible on the rectangular board.

G. P. Jelliss (1999); The Games and Puzzles Journal, vol.2 #16 (May 1999): p.273 18×18 tour with quaternary (and translatory) symmetry. p.275-276 Fillet and Field tours. p277 Incantatory and Musical Tours. pp.282-287, 289 Symmetry in knight's tours.

T. W. Marlow (1999); The Games and Puzzles Journal, vol.2 #16 (May 1999): p.288 Enumeration of closed knight tours. Includes the total 415902 for quaternary symmetry on the 10×10 board; compare W.H.Cozens (1956). pp.290-291 Longer leaper tours with quaternary symmetry.

R. H. Merson and G. P. Jelliss (1999); The Games and Puzzles Journal, vol.2 #17 (October 1999): pp.297, Non-intersecting knight path 24×24. 305-307 Historical notes and non-intersecting paths by longer leapers. pp.307-310 nonintersecting knight paths on larger boards.

G. P. Jelliss (1999); The Games and Puzzles Journal, vol.2 #17 (October 1999): p.314 Knight's tour with maximum border braid. p.315 Knightly triangles, on knight's move geometry.

Arnd Roth (1999): The Problem of the Knight. Implementation of Warnsdorf's rule in Mathematica. "Although Warnsdorff thought his rule would find a tour on any size rectangular board, exhaustive computer searches have shown that it can fail for boards larger than 76×76. A modern day improvement [on Wornsdorf] by Arnd Roth breaks ties in choosing successors by selecting the square furthest from the center of the board; this method first fails on a 428×28 board." [Hill and Tostado (2004)]

Back to top.