1.       BIOGRAPHICAL MATERIAL  _  in chronological order



          ALCUIN  (c735‑804)


Phillip Drennon Thomas.  Alcuin of York.  DSB I, 104‑105.

Robert Adamson.  Alcuin, or Albinus.  DNB, (I, 239‑240), 20.

Andrew Fleming West.  Alcuin and the Rise of the Christian Schools.  (The Great Educators _ III.)  Heinemann, 1893.  The only book on Alcuin that I found which deals with the Propositiones.

Stephen Allott.  Alcuin of York  c. A.D. 732 to 804  _  his life and letters.  William Sessions, York, 1974.


          FIBONACCI [LEONARDO PISANO]  (c1170->1240)


Fibonacci.  (1202 _ first paragraph);  1228 _ second paragraph, on p. 1.  In this paragraph he narrates almost everything we know about him.  [In the second ed., he inserted a dedication as the first paragraph.]

                    The paragraph ends with the notable sentence which I have used as a motto for this work.  "Si quid forte minus aut plus iusto vel necessario intermisi, mihi deprecor indulgeatur, cum nemo sit qui vitio careat et in omnibus undique sit circumspectus."  (If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things.  [Grimm's translation.])

Richard E. Grimm.  The autobiography of Leonardo Pisano.  Fibonacci Quarterly 11 (1973) 99-104.  He has collated six MSS of the autobiographical paragraph and presents his critical version of it, with English translation and notes.  Sigler, below, gives another translation.  I give Grimm's translation, omitting his notes.


                    After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and, in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days.  There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business, I pursued my study in depth and learned the give-and-take of disputation.  But all this even, and the algorism, as well as the art of Pythagoras I considered as almost a mistake in respect to the method of the Hindus.  Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art, I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters.  Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now.  If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things.


F. Bonaini.  Memoria unica sincrona di Leonardo Fibonacci novamente scoperta.  Giornale Storico degli Archivi Toscani 1:4 (Oct-Dec 1857) 239-246.  This reports the discovery of a 1241 memorial of the Comune of Pisa, which I reproduce as it is not well known.  This grants Leonardo an annual honorarium of 20 pounds.  In 1867, a plaque bearing this inscription and an appropriate heading was placed in the atrium of the Archivio di Stato in Pisa.


                    "Considerantes nostre civitatis et civium honorem atque profectum, qui eis, tam per doctrinam quam per sedula obsequia discreti et sapientis viri magistri Leonardi Bigolli, in abbacandis estimationibus et rationibus civitatis eiusque officialium et aliis quoties expedit, conferunter; ut eidem Leonardo, merito dilectionis et gratie, atque scientie sue prerogativa, in recompensationem laboris sui quem substinet in audiendis et consolidandis estimationibus et rationibus supradictis, a Comuni et camerariis publicis, de Comuni et pro Comuni, mercede sive salario suo, annis singulis, libre xx denariorum et amisceria consueta dari debeant (ipseque pisano Comuni et eius officialibus in abbacatione de cetero more solito serviat), presenti constitutione firmamus."


                    A translation follows, but it can probably be improved.  My thanks to Steph Maury Gannon for many improvements over my initial version.


          Considering the honour and progress of our city and its citizens that is brought to them through both the knowledge and the diligent application of the discreet and wise Maestro Leonardo Bigallo in the art of calculation for valuations and accounts for the city and its officials and others, as often as necessary,; we declare by this present decree that there shall be given to the same Leonardo, from the Comune and on behalf of the Comune, by reason of affection and gratitude, and for his excellence in science, in recompense for the labour which he has done in auditing and consolidating the above mentioned valuations and accounts for the Comune and the public bodies, as his wages or salary, 20 pounds in money each year and his usual fees (the same Pisano shall continue to render his usual services to the Comune and its officials in the art of calculation etc.).


                    Bonaini also quotes a 1506 reference to Lionardo Fibonacci.


Mario Lazzarini.  Bolletino di Bibliografia e Storia delle Scienze Matematiche  6 (1903) 98‑102  &  7 (1904) 1-7.  ??NYS _ Grimm says it reproduces the above memorial and gives more information.

Gino Loria.  Leonardo Fibonacci.  Gli Scienziati Italiana dall'inizio del medio evo ai nostri giorni.  Ed. by Aldo Mieli.  (Dott. Attilio Nardecchia Editore, Rome, 1921;)  Casa Editrice Leonardo da Vinci, Rome, 1923.  Vol. 1, pp. 4-12.  This quotes the above inscription and the opening biographical paragraph of Liber Abaci.  Says he was born around 1170.  He cites a contract of 28 Aug 1226 involving a brother of Leonardo which shows that his father was named Guglielmo and his grandfather was named Bonaccio [G. Milanesi; Documento inedito intorno a Leonardo Fibonacci; Rome, 1867 _ ??NYS].

Charles King.  Leonardo Fibonacci.  Fibonacci Quarterly 1:4 (Dec 1963) 15-19.

Gino Arrighi, ed.  Leonardo Fibonacci: La Practica di Geometria _ Volgarizzata da Cristofano di Gherardo di Dino, cittadino pisano.  Dal Codice 2186 della Biblioteca Riccardiana di Firenze.  Domus Galilaeana, Pisa, 1966.  The Frontispiece is the mythical portrait of Fibonacci.  P. 15 shows the plaque erected in the Archivio di Stato di Pisa in 1855 which reproduces the above memorial with an appropriate heading, but Arrighi has no discussion of it.  P. 19 is a photo of the statue in Pisa and p. 16 describes its commissioning in 1859.

Joseph and Francis Gies.  Leonard of Pisa and the New Mathematics of the Middle Ages.  Crowell, NY, 1969.  This is a book for school students and contains a number of dubious statements and several false statements.

K. Vogel.  Fibonacci, Leonardo, or Leonardo of Pisa.  DSB IV, 604-613.

A. F. Horadam.  Eight hundred years young.  Australian Mathematics Teacher 31 (1975) 123‑134.  Good survey of Fibonacci's life & work.  Gives English of a few problems.

Ettore Picutti.  Leonardo Pisano.  Le Scienze 164 (Apr 1982) ??NYS.  = Le Scienze, Quaderni; 1984, pp. 30-39.  (Le Scienze is a magazine;  the Quaderni are collections of articles into books.)  Mostly concerned with the Liber Quadratorum, but surveys Fibonacci's life and work.  Says he was born around 1170.  Includes photo of the plaque in the Archivo di Stato di Pisa.

Leonardo Pisano Fibonacci.  Liber quadratorum, 1225.  Translated and edited by L. E. Sigler as:  The Book of Squares; Academic Press, NY, 1987.  Introduction: A brief biography of Leonardo Pisano (Fibonacci) [1170 - post 1240], pp. xv-xx.  This is the best recent biography, summarizing Picutti's article.  Says he was born in 1170 and his father's name was Guilielmo _ cf Loria above.  Gives another translation of the biographical paragraph of the Liber Abbaci.

A. F. Horadam & J. Lahr.  Letter to the Editor.  Fibonacci Quarterly 28:1 (Feb 1990) 90.  The authors volunteer to act as coordinators for work on the life and work of Fibonacci.  Addresses:  A. F. Horadam, Mathematics etc., Univ. of New England, Armidale, New South Wales, 2351, Australia;  J. Lahr, 14 rue des Sept Arpents, L‑1139 Luxembourg, Luxembourg.


          Claude-Gaspar BACHET de Méziriac  (1581‑1638)


C.‑G. Collet & J. Itard.  Un mathématicien humaniste _ Claude‑Gaspar Bachet de Méziriac (1581‑1638).  Revue d'Histoire des Sciences et leurs Applications 1 (1947) 26‑50. 

J. Itard.  Avant-propos.  IN:  Bachet; Problemes; 1959 reprint, pp. v‑viii.  Based on the previous article. 

There is a Frontispiece portrait in the reprint.

Underwood Dudley.  The first recreational mathematics book.  JRM 3 (1970) 164‑169.  On Bachet's Problemes.

William Schaaf.  Bachet de Méziriac, Claude‑Gaspar.  DSB I, 367‑368.


          Jean LEURECHON (c1591‑1670)  and  Henrik VAN ETTEN


William Schaaf.  Leurechon, Jean.  DSB VIII, 271‑272.

Trevor H. Hall.  Mathematicall Recreations.  An Exercise in Seventeenth Century Bibliography.  Leeds Studies in Bibliography and Textual Criticism, No. 1.  The Bibliography Room, School of English, University of Leeds, 1969, 38pp.  Pp. 18‑38 discuss the question of authorship and Hall feels that van Etten probably was the author and that there is very little evidence for Leurechon being the author.  Much of the mathematical content is in Bachet's Problemes and may have been copied from it or some common source.  [This booklet is reproduced as pp. 83-119 of Hall, OCB, with the title page of the 1633 first English edition reproduced as plate 5, opp. p. 112.  Some changes have been made in the form of references since OCB is a big book, but the only other substantial change is that he spells the name of the dedicatee of the book as Verreyken rather than Verreycken.]

A. Deblaye.  Étude sur la récréation mathématique du P. Jean Leurechon, Jésuite.  Mémoires de la Société Philotechnique de Pont-à-Mousson 1 (1874) 171-183.  [MUS #314.  Schaaf.  Hall, OCB, pp. 86, 88 & 114, says the only known copy of this journal is at Harvard, which has kindly supplied me with a xerox of this article.  Hall indicates the article is in vol. II and says it is 12 pages, but only cites pp. 171 & 174.]  This simply assumes Leurechon is the author and gives a summary of his life.  The essential content is described by Hall.

G. Eneström.  Girard Desargues und D.A.L.G.  Biblioteca Mathematica (3) 14 (1914) 253‑258.  D.A.L.G. was an annotator of van Etten's book in c1630.  Although D.A.L.G. was used by Mydorge on one of his other books, it had been conjectured that this stood for  Des Argues Lyonnais Girard (or Géomètre).  Eneström can find no real evidence for this and feels that Mydorge is the most likely person.


          Jacques OZANAM (1640‑1717)


On the flyleaf of J. E. Hofmann's copy of the 1696 edition of Ozanam's Recreations is a pencil portrait labelled Ozanam _ the only one I know of.  This copy is at the Institut für Geschichte der Naturwissenschaft in Munich.  Hofmann published the picture _ see below.

Charles Hutton.  A Mathematical and Philosophical Dictionary.  1795-1796.  Vol. II, pp. 184-185.  ??NYS  [Hall, OCB, p. 166.]

Charles Hutton.  On the life and writings of Ozanam, the first author of these Mathematical Recreations.  Ozanam-Hutton.  Vol. I.  1803: xiii-xv;  1814: ix-xi.

William L. Schaaf.  Jacques Ozanam on mathematics ....  MTr 50 (1957) 385-389.  Mostly based on Hutton.  Includes a sketchy bibliography of Ozanam's works, generally ignoring the Recreations.

Joseph Ehrenfried Hofmann.  Leibniz und Ozanams Problem, drei Zahlen so zu bestimmen, dass ihre Summe eine Quadratzahl und ihre Quadratsumme eine Biquadratzahl ergibt.  Studia Leibnitiana 1:2 (1969) 103-126.  Outlines Ozanam's life, gives a bibliography of his works and reproduces the above-mentioned drawing as a plate opp. p. 124.  (My thanks to Menso Folkerts for this information and a copy of Hofmann's article.)

William L. Schaaf.  Ozanam, Jacques.  DSB X, 263‑265.


          Jean Étienne MONTUCLA  (1725-1799)


Charles Hutton.  Some account of the life and writings of Montucla.  Ozanam‑Hutton.  Vol. I.  1803: viii-xii;  1814: iv-viii.

Charles Hutton.  A Philosophical and Mathematical Dictionary.  2nd ed. of the Dictionary cited under Ozanam, 1815,  Vol. II, pp. 63-64.  ??NYS.  According to Hall, OCB, p. 167, this is not in the 1795-1796 ed. and is a reworking of the previous item.


          Lewis CARROLL  (1832-1898)


          Pseudonym of Charles Lutwidge Dodgson.  There is so much written on Carroll that I will only give some references to his specifically recreational work.


Warren Weaver.  Lewis Carroll: Mathematician.  SA 194:4 (Apr 1956) 116‑128.  +  Letters and response.  SA 194:6 (Jun 1956) 19-22.

John Fisher.  The Magic of Lewis Carroll.  (Nelson, 1973), Penguin (1975), 1980.


          Professor Louis HOFFMANN  (1839‑1919)


          Pseudonym of Angelo John Lewis.


Anonymous.  Professor Hoffmann.  Mahatma 4:1 (Jul 1900) 377-378.  A brief note, with photograph, stating that he is Mr. Angelo Lewis, M.A. and Barrister-at-Law.

Will Goldston.  Will Goldston's Who's Who in Magic.  My version is included in a compendium called: Tricks that Mystify; Will Goldston, London, nd [1934-NUC].  Pp. 106-107.  Says he was a barrister, retired to Hastings about 1903 and died in 1917.

J. B. Findlay & Thomas A. Sawyer.  Professor Hoffmann: A Study.  Published by Thomas A. Sawyer, Tustin, California, 1977.  A short book, 12 + 67 pp, with two portraits and 27pp of bibliography.  He was a barrister and wrote two books on Indian law.

Charles Reynolds.  Introduction  _  to the reprint of Hoffmann's Modern Magic, Dover, 1978, pp. v‑xiv.  This says Lewis was a barrister, which is mentioned in another reprint of a Hoffmann book and in S. H. Sharpe's translation of Ponsin on Conjuring.

Edward Hordern.  Forword to this edition.  In:  Hoffmann's Puzzles Old and New (see under Common References), 1988 reprint, pp. v‑vi.  This says he was the Reverend Lewis. 

Hall, OCB, p. 189, gives Hoffmann's address as Ireton Lodge, Cromwell Ave., N. _ presumably the Cromwell Ave. in Highgate.

Toole Stott 386 gives a little information about Hoffmann and Modern Magic, including an address in Mornington Crescent in 1877.

No DNB or DSB entry _ I have suggested a DNB entry.


          Sam LOYD  (1841‑1911)  and  Sam LOYD JR.  (1873‑1934)


[W. R. Henry.]  Samuel Loyd.  [Biography.]  Dubuque Chess Journal, No. 66 (Aug-Sep 1875) 361-365.  ??NX _ o/o (11 Jul 91).

Loyd.  US Design 4793 _ Design for Puzzle-Blocks.  11 April 1871.  These are solid pieces, but unfortunately the drawing did not come with this, so I am not clear what they are.  ??Need drawing _ o/o (11 Jul 91).

Anonymous & Sam Loyd.  Loyd's puzzles (Introductory column).  Brooklyn Daily Eagle (22 Mar 1896) 23.  Says he lives at 153 Halsey St., Brooklyn.

L. D. Broughton Jr.  Samuel Loyd.  [A Biography.]  Lasker's Chess Magazine 1:2 (Dec 1904) 83-85.  About his chess problems with a mention of some of his puzzles.

G. G. Bain.  The prince of puzzle‑makers.  An interview with Sam Loyd.  Strand Magazine 34 (No. 204) (Dec 1907) 771‑777.  Solutions of Sam Loyd's puzzles.  Ibid. 35 (No. 205) (Jan 1908) 110.

Walter Prichard Eaton.  My fifty years in puzzleland _ Sam Loyd and his ten thousand brain‑teasers.  The Delineator (New York) (April 1911) 274 & 328.  Drawn portrait of Loyd, age 69.

Anon.  Puzzle inventor dead.  New-York Daily Tribune (12 Apr 1911) 7.  Says he died at his house, 153 Halsey St.  "He declared no one had ever succeeded in solving [the "Disappearing Chinaman"]."  Says he is survived by a son and two daughters (!! _ has anyone ever tracked the daughters and their descendents??).

Anon.  Sam Loyd, puzzle man, dies.  New York Times (12 Apr 1911) 13.  Says he was for some time editor of The Sanitary Engineer and a shrewd operator on Wall Street.

Anon.  Sam Loyd.  SA (22 Apr 1911) 40-41??  Says he was for some years chess editor of SA and was puzzle editor of Woman's Home Companion when he died.

W. P. Eaton.  Sam Loyd.  The American Magazine 72 (May 1911) 50, 51, 53.  Abridged version of Eaton's earlier article.  Photo of Loyd on p. 50.

P. J. Doyle.  Letter to the Chess column.  The Sunday Call [Newark, NJ] (21 May 1911), section III, p. 10.

A. C. White.  Sam Loyd and His Chess Problems.  Whitehead and Miller, Leeds, UK, 1913;  corrected, Dover, 1962.

Alain C. White.  Supplement to Sam Loyd and His Chess Problems.  Good Companion Chess Problem Club, Philadelphia, vol. I, nos. 11-12 (Aug 1914), 12pp.  This is mostly corrections of the chess problems, but adds a few family details with a picture of the Loyd Homestead and Grist Mill in Moylan, Pennsylvania.

Alain C. White.  Reminiscences of Sam Loyd's family.  The Problem [Pittsburg] (28 Mar 1914) 2, 3, 6, 7.

Louis C. Karpinski.  Loyd, Samuel.  Dictionary of American Biography, Scribner's, NY, vol. XI, 1933, pp. 479‑480.

Loyd Jr.  SLAHP.  1928.  Preface gives some details of his life, making little mention of his father, "who was a famous mathematician and chess player".  He claims to have created over 10,000 puzzles. There are some vague biographical details on pp. 1‑22, e.g. 'Father conducted a printing establishment.'  'My "Missing Chinaman Puzzle"'.  (It may have been some such assertion that led me to estimate his birthdate as 1865, but I now see it is well known to be 1873.)

Anonymous.  Sam Loyd dead; puzzle creator.  New York Times (25 Feb 1934).  Obituary of Sam Loyd Jr.  Says he resided at 153 Halsey St., Brooklyn _ the same address as his father _ see the Brooklyn Daily Eagle article of 1896, above.  He worked from a studio at 246 Fulton St., Brooklyn.  It says Jr. invented 'How Old is Ann?'.

Clark Kinnaird.  Encyclopedia of Puzzles and Pastimes.  Grosset & Dunlap, NY, 1946.  Pp. 263‑267: Sam Loyd.  Asserts that Loyd Jr. invented 'How Old is Ann?'

Will Shortz is working on a biography.

No DSB entry.


          François Anatole Edouard LUCAS  (1842‑1891)


Obituary.  Nature 44 (1891) 574-575.  ??NYS _ cited by Campbell.

Nécrologie:  Édouard Lucas.  La Nature 19 (1891) II, 302.  ??NYS _ cited by Campbell.

Duncan Harkin.  On the mathematical work of François‑Édouard‑Anatole Lucas.  L'Enseignement Math. (2) 3 (1957) 276‑288.  Pp. 282‑288 is a bibliography of 184 items.  I have discovered there are many items in Dickson's History of the Theory of Numbers and elsewhere which are not given by Harkin.  In Dec 1998, I made a 14 pp file with all of Harkin's entries and all the new material I have found, making 248 items, though many of these are multiple items so the actual count is perhaps 275.  However, Dickson does not give article titles, and may not give the pages of the entire article (so the same article may be cited more than once, at different pages).  I hope to fill in the missing information at some time.

P. J. Campbell.  Lucas' solution to the non‑attacking rooks problem.  JRM 9 (1976/77) 195‑200.  Gives life of Lucas.

A photo of Lucas is available from Bibliothèque Nationale, Service Photographique, 58 rue Richelieu, F‑75084 Paris Cedex 02, France.  Quote Cote du Document Ln27 . 43345 and Cote du Cliche 83 A 51772.  (??*)  I have obtained a copy, about 55 x 85 mm, with the photo in an oval surround.  It looks like a carte-de-visite, but has  Édouard LUCAS (1842-1891).  _  Phot. Zagel.  underneath.  (Thanks to H. W. Lenstra for the information.)

Norman T. Gridgeman.  Lucas, François‑Édouard‑Anatole.  DSB VIII, 531‑532.

Susanna S. Epp.  Discrete Mathematics with Applications.  Wadsworth, Belmont, Calif., 1990, p. 477 gives a small photo of Lucas which looks nothing like the photo from the BN.  I have since received a note from Epp via Paul Campbell that a wrong photo was used in the first edition, but this was corrected in later editions.

Alain Zalmanski.  Edouard Lucas  Quand l'arithmétique devient amusante.  Jouer Jeux Mathématiques 3 (Jul/Sep 1991) 5.  Brief notice of his life and work.

Andreas M. Hinz.  Pascal's triangle and the Tower of Hanoi.  AMM 99 (1992) 538-544.  Sketches Lucas' life and work, giving details that are not in the above items.


          Hermann SCHUBERT  (1848-1911)


Acta Mathematica  1882-1912.  Table Générale des Tomes 1-35.  1913.  P. 169.  Portrait of Schubert.

Werner Burau.  Schubert, Hermann Cäsar Hannibal.  DSB XII, 227‑229.


          Walter William Rouse BALL  (1850‑1925)


Anon.  Obituary:  Mr. Rouse Ball.  The Times (6 Apr 1925) 16. 

Anon.  Funeral notice:  Mr. W. W. R. Ball.  The Times (9 Apr 1925) 13.

(Lord) Phillimore.  Letter:  Mr. Rouse Ball.  The Times (9 Apr 1925) 15.

"An old pupil".  The late Mr. Rouse Ball.  The Times (13 Apr 1925) 12.

J. J. Thomson.  W. W. Rouse Ball.  The Cambridge Review (24 Apr 1925) 341-342.

Anon.  Obituary of W. W. Rouse Ball.  Nature 115 (23 May 1925) 808‑809. 

Anon.  The late Mr. W. W. Rouse Ball.  The Trinity Magazine (Jun 1925) 53-54.

Anon.  Entry in Who's Who, 1925, p. 127.

Anon.  Wills and bequests:  Mr. Walter Willliam Rouse Ball.  The Times (7 Sep 1925) 15.

E. T. Whittaker.  Obituary.  W. W. Rouse Ball.  Math. Gaz. 12 (No. 178) (Oct 1925) 449-454, with photo opp. p. 449.

F. Cajori.  Walter William Rouse Ball.  Isis 8 (1926) 321‑324.  Photo on plate 15, opp. p. 321.  Copy of Ball's 1924 Xmas card on p. 324.

J. A. Venn.  Alumni Cantabrigienses.  Part II:  From 1752 to 1900.  Vol. I, p. 136.  CUP, 1940.

David Singmaster.  Walter William Rouse Ball (1850-1925).  6pp handout for 1st UK Meeting on the History of Recreational Mathematics, 24 Oct 1992.  Plus extended biographical (6pp) and bibliographical (8pp) notes which repeat some of the material in the handout.

No DNB or DSB entry _ however I have offered to write a DNB entry.  I have since seen the proposed list of names for the next edition and Ball is already on it.


          Henry Ernest DUDENEY  (1857‑1930)


Anon. & Dudeney.  A chat with the puzzle king.  The Captain 2 (Dec? 1899) 314‑320, with photo.  Partly an interview.  Includes photos of Littlewick Meadow.

Anon.  Solutions to "Sphinx's puzzles".  The Captain 2:6 (Mar 1900) 598‑599  &  3:1 (Apr 1900) 89.

Anon.  Master of the breakfast table problem.  Daily Mail (1 Feb 1905) 7.  An interview with Dudeney in which he gives the better version of his spider and fly problem.

Fenn Sherie.  The Puzzle King: An Interview with Henry E. Dudeney.  Strand Magazine 71 (Apr 1926) 398‑4O4.

Alice Dudeney.  Preface to PCP, dated Dec 1931, pp. vii‑x.  The date of his death is erroneously given as 1931.

Angela Newing.  The Life and Work of H. E. Dudeney.  MS 21 (1988/89) 37‑44.

Angela Newing is working on a biography.

No DNB or DSB entry.  I have suggested a DNB entry.


          Wilhelm AHRENS  (1872‑1927)


Wilhelm Lorey.  Wilhelm Ahrens zum Gedächtnis.  Archiv für Geschichte der Mathematik, der Naturwissenschaften und der Technik 10 (1927/28) 328‑333.  Photo on p. 328.

O. Staude.  Dem Andenken an Dr. Wilhelm Ahrens.  Jahresbericht DMV 37 (1928) 286-287.

No DSB entry.


          Hubert PHILLIPS  (1891-1964)


Hubert Phillips.  Journey to Nowhere.  A Discursive Autobiography.  Macgibbon & Kee, London, 1960.  ??NYR

No DNB entry _ I have suggested one.





H. E. Dudeney.  Great puzzle crazes.  London Magazine 13?? (Nov 1904) 478‑482.  Fifteen Puzzle.  Pigs in Clover, Answers, Pick-me-up (spiral ramp) and other dexterity puzzles.  Get Off the Earth.  Conjurer's Medal (ring maze).  Chinese Rings.  Chinese Cross (six piece burr).  Puzzle rings.  Solitaire.  The Mathematician's Puzzle (square, circle, triangle).  Imperial Scale.  Heart and Balls.

H. E. Dudeney.  Puzzles from games.  Strand Magazine 35 (No. 207) (Mar 1908) 339‑344.  Solutions.  Ibid. 35 (No. 208) (Apr 1908) 455‑458.

H. E. Dudeney.  Some much‑discussed puzzles.  Strand Magazine 35 (No. 209) (May 1908) 580‑584.  Solutions.  Ibid. 35 (No. 210) (Jun 1908) 696.

H. E. Dudeney.  The world's best puzzles.  Strand Magazine 36 (No. 216) (Dec 1908) 779‑787.  Solutions.  Ibid. 37 (No. 217) (Jan 1909) 113‑116.

H. E. Dudeney.  The psychology of puzzle crazes.  The Nineteenth Century 100:6 (Dec 1926) 868‑879.  Repeats much of his 1904 article.

Sam Loyd Jr.  Are you good at solving puzzles?  The American Magazine (Sep 1931) 61‑63, 133‑137.

Orville A. Sullivan.  Problems involving unnusual situations.  SM 9 (1943) 114‑118  &  13 (1947) 102‑104.





          I have tried to divide this material into historical and bibliographical parts, but the two overlap considerably. 




Raffaella Franci.  Giochi matematici in trattati d'abaco del medioevo e del rinascimento.  Atti del Convegno Nazionale sui Giochi Creative, Siena, 11-14 Jun 1981.  Tipografia Senese for GIOCREA (Società Italiana Giochi Creativi), 1981.  Pp. 18-43.  Describes and quotes many typical problems.  17 references, several previously unknown to me.

Heinrich Hermelink.  Arabische Unterhaltungsmathematik als Spiegel Jahrtausendealter Kulturbeziehungen zwischen Ost und West.  Janus 65 (1978) 105-117, with English summary.  An English translation appeared as:  Arabic recreational mathematics as a mirror of age-old cultural relations between Eastern and Western civilizations; in:  Ahmad Y. Al-Hassan, Ghada Karmi & Nizar Namnum, eds.; Proceedings of the First International Symposium for the History of Arabic Science, April 1976 _ Vol. Two: Papers in European Languages; Institute for the History of Arabic Science, Aleppo, 1978, pp. 44-52.  (There are a few translation and typographical errors, which make it clear that the English version is a translation of the German.)

D. E. Smith.  On the origin of certain typical problems.  AMM 24 (1917) 64‑71.  (This is mostly contained in his History, vol. II, pp. 536‑548.)




          Many of the items cited in the Common References have extensive bibliographies.  In particular:  BLC;  BMC;  BNC;  DNB;  DSB;  NUC;  Schaaf;  Smith & De Morgan: Rara;  Suter  are basic bibliographical sources.  Datta & Singh;  Dickson;  Heath: HGM;  Murray;  Sanford: H&S  &  Short History;  Smith: History  &  Source Book;  Struik;  Tropfke  are histories with extensive bibliographical references.  AR;  BR  are editions of early texts with substantial bibliographical material.  Ahrens: MUS;  Ball: MRE;  Berlekamp, Conway & Guy: Winning Ways;  Gardner;  Lucas: RM  are recreational books with some useful bibliographical material.  Of these, the material in Ahrens is by far the most useful.  The magic bibliographies of Christopher, Clarke & Blind, Hall, Heyl, Price (see HPL), Toole Stott and Volkmann & Tummers have considerable overlap with the present material, particularly for older books, though Hall, Heyl and Toole Stott restrict themselves to English material, while Volkmann & Tummers only considers German.  Santi is also very useful.  Below I give some additional bibliographical material which may be useful, arranged in author order.


Anonymous.  Mathematical bibliography.  SSM 48 (1948) 757‑760.  Covers recreations.


Wilhelm Ahrens.  Mathematische Spiele.  Section I G 1 of Encyklopadie der Math. Wiss., Vol. I, part 2, Teubner, Leipzig, 1900‑1904, pp. 1080‑1093.

Raymond Clare Archibald.  Notes on some minor English mathematical serials.  MG 14 (1928-29) 379-400.

Elliott M. Avedon  &  Brian Sutton‑Smith.  The Study of Games.  (Wiley, NY, 1971);  Krieger, Huntington, NY, 1979.

Anthony S. M. Dickins.  A Catalogue of Fairy Chess Books and Opuscules Donated to Cambridge University Library, 1972‑1973, by Anthony Dickins M.A.  Third ed., Q Press, Kew Gardens, UK, 1983.

Underwood Dudley.  An annotated list of recreational mathematics books.  JRM 2:1 (Jan 1969) 13-20.  61 titles, in English and in print at the time.

Aviezri S. Fraenkel.  Selected Bibliography on Combinatorial Games and Some Related Material.  There have been several versions with slightly varying titles.  The most recent printed version is:  400 items, 28 pp., including 4 pp of text, Sep 1990.  Technical Report CS90‑23, Weizmann Institute of Science, Rehovot, Israel.  = Proc. Symp. Appl. Math. 43 (1991) 191-226.  Fraenkel has since produced Update 1 to this which lists 430 items on 31pp, Aug 1992;  and Update 2, 480 items on 33pp, with 5 pp of text, accidentally dated Aug 1992 at the top but produced in Feb 1994.  On 22 Nov 1994, it became a dynamic survey on the Electronic J. Combinatorics and can be accessed from:


          It can also be accessed via anonymous ftp from  ftp.wisdom.weizmann.ac.il.  After logging in, do  cd pub/fraenkel  and then  get  one of the following three compressed files:  games.tex.z;  games.dvi.z;  games.ps.z.

Martin P. Gaffney  &  Lynn Arthur Steen.  Annotated Bibliography of Expository Writing in the Mathematical Sciences.  MAA, 1976.

JoAnne S. Growney.  Mathematics and the arts _ A bibliography.  Humanistic Mathematics Network Journal 8 (1993) 22-36.  General references.  Aesthetic standards for mathematics and other arts.  Biographies/autobiographies of mathematicians.  Mathematics and display of information (including mapmaking).  Mathematics and humor.  Mathematics and literature (fiction and fantasy).  Mathematics and music.  Mathematics and poetry.  Mathematics and the visual arts. 

JoAnne S. Growney.  Mathematics in Literature and Poetry.  Humanistic Mathematics Network Journal 10 (Aug 1994) 25-30.  Short survey.  3 pages of annotated references to 29 authors, some of several books.

R. C. Gupta.  A bibliography of selected book [sic] on history of mathematics.  The Mathematics Education 23 (1989) 21-29.

Trevor H. Hall.  Mathematicall Recreations.  Op. cit. in 1.  This is primarily concerned with the history of the book by van Etten.  [This booklet is revised as pp. 83-119 of Hall, OCB _ see Section 1.]

Catherine Perry Hargrave.  A History of Playing Cards and a Bibliography of Cards and Gaming.  (Houghton Mifflin, Boston, 1930);  Dover, 1966.

Susan Hill.  Catalogue of the Turner Collection of the History of Mathematics Held in the Library of the University of Keele.  University Library, Keele, 1982.  (Sadly this collection was secretly sold by Keele University in 1998 and has now been dispersed.)

Honeyman Collection _ see: Sotheby's.

Horblit Collection _ see: Sotheby's and H. P. Kraus.

Else Høyrup.  Books about Mathematics.  Roskilde Univ. Center, PO Box 260, DK‑4000, Roskilde, Denmark, 1979.

D. O. Koehler.  Mathematics and literature.  MM 55 (1982) 81-95.  64 references.  See Utz for some further material.

H. P. Kraus (16 East 46th Street, New York, 10017).  The History of Science including Navigation.

Catalogue 168.  A First Selection of Books from the Library of Harrison D. Horblit.  Nd [c1976].

Catalogue 169.  A Further Selection of Books, 1641-1700 (Wing Period) from the Library of Harrison D. Horblit.  Nd [c1976].

Catalogue 171.  Another Selection of Books from the Library of Harrison D. Horblit.  Nd [c1976].

          These are the continuations of the catalogues issued by Sotheby's, qv.

John S. Lew.  Mathematical references in literature.  Humanistic Mathematics Network Journal 7 (1992) 26-47.

Antonius van der Linde.  Das erst Jartausend [sic] der Schachlitteratur _ (850‑1880).  (1880);  Facsimile reprint by Caissa Limited Editions, Yorklyn, Delaware, 1979, HB.

Andy Liu.  Appendix III:  A selected bibliography on popular mathematics.  Delta-k 27:3 (Apr 1989)  _  Special issue:  Mathematics for Gifted Students, 55-83.

Édouard Lucas.  Récréations mathématiques, vol 1 (i.e. RM1), pp. 237-248 is an Index Bibliographique.

Felix Müller.  Führer durch die mathematische Literature mit besonderer Berücksichtigung der historisch wichtigen Schriften.  Abhandlungen zur Geschichte der Mathematik 27 (1903).

Charles W. Newhall.  "Recreations" in secondary mathematics.  SSM 15 (1915) 277‑293.

Mathematical Association.  259 London Road, Leicester, LE2 3BE.

Catalogue of Books and Pamphlets in the Library.  No details, [c1912], 19pp, bound in at end of Mathematical Gazette, vol. 6 (1911‑1912).

A First List of Books & Pamphlets in the Library of the Mathematical Association _ Books and Pamphlets acquired before 1924.  Bell, London, 1926.

A Second List of Books & Pamphlets in the Library of the Mathematical Association _ Books and Pamphlets acquired during 1924 and 1925.  Bell, London, 1929.

A Third List of Books & Pamphlets in the Library of the Mathematical Association _ Books and Pamphlets added from 1926 to 1929.  Bell, London, 1930.

A Fourth List of Books & Pamphlets in the Library of the Mathematical Association _ Books and Pamphlets added from 1930 to 1935.  Bell, London, 1936. 

                    Lists 1‑4 edited by E. H. Neville.

Books and Periodicals in the Library of the Mathematical Association.  Ed. by R. L. Goodstein.  MA, 1962.  Includes the four previous lists and additions through 1961.

SEE ALSO:  Riley; Rollett; F. R. Watson.

Poggendorff, J. C.  Biographisch-Literarisches Handwörterbuch zur Geschichte der Exacten Wissenschaften enthaltend Nachweisungen über Lebensverhältnisse und Leistungen von Mathematikern, Astronomen, Physikern, Chemikern, Mineralogen, Geologen usw aller Völker und Zeiten.  Johann Ambrosius Barth, Leipzig.  Facsimile by Maurizio Martino, Storrs-Mansfield (later Mansfield Center), Connecticut.

                    Vols. I (A - L) and II (M - Z).  (1863), nd [bought in 1996].

                    Vol. III (1858-1883), edited by B. W. Feddersen & A. J. von Oettingen;

                              Parts I (A - L) and II (M - Z).  (1898), nd [bought in 1998].

                    Vol. IV (1883-1904), edited by Arthur von Oettingen;

                              Parts I (A - L) and II (M - Z).  (1904), nd [bought in 1998].

          [Vol. V covers 1904-1922.  Vol. VI covers 1922-1949.  One can get I-VI on microfiche.  Vol. VIIa covers 1932-1953 and apparently comprises 5 volumes.  There is also a VIIa Supplement which gives material supplementary to vols. I-VI.]

Stanley Rabinowitz.  Index to Mathematical Problems  1980-1984.  MathPro Press, Westford, Massachusetts, 1992.

Cecil B. Read  &  James K. Bidwell.

Selected articles dealing with the history of elementary Mathematics.  SSM 76 (1976) 477-483.

Periodical articles dealing with the history of advanced mathematics _ Parts I & II.  SSM 76 (1976) 581-598  &  687-703.

Rudolf H. Rheinhardt.  Bibliography on Whist and Playing Cards.  From:  Whist Scores and Card-table Talk, Chicago, 1887.  Reprinted by L. & P. Parris, Llandrindod Wells, nd [1980s].

Pietro Riccardi.  Biblioteca Matematica Italiana dalla Origine della Stampa ai Primi Anni del Secolo XIX.  G. G. Görlich, Milan, 1952, 2 vols.  This work appeared in several parts and supplements in the late 19C and early 20C, mostly published by the Società Tipografica Modense, Modena, 1878-1893.  Because it appeared in parts, the contents of early copies are variable and even the reprints may vary.  The contents of this set are as follows.

          I.        20pp prelims   +  Col. 1 - 656 (Abaco - Kirchoffer).  [= original Vol. I.]

                    Col. 1 - 676 (La Cometa - Zuzzeri)  +  2pp correzioni.  [= original Vol. II.]

          II.      4pp titles and reverses.  Correzioni ed Aggiunte.  [= original Appendice.]

                    Serie I.a  Col. 1 - 78  +  1½pp  Continuazione delle Correzioni (note that these

                              have  Pag.  when they mean  Col.).

                    Serie II.a.  Col. 81 - 156.

                    Serie III.a.  Col. 157 - 192  +  Aggiunte al Catalogo delle Opere di sovente citate,

                              col. 193-194  +  1p Continuazione delle Correzioni (note that these have 

                              Pag.  when they mean  Col.). 

                    Serie IV.a.  Col. 197 - 208  +  Seconda Aggiunta al Catalogo delle Opere più di

                              sovente citate, col. 209 - 212  +  Continuazione delle Correzioni in

                              col. 211-212.

                    Serie V.a.  Col. 1 - 180.

                    Serie VI.a.  Col. 179 - 200.

                              Serie V & VI must have been published as one volume as Serie V ends

                                        halfway down a page and then Serie VI begins on the same page.

                    Serie VII.a.  2pp introductory note by Ettore Bortolotti in 1928 saying that this

                              material was left as a manuscript by Riccardi and never previously

                              published  +  Col. 1 - 106.

                    Indice Alfabetico, of authors, covering the original material and all seven Series

                              of Correzioni ed Aggiunte, in 34 unnumbered columns.

                    Parte Seconda.  Classificazione per materie delle opere nella Parte I.  18pp

                              (including a chronological table)  +  subject index, pp. 1 - 294.

                    Catalogo  Delle opere più di sovente citate, col. 1 - 54.

                    [I have seen an early version which had the following parts:  Vol. I, 1893, col. 1‑656;  Vol. II, 1873, col. 1-676;  Appendice, 1878-1880-1893, col. 1-228.  Appendice, nd, col. 1-212.  Serie V, col. 1-228.  Parte 2, Vol. 1, 1880, pp. 1-294.  Renner Katalog 87 describes it as 5 in 2 vols.]

A. W. Riley.  School Library Mathematics List _ Supplement No. 1.  MA, 1973. 

          SEE ALSO:  Rollett.

Tom Rodgers.  Catalog of his collection of books on recreational mathematics, etc.  The author, Atlanta, May 1991, 40pp.

Leo F. Rogers.  Finding Out in the History of Mathematics.  Produced by the author, London, c1985, 52pp.

A. P. Rollett.  School Library Mathematics List.  Bell, London, for MA, 1966. 

          SEE ALSO:  Riley.

Charles L. Rulfs.  Origins of some conjuring works.  Magicol 24 (May 1971) 3-5.

José A. Sánchez Pérez.  Las Matematicas en la Biblioteca del Escorial.  Imprenta de Estanislao Maestre, Madrid, 1929.

William L. Schaaf. 

List of works on recreational mathematics.  SM 10 (1944) 193-200. 

                    PLUS:  A. Gloden; Additions to Schaaf's "List of works on mathematical recreations"; SM 13 (1947) 127.

A Bibliography of Recreational Mathematics.  Op. cit. in Common References, 4 vols., 1955-1978.  In these volumes he gives several lists of relevant books.

                    Books for the periods 1900-1925 and 1925-c1956 are given as Sections 1.1 (pp. 2-3) and 1.2 (pp. 4-12) in Vol. 1. 

                    Chapter 9, pp. 144-148, of Vol. 1, is a Supplement, generally covering c1954-c1962, but with some older items. 

                    In Vol. 2, 1970, the Appendix, pp. 181-191, extends to c1969, including some older items and repeating a few from the Supplement of Vol. 1. 

                    Appendix A of Vol. 3, 1973, pp. 111-113, adds some more items up through 1972. 

                    Appendix A, pp. 134-137, of Vol. 4, 1978, extends up through 1977.

  The following VESTPOCKET BIBLIOGRAPHIES are extensions of the material

                    in his Bibliographies.

No. 1: Pythagoras and rational triangles;  Geoboards and lattices.  JRM 16:2 (1983-84) 81-88.

No. 2: Combinatorics;  Gambling and sports.  JRM 16:3 (1983-84) 170-181.

No. 3: Tessellations and polyominoes;  Art and music.  JRM 16:4 (1983-84) 268‑280.

No. 4: Recreational miscellany.  JRM 17:1 (1984-85) 22-31.

No. 5: Polyhedra;  Topology;  Map coloring.  LRM 17:2 (1984-85) 95-105.

No. 6: Sundry algebraic notes.  JRM 17:3 (1984-85) 195-203.

No. 7: Sundry geometric notes.  JRM 18:1 (1985-86) 36-44.

No. 8: Probability;  Gambling.  JRM 18:2 (1985-86) 101-109.

No. 9: Games and puzzles.  JRM 18:3 (1985-86) 161-167.

No. 10:         Recreational mathematics;  Logical puzzles;  Expository mathematics.  JRM 18:4 (1985-86) 241-246.

No. 11:         Logic, Artificial intelligence, and Mathematical foundations.  JRM 19:1 (1987) 3-9.

No. 12:         Magic squares and cubes;  Latin squares;  Mystic arrays and Number patterns.  JRM 19:2 (1987) 81-86.


The High School Mathematics Library.  NCTM, (1960, 1963, 1967, 1970, 1973);  6th ed., 1976;  7th ed., 1982;  8th ed., 1987. 

          SEE ALSO:  Wheeler; Wheeler & Hardgrove.

Early Books on Magic Squares.  JRM 16:1 (1983-84) 1-6.

William L. Schaaf  &  David Singmaster.  Books on Recreational Mathematics.  A Supplement to the Lists in William L. Schaaf's A Bibliography of Recreational Mathematics.  Collected by William L. Schaaf;  typed and annotated by David Singmaster.  School of Computing, Information Systems and Mathematics, South Bank University, London, SE1 0AA.  18pp, Dec 1992 and revised several times afterwards.

Peter Schreiber.

Mathematik und belletristik [1.] & 2. Teil.  Mitteilungen der Mathematischen Gesellschaft der Deutschen Demokratischer Republik.  (1986), no. 4, 57-71  &  (1988), no. 1-2, 55-61.  Good on German works relating mathematics and arts.

Mathematiker als Memoirenschreiber.  Alpha (Berlin) (1991), no. 4, no page numbers on copy received from author.  Extends previous work.

S. N. Sen.  Scientific works in Sanskrit, translated into foreign languages and vice‑versa in the 18th and 19th century A.D.  Indian J. History of Science 7 (1972) 44‑70.

Will Shortz.  Puzzleana [catalogue of his puzzle books].  Produced by the author.  14 editions have appeared.  The latest is:  May 1992, 88pp with 1175 entries in 26 categories, with indexes of authors and anonymous titles.  Some entries cover multiple items.  In Jan 1995, he produced a 19pp Supplement extending to 1451 entries.

David Singmaster.

The Bibliography of Some Recreational Mathematics Books.  (Report SBU-CISM-94-09, Nov 1994, 39pp.)  2nd ed., Technical Report SBU-CISM-95-08; School of Computing, Information Systems, and Mathematics; South Bank University, London, Aug 1995, 41pp.  3rd ed., as above: SBU-CISM-96-12, Jun 1996, 42pp.  4th ed., as above: SBU-CISM-98-02, 44pp.

Books on Recreational Mathematics, etc. [Catalogue of my collection.]  Produced by the author, London, various editions.  Current version is Technical Report SBU-CISM-99-14; School of Computing, Information Systems, and Mathematics; South Bank University, London, May 1999, with c6015 items and a supplement of 133 old books, 166pp.

Index to Martin Gardner's Columns and Cross Reference to His Books.  (Oct 1993.)  Slightly revised as:  Technical Report SBU-CISM-95-09; School of Computing, Information Systems, and Mathematics; South Bank University, London, Aug 1995, 22pp.

Harold Adrian Smith.  Dick and Fitzgerald Publishers.  Books at Brown 34 (1987) 108-114. 

Sotheby's [Sotheby Parke Bernet]. 

Catalogue of the J. B. Findlay Collection  Books and Periodicals on Conjuring and the Allied Arts.  Part I: A-O  5-6 Jul 1979.  Part II: P-Z  plus:  Mimeographed Books and Instructions; Flick Books  Catalogues of Apparatus and Tricks  Autograph Letters, Manuscripts, and Typescripts  4-5 Oct 1979.  Part III: Posters and Playbills  3-4 Jul 1980.  Each with estimates and results lists.

The Celebrated Library of Harrison D. Horblit Esq.  Early Science  Navigation & Travel  Including Americana  with a few medical books.  Part I  A - C  10/11 Jun 1974.  Part II  D - G  11 Nov 1974.  HB.  The sale was then cancelled and the library was sold to E. P. Kraus, qv, who issued three further catalogues, c1976.

The Honeyman Collection of Scientific Books and Manuscripts.  Seven volumes, each

                    with estimates and results booklets.

          Part I: Printed Books A-B, 30-31 Oct 1978.

          Part II:  Printed Books C-E, 30 Apr - 1 May 1979.

          Part III: Manuscripts and Aurograph Letters of the 12th to the 20th Centuries.

          Part IV: Printed Books F-J, 5-6 Nov 1979.

          Part V: Printed Books K-M, 12-13 May 1980.

          Part VI: Printed Books N-Sa, 10-11 Nov 1980.

          Part VII:  Printed Books Sc-Z and Addenda, 19-20 May 1981.

Lynn A. Steen, ed. 

Library Recommendations for Undergraduate Mathematics.  MAA Reports No. 4, 1992.

Two-Year College Mathematics Library Recommendations.  MAA Reports No. 5, 1992.

Strens/Guy Collection.  Author/Title Listing.  Univ. of Calgary.  Preliminary Catalogue, 319 pp., July 1986.  [The original has a lot of blank space.  I have a computer version which is reduced to 67pp.]

Eva Germaine Rimington Taylor.  The Mathematical Practitioners of Tudor & Stuart England  1485-1714.  CUP for the Institute of Navigation, 1970.

Eva Germaine Rimington Taylor.  The Mathematical Practitioners of Hanoverian England  1714‑1840.  CUP for the Institute of Navigation, 1966. 

                    PLUS:  Kate Bostock, Susan Hurt & Michael Hart; An Index to the Mathematical Practitioners of Hanoverian England  1714-1840; Harriet Wynter Ltd., London, 1980.

W. R. Utz.  Letter:  Mathematics in literature.  MM 55 (1982) 249‑250.  Utz has sent his 3pp original more detailed version along with 4pp of further citations.  This extends Koehler's article.

George Walker.  The Art of Chess-Play: A New Treatise on the Game of Chess.  4th ed., Sherwood, Gilbert & Piper, London, 1846.  Appendix:  Bibliographical Catalogue of the chief printed books, writers, and miscellaneous articles on chess, up to the present time, pp. 339-375.

Frank R. [Joe] Watson, ed.  Booklists.  MA.

Puzzles, Problems, Games and Mathematical Recreations.  16pp, 1980.

Selections from the Recommended Books.  18pp, 1980.

Full List of Recommended Books.  105pp, 1984.

Margariete Montague Wheeler.  Mathematics Library _ Elementary and Junior High School.  5th ed., NCTM, 1986. 

          SEE ALSO:  Schaaf; Wheeler & Hardgrove.

Margariete Montague Wheeler  &  Clarence Ethel Hardgrove.  Mathematics Library _ Elementary and Junior High School.  NCTM, (1960;  1968;  1973);  4th ed., 1978. 

          SEE ALSO:  Schaaf; Wheeler.

Ernst Wölffing.  Mathematischer Bücherschatz.  Systematisches Verzeichnis der wichtigsten deutschen und ausländischen Lehrbücher und Monographien des 19. Jahrhunderts auf dem Gebiete der mathematischen Wissenschaften.  I:  Reine Mathematik;  (II:  Angewandte Mathematik never appeared).  AGM 16, part I (1903).





Aviezri S. Fraenkel.  Selected Bibliography on Combinatorial Games and Some Related Material.  Op. cit. in 3.B.




          Conway's extension of this theory is well described in Winning Ways and later work is listed in Frankel's Bibliography _ see section 3.B & 4 _ so I will not cover such material here.


          4.A.1. ONE PILE GAME


          See MUS I 145-147.

          (a, b)  denotes the game where one can take  1, 2, ...,  or  a  away from one pile, starting with  b  in the pile, with the last player winning.  The version  (10, 100)  is sometimes called Piquet des Cavaliers or Piquet à Cheval, a name which initially perplexed me.  Piquet is one of the older card games, being well known to Rabelais (1534) and was known in the 16C as Cent (or Saunt or Saint) because of its goal of 100 points.  See:  David Parlett; (Oxford Guide to Card Games, 1990  =)  A History of Card Games; OUP, 1991, pp. 24 & 175-181.  The connection with horses undoubtedly indicates that  (10, 100)  was viewed as a game which could be played without cards, while riding _ see Les Amusemens.



(  3,   13)                Dudeney,

(  3,   15)                Hoffmann, Mr. X, Dudeney, Blyth,

(  3,   17)                Fourrey,

(  3,   21)                Blyth,

(  6,   30)                Pacioli, Leske, Ducret,

(  6,   31)                Baker,

(  6,   50)                Ball-Fitzpatrick,

(  7,   41)                Sprague,

(  7,   60)                Fourrey,

(  8, 100)                Bachet,

(  9, 100)                Bachet, Ozanam, Alberti

(10, 100)                Bachet, Henrion, Ozanam, Alberti, Les Amusemens, Hooper, Badcock,

                                        Jackson, Manuel des Sorciers, Boy's Own Book, Nuts to Crack,

                                        Young Man's Book, Magician's Own Book, Book of 500 Puzzles,

                                        Boy's Own Conjuring Book, Riecke, Secret Out, Fourrey, Ducret,


(10, 120)                Bachet,

          General case:  Bachet, Ozanam, Alberti, Boy's Own Book, Young Man's Book, (others ?? check)

          Versions with limited numbers of each value or using a die _ see 4.A.1.a.

          Version where an odd number in total has to be taken:  Dudeney, Grossman & Kramer, Sprague.


Pacioli.  De Viribus.  c1500.  Prob. 34: A finire qualunch' numero nanze al compagno a non prendere più de un terminato numero.  Phrases it as an addition problem.  Considers  (6, 30)  and the general problem.

David Parlett.  (Originally:  The Oxford Guide to Card Games; OUP, 1990);  reissued as:  A History of Card Games.  Penguin, 1991, pp. 174-175.  "Early references to 'les luettes', said to have been played by Anne de Bretagne and Archduke Philip the Fair in 1503, and by Gargantua in 1534, seem to suggest a game of the Nim family (removing numbers of objects from rows and columns)."

Cardan.  Practica Arithmetice.  1539.  Chap. 61, section 18, ff. T.iiii.v - T.v.r (p. 113).  "Ludi mentales".  One has  1, 3, 6  and the other has  2, 4, 5;  or one has  1, 3, 5, 8, 9  and the other has  2, 4, 6, 7, 10;  one one wants to make  100.  "Sunt magnæ inventionis, & ego inveni æquitando & sine aliquo auxilio cum socio potes ludere & memorium exercere ...."

Baker.  Well Spring of Sciences.  1562?  Prob. 5: To play at 31 with Numbers, 1670: pp. 353‑354.  ??NX.  (6, 31).

Bachet.  Problemes.  1612.  Prob. XIX: 1612, 99-103.  Prob. XXII, 1624: 170-173;  1884: 115‑117.  Phrases it as an addition problem.  First considers  (10, 100),  then  (10, 120),  (8, 100),  (9, 100),  and the general case.  Labosne omits the demonstration.

Dennis Henrion.  Nottes to van Etten.  1630.  Pp. 19-20.  (10, 100)  as an addition problem, citing Bachet.

Ozanam.  1694.  Prob. 21, 1696: 71-72;  1708: 63‑64.  Prob. 25, 1725: 182‑184.  Prob. 14, 1778: 162-164;  1803: 163-164;  1814: 143-145.  Prob. 13, 1840: 73-74.  Phrases it as an addition problem.  Considers  (10, 100)  and  (9, 100)  and remarks on the general case.

Alberti.  1747.  Due persone essendo convenuto ..., pp. 105‑108 (66‑67).  This is a slight recasting of Ozanam.

Les Amusemens.  1749.  Prob 10, p. 130: Le Piquet des Cavaliers.  (10, 100)  in additive form.  "Deux amis voyagent à cheval, l'un propose à l'autre un cent de Piquet sans carte." 

William Hooper.  Rational Recreations, In which the Principles of Numbers and Natural Philosophy Are clearly and copiously elucidated, by a series of Easy, Entertaining, Interesting Experiment.  Among which are All those commonly performed with the cards. [Taken from my 2nd ed.]  4 vols., L. Davis et al., London, 1774;  2nd ed., corrected, L. Davis et al., London, 1783-1782 (vol. 1 says 1783, the others say 1782; BMC gives 1783-82);  3rd ed., corrected, 1787;  4th ed., corrected, B. Law et al., London, 1794.  [Hall, BCB 180-184  &  Toole Stott 389-392.  Hall says the first four eds. have identical pagination.  I have not seen any difference in the first four editions.  Hall, OCB, p. 155.  Heyl 177 notes the different datings of the 2nd ed,  Hall, BCB 184 and Toole Stott 393 is a 2 vol. 4th ed., corrected, London, 1802.  Toole Stott 394 is a 2 vol. ed. from Perth, 1801.  I have a note that there was an 1816 ed, but I have no details.  Since all relevant material seems the same in all volumes, I will cite this as 1774.]  Vol. 1, recreation VIII: The magical century.  (10, 100)  in additive form.  Mentions other versions and the general rule.

Badcock.  Philosophical Recreations, or, Winter Amusements.  [1820].  Pp. 33-34, no. 48: A curious recreation with a hundred numbers, usually called the magical century.  (10, 100)  as an additive problem where each person starts with  50  counters.  Discusses general case, but doesn't notice that the limitation to  50  counters each considerably changes the game!

Jackson.  Rational Amusement.  1821.  Arithmetical Puzzles, no. 47, pp. 11 & 64.  Additive form of  (10, 100).

Manuel des Sorciers.  1825.  Pp. 57-58, art. 30: Le piquet sans cartes.  ??NX  (10, 100) done subtractively.

The Boy's Own Book.

The certain game.  1828: 177;  1828-2: 236;  1829 (US): 104;  1855: 386‑387;  1868: 427.

The magical century.  1828: 180;  1828-2: 236‑237;  1829 (US): 104-105;  1855: 391‑392. 

          Both are additive phrasings of  (10, 100).  The latter mentions using other numbers and how to win then.

Nuts to Crack V (1836), no. 70.  An arithmetical problem.  (10, 100).

Young Man's Book.  1839.  Pp. 294-295.  A curious Recreation with a Hundred Numbers, usually called the Magical Century.  Almost identical to Boy's Own Book.

Magician's Own Book.  1857.

The certain game, p. 243.  As in Boy's Own Book.

The magical century, pp. 244-245.  As in Boy's Own Book.

Book of 500 Puzzles.  1859.

The certain game, p. 57.  As in Boy's Own Book.

The magical century, pp. 58-59.  As in Boy's Own Book.

Boy's Own Conjuring Book.  1860.

The certain game, pp. 213‑214.  As in Boy's Own Book.

Magical century, pp. 215.  As in Boy's Own Book.

Leske.  Illustriertes Spielbuch für Mädchen.  1864?  Prob. 563-III, pp. 247: Wer von 30 Rechenpfennigen den letzen wegnimmt, hat gewonnen.  (6, 30).

F. J. P. Riecke.  Mathematische Unterhaltungen.  3 vols., Karl Aue, Stuttgart, 1867, 1868 & 1873;  reprint in one vol., Sändig, Wiesbaden, 1973.  Vol. 3, art 22.2, p. 44.  Additive form of  (10, 100).

The Secret Out or, One Thousand Tricks in Drawing-room or White Magic, with an Endless Variety of Entertaining Experiments.  By the author of "The Magician's Own Book."  Translated and edited by W. H. Cremer, Junr.  With three hundred illustrations.  [Toole Stott lists all versions under Cremer.]

                    (Dick & Fitzgerald, NY, 1859 [Toole Stott 191].  C&B, under Frikell, have New York, 1859)

                    John Camden Hotten, London, 1871? [NUC; Toole Stott 192; C&B, under Cremer, have London & New York, 1871, and, under Frikel, have 1870]. 

                    Chatto and Windus, nd [1871? _ NUC lists several dates; Toole Stott 1013 is 1870]. 

                    (John Grant, Edinburgh, 1872 [Toole Stott 1014].)

          [Toole Stott 192 discusses the authorship, saying that Wiljalba (or Gustave) Frikell is named on the TP of some editions, but that most of the tricks are taken from the US ed of The Magician's Own Book, qv for more discussion of the authorship.  In the US ed, Cremer acknowledges this, but in the Preliminary to the UK ed he says he is indebted to Le Magicien des Salons, and an 1874 ad by Chatto & Windus indicates that The Secret Out is translated from that work (this may be Le Magicien de Société, Delarue, Paris, c1860).  The back of the TP of Bellew's The Art of Amusing, Hotten, 1866?, (op. cit. in 5.E) says The Secret Out is a companion volume, just issued, by Hermann Frikell.  BM, Toole Stott & C&B say it is also attributed to Henry L. Williams.  Toole Stott 481 cites a 1910 letter from Harris B. Dick, of the publishers Dick & Fitzgerald, who thinks their version of The Secret Out "was a reprint of an English book by W. H. Cremer" _ but there seems to be no record of a UK ed before the US one.  NUC says an 1871 ed. gives author as Gustave Frikell.  I have seen an earlier (1859?) US ed which is quite different, much more about magic, consequently I will treat this as 1871?.  Christopher 240-242 are two copies from Dick & Fitzgerald, c1859, and a C&W, 1878?  He repeats most of the above comments from Toole Stott and cites the following article on this series.

                    Charles L. Rulfs.  Origins of some conjuring works.  Magicol 24 (May 1971) 3-5.  He discusses the various books, saying that Cremer essentially pirated the Dick & Fitzgerald productions.  He says The Secret Out is largely taken, illustrations and all, from Blismon de Douai's Manuel du Magicien (1849) and Richard & Delion's Magicien des salons ou le diable couleur de rose (1857 and earlier).  C&B, under Gustave Frikell, say it is a translation of Richard & Delion.  C&B, under Herrman Frikell, list London, 1870.  C&B, under Secret, list New York, nd.  C&B also list it under Williams, as London, 1871.

                    Piquet on horseback, pp. 130‑131 _ additive  (10, 100).

Hoffmann.  1893.  Chap VII, no. 19: The fifteen matches puzzle, pp. 292 & 300‑301.  (3, 15).  c= Benson, 1904, The fifteen match puzzle, pp. 241‑242.

Ball-FitzPatrick.  1st ed., 1898.  Deuxième exemple, pp. 29-30.  (6, 50).

E. Fourrey.  Récréations Arithmétiques.  (Nony, Paris, 1899;  2nd ed., 1901);  3rd ed., Vuibert & Nony, Paris, 1904;  (4th ed., 1907);  8th ed., Librairie Vuibert, Paris, 1947.  [The 3rd and 8th eds are identical except for the title page, so presumably are identical to the 1st ed.]  Sections 65‑66: Le jeu du piquet à cheval, pp. 48‑49.  Additive forms of  (10, 100)  and  (7, 60).  Then gives subtractive form for a pile of matches for  (3, 17).

Étienne Ducret.  Récréations Mathématiques.  Garnier Frères, Paris, nd [not in BN, but a similar book, nouv. ed., is 1892].  Pp. 102‑104: Le piquet à cheval.  Additive version of  (10, 100)  with some explanation of the use of the term piquet.  Discusses  (6, 30).

Mr. X [possibly J. K. Benson _ see entry for Benson in Abbreviations).  His Pages.  The Royal Magazine 9:3 (Jan 1903) 298-299.  A good game for two.  (3, 15) as a subtraction game.

David Devant.  Tricks for Everyone.  Clever Conjuring with Everyday Objects.  C. Arthur Pearson, London, 1910.  A counting race, pp. 52-53.  (10, 100).

Dudeney.  AM.  1917.  Prob. 392: The pebble game, pp. 117 & 240.  (3, 15)  &  (3, 13)  with the object being to take an odd number in total.  For 15, first player wins;  for 13, second player wins.  (Barnard (50 Telegraph ..., 1985) gives the case  (3, 13).)

Blyth.  Match-Stick Magic.  1921. 

Fifteen matchstick game, pp. 87-88.  (3, 15).

Majority matchstick game, p. 88.  (3, 21).

H. D. Grossman & David Kramer.  A new match-game.  AMM 52 (1945) 441‑443.  Cites Dudeney and says Games Digest (April 1938) also gave a version, but without solution.  Gives a general solution whether one wants to take an odd total or an even total.

Ronald Sprague.  Unterhaltsame Mathematik.  Vieweg, Braunschweig, 1961.  Translated by T. H. O'Beirne as:  Recreations in Mathematics, Blackie, London, 1963.  Problem 24: "Ungerade" gewinnt, pp. 16 & 44‑45.  (= 'Odd' is the winner, pp. 18 & 53‑55.)  (7, 41)  with the winner being the one who takes an odd number in total.  Solves  (7, b)  and states the structure for  (a, b). 

                    I also have some other recent references to this problem.  Lewis (1983) gives a general solution which seems to be wrong. 


          4.A.1.a.        THE 31 GAME


          Numerical variations:  Gibson, McKay.

          Die versions:  Secret Out, Loyd, Mott-Smith, Murphy.


John Fisher.  Never Give a Sucker an Even Break.  (1976);  Sphere Books, London, 1978.  Thirty-one, pp. 102-104.  (6, 31)  additively, but played with just 4 of each value, the 24 cards of ranks  1 _ 6,  and the first to exceed 31 loses.  He says it is played extensively in Australia and often referred to as "The Australian Gambling Game of 31".  Cites the 19C gambling expert Jonathan Harrington Green who says it was invented by Charles James Fox (1749‑1806).  Gives some analysis.

See Badcock, above for a version.

Nuts to Crack V (1836), no. 71.  (6, 31) additively, with four of each value.  "Set down on a slate, four rows of figures, thus:_  ...  You agree to rub out one figure alternately, to see who shall first make the number thirty-one."

Magician's Own Book.  1857.  Art. 31: The trick of thirty‑one, pp. 70‑71.  (6, 31)  additively, but played with just 4 of each value _ e.g. the 24 cards of ranks  1 _ 6.  The author advises you not to play it for money with "sporting men" and says it it due to Mr. Fox.  Cf. Fisher.  = Boy's Own Conjuring Book; 1860; Art. 29: The trick of thirty‑one, pp. 78‑79.

The Secret Out.  Op. cit. in 4.A.1.  1871?.  To throw thirty‑one with a die before your antagonist, p. 7.  This is incomprehensible, but is probably the version discussed by Mott-Smith.

Larry Freeman.  Yesterday's Games.  Taken from "an 1880 text" of games.  (American edition by H. Chadwick.)  Century House, Watkins Glen, NY, 1970.  P. 107: Thirty-one.  (6, 31)  with 4 of each value _ as in Magician's Own Book.

Algernon Bray.  Letter:  "31" game.  Knowledge 3 (4 May 1883) 268, item 806.  "... has lately made its appearance in New York, ...."  Seems to have no idea as how to win.

Loyd.  Problem 38: The twenty‑five up puzzle.  Tit‑Bits 32 (12 Jun  &  3 Jul 1897) 193  &  258.  = Cyclopedia.  1914.  The dice game, pp. 243 & 372.  = SLAHP: How games originate, pp. 73 & 114.  The first play is arbitrary.  The second play is by throwing a die.  Further values are obtained by rolling the die by a quarter turn.

Ball-FitzPatrick.  1st ed., 1898.  Généralization récente de cette question, pp. 30-31.  (6, 50)  with each number usable at most 3 times.  Some analysis.

Ball.  MRE, 4th ed., 1905, p. 20.  Some analysis of  (6, 50)  where each player can play a value at most 3 times _ as in Ball-FitzPatrick, but with the additional sentence:  "I have never seen this extension described in print ...."  He also mentions playing with values limited to two times.  In the 5th ed., 1911, pp. 19-21, he elaborates his analysis.

Dudeney.  CP.  1907.  Prob. 79: The thirty-one game, pp. 125-127 & 224.  Says it used to be popular with card-sharpers at racecourses, etc.  States the first player can win if he starts with 1, 2 or 5, but the analysis of cases 1 and 2 is complicated.

Devant.  Tricks for Everyone.  Op. cit. in 4.A.1.  1910.  The thirty-one trick, pp. 53-54.  Says to get to  3, 10, 17, 24.

Loyd Jr.  SLAHP.  1928.  The "31 Puzzle Game", pp. 3 & 87.  Loyd Jr says that as a boy, he often had to play it against all comers with a $50 prize to anyone who could beat 'Loyd's boy'.  This is the game that Loyd Sr called 'Blind Luck', but I haven't found it in the Cyclopedia.  States the first player wins with 1, 2 or 5, but only sketches the case for opening with 5.

McKay.  Party Night.  1940.  The 21 race, pp. 166.  Using the numbers  1, 2, 3, 4,  at most four times, achieve 21.  Says to get  1, 6, 11, 16.  He doesn't realise that the sucker can be mislead into playing first with a 1 and losing!  Says that with  1, ..., 5  at most four times, one wants to achieve 26 and that with  1, ..., 6  at most four times, one wants to achieve 31.  Gives just the key numbers each time.

Geoffrey Mott-Smith.  Mathematical Puzzles for Beginners and Enthusiasts.  (Blakiston, 1946);  revised 2nd ed., Dover, 1954.

Prob. 179: The thirty-one game, pp. 117-119  &  231-232.  As in Dudeney.

Prob. 180: Thirty-one with dice, p. 119  &  232-233.  Throw a die, then make quarter turns to produce a total of 31.  Analysis based on digital roots (i.e. remainders (mod 9)).  First player wins if the die comes up 4, otherwise the second player can win.  He doesn't treat any other totals.

"Willane".  Willane's Wizardry.  Academy of Recorded Crafts, Arts and Sciences, Croydon, 1947.  "Trente et un", pp. 56-57.  Says he doesn't know any name for this.  Get 31 using 4 each of the cards  A, 2, ..., 6.  Says first player loses easily if he starts with  4, 5, 6  (not true according to Dudeney) and that gamblers dupe the sucker by starting with 3 and winning enough that the sucker thinks he can win by starting with 3.  But if he starts with a 1 or 2, then the second player must play low and hope for a break.

Walter B. Gibson.  Fell's Guide to Papercraft Tricks, Games and Puzzles.  Frederick Fell, NY, 1963.  Pp. 54-55: First to fifty.  First describes  (50, 6),  but then adds a version with slips of paper:  eight marked 1 and seven marked with  2, 3, 4, 5, 6  and you secretly extract a 6 slip when the other player starts.

Harold Newman.  The 31 Game.  JRM 23:3 (1991) 205-209.  Extended analysis.  Confirms Dudeney.  Only cites Dudeney & Mott-Smith.

Bernard Murphy.  The rotating die game.  Plus 27 (Summer 1994) 14-16.  Analyses the die version as described by Mott-Smith and finds the set,  S(n),  of winning moves for achieving a count of  n  by the first player, is periodic with period 9 from  n = 8,  i.e.  S(n+9) = S(n)  for  n ³ 8.  There is no first player winning move if and only if  n  is a multiple of 9.  [I have confirmed this independently.]

Ken de Courcy.  The Australian Gambling Game of 31.  Supreme Magic Publication, Bideford, Devon, nd [1980s?].  Brief description of the game and some indications of how to win.  He then plays the game with face-down cards!  However, he insures that the cards by him are one of of each rank and he knows where they are.


          4.A.2.           SYMMETRY ARGUMENTS


Loyd??  Problem 43: The daisy game.  Tit‑Bits 32 (17 Jul  &  7 Aug 1897) 291  &  349.  (= Cyclopedia.  1914.  A daisy puzzle game, pp. 85 & 350.  c= MPSL2, prob. 57, pp. 40‑41 & 140.  c= SLAHP: The daisy game, pp. 42 & 99.)  Circular version of Kayles with 13 objects.  Solution uses a symmetry argument _ but the Tit‑Bits solution was written by Dudeney.

Dudeney.  Problem 500: The cigar puzzle.  Weekly Dispatch (7 Jun, 21 Jun, 5 Jul, 1903) all p. 16.  (= AM, prob. 398, pp. 119, 242.)  Symmetry in placement game, using cigars on a table.

Loyd.  Cyclopedia.  1914.  The great Columbus problem, pp. 169 & 361.  (= MPSL1, prob. 65, pp. 62 & 144.  = SLAHP: When men laid eggs, pp. 75 & 115.)  Placing eggs on a table.

M. Kraitchik.  Section XII, prob. 1, p. 296.  La Mathématique des Jeux.  Stevens, Bruxelles, 1930.  (= Problem 1, pp. 13‑14.  Mathematical Recreations.  Allen & Unwin, London, 1943.)  Child plays black and white against two chess players and guarantees to win one game.  [MJ cites L'Echiquier (1925) 84, 151.]

                    CAUTION.  The 2nd edition of Math. des Jeux, 1953, is a translation of Mathematical Recreations and hence omits much of the earlier edition.

Jules Leopold.  At Ease!  Whittlesey House (McGraw-Hill), NY, 1943.  [This appears to be largely drawn from Yank, The Army Weekly over the previous few years.]  Chess wizardry in two minutes, pp. 105‑106.  Same as Kraitchik.


          4.A.3. KAYLES


          This has objects in a line or a circle and one can remove one object or two adjacent objects (or more adjacent objects in a generalized version of the game).  This derives from earlier games with an array of pins at which one throws a ball or stick.

          Murray 442 cites Act 17 of Edward IV, c.3 (1477):  "Diversez novelx ymagines jeuez appellez Cloishe Kayles ..."  This outlawed such games.  A version is shown in Pieter Bruegel's painting "Children's Games" of 1560 with balls being thrown at a row of pins by a wall, in the back right of the scene.  Versions of the game are given in the works of Strutt and Gomme cited in 4.B.1.  Gomme II 115‑116 discusses it under Roly‑poly, citing Strutt and some other sources.  Strutt 270‑271 (= Strutt-Cox 219-220) calls it "Kayles, written also cayles and keiles, derived from the French word quilles".  He has redrawings of two 14C engravings showing lines of pins at which one throws a stick (= plate opp. 220 in Strutt-Cox).  He also says Closh or Cloish seems to be the same game and cites prohibitions of it in c1478 et seq.  Loggats was analogous and was prohibited under Henry VIII and is mentioned in Hamlet.


14C MS in the British Museum, Royal Library, No. 2, B. vii.  Reproduced in Strutt, p. 271.  Shows a monk(?) standing by a line of eight conical pins and another monk(?) throwing a stick at the pins.

Anonymous.  Games of the 16th Century.  The Rockliff New Project Series.  Devised by Arthur B. Allen.  The Spacious Days of Queen Elizabeth.  Background Book No. 5.  Rockcliff Publishing, London, ©1950, 4th ptg.  The Background Books seem to be consecutively paginated as this booklet is paginated 129-152.  Pp. 133-134 describes loggats, quoting Hamlet and an unknown poet of 1611.  P. 137 is a photograph of the above 14C illustration.  The caption is "Skittles, or "Kayals", and Throwing a Whirling Stick".

van Etten.  1624.  Prob. 72 (misnumbered 58) (65), pp 68‑69 (97‑98): Du jeu des quilles (Of the play at Keyles or Nine-Pins).  Describes the game as a kind of ninepins.

Loyd.  Problem 43: The daisy game.  Tit‑Bits 32 (17 Jul  &  7 Aug 1897) 291  &  349.  (= Cyclopedia.  1914.  A daisy puzzle game, pp. 85 & 350.  c= MPSL2, prob. 57, pp. 40‑41 & 140.  c= SLAHP: The daisy game, pp. 42 & 99.)  Circular version of Kayles with 13 objects.  See also 4.A.2.

Dudeney.  Sharpshooters puzzle.  Problem 430.  Weekly Dispatch (26 Jan, 9 Feb, 1902) both p. 13.  Simple version of Kayles.

Ball.  MRE, 4th ed., 1905, pp. 19-20.  Cites Loyd in Tit‑Bits.  Gives the general version:  place  p  counters in a circle and one can take not more than  m  adjacent ones.

Dudeney.  CP.  1907.  Prob. 73: The game of Kayles, pp. 118‑119 & 220.  Kayles with 13 objects.

Loyd.  Cyclopedia.  1914.  Rip van Winkle puzzle, pp. 232 & 369‑370.  (c= MPSL2, prob. 6, pp. 5 & 122.)  Linear version with 13 pins and the second knocked down.  Gardner asserts that Dudeney invented Kayles, but it seems to be an abstraction from the old form of the game.

Rohrbough.  Puzzle Craft, later version, 1940s?.  Daisy Game, p. 22.  Kayles with 13 petals of a daisy.

Doubleday - II.  1971.  Take your pick, pp. 63-65.  This is Kayles with a row of 10, but he says the first player can only take one.


          4.A.4. NIM


          Nim is the game with a number of piles and a player can take any number from one of the piles.  Normally the last one to play wins.


David Parlett.  (Originally:  The Oxford Guide to Card Games; OUP, 1990);  reissued as:  A History of Card Games.  Penguin, 1991.  Pp. 174-175.  "Early references to 'les luettes', said to have been played by Anne de Bretagne and Archduke Philip the Fair in 1503, and by Gargantua in 1534, seem to suggest a game of the Nim family (removing numbers of objects from rows and columns)."

Charles L. Bouton.  Nim: a game with a complete mathematical theory.  Annals of Math. (2) 3 (1901/02) 35‑39.  He says Nim is played at American colleges and "has been called Fan‑Tan, but as it is not the Chinese game of that name, the name in the title is proposed for it."  He says Paul E. More showed him the misère (= last player loses)  version in 1899, so it seems that Bouton did not actually invent the game himself.

Ahrens.  "Nim", ein amerikanisches Spiel mit mathematischer Theorie.  Naturwissenschaftliche Wochenschrift  17:22 (2 Mar 1902) 258‑260.  He says that Bouton has admitted that he had confused Nim and Fan‑Tan.  Fan‑Tan is a Chinese game where you bet on the number of counters (mod 4) in someone's hand.  Parker, Ancient Ceylon, op. cit. in 4.B.1, pp. 570-571, describes a similar game, based on odd and even, as popular in Ceylon and "certainly one of the earliest of all games".

                    For more about Fan-Tan, see the following.

Stewart Culin.  Chess and playing cards.  Catalogue of games and implements for divination exhibited by the United States National Museum in connection with the Department of Archæology and Paleontology of the University of Pennsylvania at the Cotton States and International Exposition, Atlanta, Georgia, 1895.  IN: Report of the U. S. National Museum, year ending June 30, 1896.  Government Printing Office, Washington, 1898, HB, pp. 665-942.  [There is a reprint by Ayer Co., Salem, Mass., c1990.]  Fan-Tan (= Fán t‘án = repeatedly spreading out) is described on pp. 891 & 896, with discussion of related games on pp. 889-902.

Alan S. C. Ross.  Note 2334:  The name of the game of Nim.  MG 37 (No. 320) (May 1953) 119‑120.  Conjectures Bouton formed the word 'nim' from the German 'nimm'.  Gives some discussion of Fan‑Tan and quotes MUS I 72.

J. L. Walsh.  Letter:  The name of the game of Nim.  MG 37 (No. 322) (Dec 1953) 290.  Relates that Bouton said that he had chosen the word from the German 'nimm' and dropped one 'm'.

W. A. Wythoff.  A modification of the game of Nim.  Nieuw Archief voor Wiskunde (Groningen) (2) 7 (1907) 199‑202.  He considers a Nim game with two piles allows the extra move of taking the same amount from both piles.  [Is there a version with more piles where one can take any number from one pile or equal amounts from two piles??  See Barnard, below for a three pile version.]

Ahrens.  MUS I.  1910.  III.3.VII: Nim, pp. 72‑88.  Notes that Nim is not the same as Fan‑Tan, has been known in Germany for decades and is played in China.  Gives a thorough discusion of the theory of Nim and of an equivalent game and of Wythoff's game.

E. H. Moore.  A generalization of the game called Nim.  Annals of Math.  (2) 11 (1910) 93‑94.  He considers a Nim game with  n  piles and one is allowed to take any number from at most  k  piles.

Ball.  MRE, 5th ed., 1911, p. 21.  Sketches the game of Nim and its theory.

A. B. Nordmann.  One Hundred More Parlour Tricks and Problems.  Wells, Gardner, Darton & Co., London, nd [1927 _ BMC].  No. 13: The last match, pp. 10-11.  Thirty matches divided at random into three heaps.  Last player loses.  Explanation of how to win is rather cryptic:  "you must try and take away ... sufficient ... to leave the matches in the two or three heaps remaining, paired in ones, twos, fours, etc., in respect of each other."

Loyd Jr.  SLAHP.  1928.  A tricky game, pp. 47 & 102.  Nim (3, 4, 8).

Emanuel Lasker.  Brettspiele der Völker.  1931.  See comments in 4.A.5.

Lynn Rohrbough, ed.  Fun in Small Spaces.  Handy Series, Kit Q, Cooperative Recreation Service, Delaware, Ohio, nd [c1935].  Take Last, p. 10.  Last player loses Nim (3, 5, 7).

Rohrbough.  Puzzle Craft.  1932.

Japanese Corn Game, p. 6 (= p. 6 of 1940s?).  Last player loses Nim (1, 2, 3, 4, 5).

Japanese Corn Game, p. 23.    Last player loses Nim (3, 5, 7).

René de Possel.  Sur la Théorie Mathématique des Jeux de Hasard et de Réflexion.  Actualités Scientifiques et Industrielles 436.  Hermann, Paris, 1936.  Gives the theory of Nim and also the misère version.

Depew.  Cokesbury Game Book.  1939.  Make him take it, pp. 187-188.  Nim (3, 4, 5),  last player loses.

Edward U. Condon, Gereld L. Tawney & Willard A. Derr.  US Patent 2,215,544 _ Machine to Play Game of Nim.  Filed 26 Apr 1940;  patented 24 Sep 1940.  10pp + 11pp diagrams.

E. U. Condon.  The Nimatron.  AMM 49 (1942) 330‑332.  Has photo of the machine.

Benedict Nixon & Len Johnson.  Letters to the Notes & Queries Column.  The Guardian (4 Dec 1989) 27.  Reprinted in:  Notes & Queries, Vol. 1; Fourth Estate, London, 1990, pp. 14-15.  These describe the Ferranti Nimrod machine for playing Nim at the Festival of Britain, 1951.  Johnson says it played  Nim (3, 5, 6)  with a maximum move of  3.  The Catalogue of the Exhibition of Science shows this as taking place in the Science Museum.

H. S. M. Coxeter.  The golden section, phyllotaxis, and Wythoff's game.  SM 19 (1953) 135‑143.  Sketches history and interconnections.

H. S. M. Coxeter.  Introduction to Geometry.  Wiley, 1961.  Chap. 11: The golden section and phyllotaxis, pp. 160-172.  Extends his 1953 material.

A. P. Domoryad.  Mathematical Games and Pastimes.  (Moscow, 1961).  Translated by Halina Moss.  Pergamon, Oxford, 1963.  Chap. 10: Games with piles of objects, pp. 61‑70.  On p. 62, he asserts that Wythoff's game is 'the Chinese national game tsyanshidzi ("picking stones")'.  However M.‑K. Siu cannot recognise such a Chinese game, unless it refers to a form of jacks, which has no obvious connection with Wythoff's game or other Nim games.  He says there is a Chinese character, 'nian', which is pronounced 'nim' in Cantonese and means to pick up or take things.

N. L. Haddock.  Note 2973:  A note on the game of Nim.  MG 45 (No. 353) (Oct 1961) 245‑246.  Wonders if the game of Nim is related to Mancala games.

T. H. O'Beirne.  Puzzles and Paradoxes.  OUP, 1965.  Section on misère version of Wythoff's game, p. 133.

Winning Ways.  1982.  P. 407 says Wythoff's game is also called Chinese Nim or Tsyan‑shizi.  No reference given.  See comment under Domoryad above.

D. St. P. Barnard.  50 Daily Telegraph Brain‑Twisters.  Javelin Books, Poole, Dorset, 1985.  Prob. 30: All buttoned up, pp. 49‑50, 91 & 115.  He suggests three pile game where one can take any number from one pile or an equal number from any two or all three piles.  [See my note to Wythoff, above.]

Matthias Mala.  Schnelle Spiele.  Hugendubel, Munich, 1988.  San Shan, p. 66.  This describes a nim-like game named San Shan and says it was played in ancient China.

Jagannath V. Badami.  Musings on Arithmetical Numbers  Plus Delightful Magic Squares.  Published by the author, Bangalore, India, nd [Preface dated 9 Sep 1999].  Section 4.16: The game of Nim, pp. 124-125.  This is a rather confused description of one pile games (21, 5) and (41, 5), but he refers to solving them by (mentally) dividing the pile into piles.  This makes me think of combining the two games, i.e. playing Nim with several piles but with a limit on the number one can take in a move.


          4.A.5. GENERAL THEORY


Charles Babbage.  The Philosophy of Analysis _ unpublished collection of MSS in the BM as Add. MS 37202, c1820.  ??NX.  Ff. 134-144 are:  Essay 10 Part 5.  See 4.B.1 for more details.  At the top of f. 134.r, he has added a note:  "This is probably my earliest Note on Games of Skill.  I do not recollect the date.  3 March 1865".  He then describes Tit Tat To and makes some simple analysis, but he never uses a name for it. 

Charles Babbage.  Notebooks _ unpublished collection of MSS in the BM as Add. MS 37205.  ??NX.  See 4.B.1 for more details.  On f. 304, he starts on analysis of games.  Ff. 310‑383 are almost entirely devoted to Tit-Tat-To, with some general discussions.  F. 321.r, 10 Sep 1860, is the beginning of a summary of his work on games of skill in general.  F. 324-333, Oct 1844, studies "General laws for all games of Skill between two players" and draws flow charts showing the basic recursive analysis of a game tree (ff. 325.v & 325.r).  On f. 332, he counts the number of positions in Tit Tat To as  9! + 8! + ... + 1!  =  409,113.  F. 333 has an idea of the tree structure of a game.

John M. Dubbey.  The Mathematical Work of Charles Babbage.  CUP, 1978, pp. 96‑97 & 125‑130.  See 4.B.1 for more details.  He discusses the above Babbage material.  On p. 127, Dubbey has:  "The basic problem is one that appears not to have been previously considered in the history of mathematics."  Dubbey, on p. 129, says:  "This analysis ... must count as the first recorded stochastic process in the history of mathematics."  However, it is really a deterministic two-person game.

E. Zermelo.  Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels.  Proc. 5th ICM (1912), CUP, 1913,  vol. II, 501‑504.  Gives general idea of first and second person games.

Ahrens.  A&N.  1918.  P. 154, note.  Says that each particular Dots and Boxes board, with rational play, has a definite outcome.

W. Rivier.  Archives des Sciences Physiques et Naturelles (Nov/Dec 1921).  ??NYS _ cited by Rivier (1935) who says that the later article is a new and simpler version of this one.

H. Steinhaus.  Difinicje potrzebne do teorji gry i po_cigu (Definitions for a theory of games and pursuit).  My_l Akademicka (Lwów) 1:1 (Dec 1925) 13‑14 (in Polish).  Translated, with an introduction by Kuhn and a letter from Steinhaus in:  Naval Research Logistics Quarterly 7 (1960) 105‑108.

Dénès König.  Über eine Schlussweise aus dem Endlichen ins Unendliche.  Mitteilungen der Universitä Szeged 3 (1927) 121-130.  ??NYS _ cited by Rivier (1935).  Kalmár cites it to the same Acta as his article. 

László Kalmár.  Zur Theorie der abstracten Spiele.  Acta Litt. Sci. Regia Univ. Hungaricae Francisco‑Josephine (Szeged) 4 (1927) 62‑85.  Says there is a gap in Zermelo which has been mended by König.  Lengthy approach, but clearly gets the idea of first and second person games.

Max Euwe.  Proc. Koninklijke Akadamie van Wetenschappen te Amsterdam 32:5 (1929).  ??NYS _ cited by Rivier (1935).

Emanuel Lasker.  Brettspiele der Völker.  Rätsel‑ und mathematische Spiele.  A. Scherl, Berlin, 1931, pp. 170‑203.  Studies the one pile game  (100, 5)  and the sum of two one‑pile games:  (100, 5) + (50, 3).  Discusses Nimm, "an old Chinese game according to Ahrens" and says the solver is unknown.  Gives Lasker's Nim _ one can take any amount from a pile or split it in two _ and several other variants.  Notes that  2nd person + 2nd person  is  2nd person  while  2nd person + 1st person  is  1st person.  Gives the idea of equivalent positions.  Studies three (and more) person games, assuming the pay‑offs are all different.  Studies some probabilistic games.

W. Rivier.  Une theorie mathématique des jeux de combinaisions.  Comptes-Rendus du Premier Congrès International de Récréation Mathématique, Bruxelles, 1935.  Sphinx, Bruxelles, 1935, pp. 106‑113.  A revised and simplified version of his 1921 article.  He cites and briefly discusses Zermelo, König and Euwe.  He seems to be classifying games as first player or second player.

René de Possel.  Sur la Théorie Mathématique des Jeux de Hasard et de Réflexion.  Actualités Scientifiques et Industrielles 436.  Hermann, Paris, 1936.  Gives the theory of Nim and also the misère version.  Shows that any combinatorial game is a win, loss or draw and describes the nature of first and second person positions.  He then goes on to consider games with chance and/or bluffing, based on von Neumann's 1927 paper.

R. Sprague.  Über mathematische Kampfspiele.  Tôhoku Math. J. 41 (1935/36) 438‑444.

P. M. Grundy.  Mathematics and games.  Eureka 2 (1939) 6‑8.  Reprinted, ibid. 27 (1964) 9‑11.  These two papers develop the Sprague-Grundy Number of a game.

D. W. Davies.  A theory of chess and noughts and crosses.  Penguin Science News 16 (Jun 1950) 40-64.  Sketches general ideas of tree structure, Sparague-Grundy number, rational play, etc.

H. Steinhaus.  Games, an informal talk.  AMM 72 (1965) 457‑468.  Discusses Zermelo and says he wasn't aware of Zermelo in 1925.  Gives Mycielski's formulation and proof via de Morgan's laws.  Goes into pursuit and infinite games and their relation to the Axiom of Choice.

H. Steinhaus.  (Proof that a game without ties has a strategy.)  In:  M. Kac; Hugo Steinhaus _ a reminiscence and a tribute; AMM 81 (1974) 572‑581.  Repeats idea of his 1965 talk.


          4.B.    PARTICULAR GAMES


          See 5.M for Sim and 5.R.5 for Fox and Geese, etc.

          Most of the board games described here are classic and have been extensively described and illustrated in the various standard books on board games, particularly the works of Robert C. Bell, especially his Board and Table Games from Many Civilizations;  OUP, vol. I, 1960, vol. II, 1969;  combined and revised ed., Dover, 1979 and the older work of Edward G. Falkener; Games Ancient and Oriental and How to Play Them;  Longmans, Green, 1892;  Dover, 1961.  The works by Culin (see 4.A.4, 4.B.5 and 4.B.9) are often useful.  Several general works on games are cited in 4.B.1 and 4.B.5 _ I have not yet read Murray's History of Board Games Other than Chess.  Note that many of these works are more concerned with the game than with its history and have a tendency to exaggerate the ages of games by assuming, e.g. that a  3 x 3  board must have been used for Tic-Tac-Toe.  I will not try to duplicate the descriptions by Bell, Falkener and others, but will try to outline the earliest history, especially when it is at variance with common belief.  The most detailed mathematical analyses are generally in Winning Ways.




          Popular belief is that the game is ancient and universal _ e.g. see Brandreth, 1976.  However the game appears to have evolved from earlier three‑in‑a‑row games, e.g. Nine Holes or Three Men's Morris, in the early 19C.  See also the historical material in 4.B.5.  The game is not mentioned in Strutt nor most other 19C books on games, not even in Kate Greenaway's Book of Games (1889), nor in Halliwell's section on slate games (op. cit. in 7.L.1, 1849, pp. 103-104), but there may be an 1875 description in Strutt-Cox of 1903.  Babbage refers to it in his unpublished MSS of c1820 as a children's game, but without giving it a name.  In 1842, he calls it Tit Tat To and he uses slight variations on this name in his extended studies of the game _ see below.  The OED's earliest references are:  1849 for Tip‑tap‑toe;  1855 for Tit‑tat‑toe;  1861 for Oughts and Crosses.  However, the first two entries may be referring to some other game _ e.g. the entries for Tick‑tack‑toe for 1884 & 1899 are clearly to the game that Gomme calls Tit‑tat‑toe.  Von der Lasa cites a 1838-39 Swedish book for Tripp, Trapp, Trull.  Van der Linde (1874, op. cit. in 5.F.1) gives Tik, Tak, Tol as the Dutch name.  Using the works of Strutt, Gomme, Strutt-Cox, Fiske, Murray, the OED and some personal communications, I have compiled a separate index of  121  variant names which refer to  5  basic games, with a few variants and a few unknown games.  The Murray and Parker material is given first, as it deals generally with the ancient history.  Then I list several standard sources and then summarize their content.  Other material follows that.  Fiske says that van der Linde and von der Lasa (see 5.F.1) mention early appearances of Morris games, but rather briefly and I don't always have that material.

          The usual  #  shape board will be so indicated.  If one is setting down pieces, then the board is often drawn as a 'crossed square', i.e. a square with its horizontal and vertical midlines drawn, and one plays on the intersections.  Fiske 127 says this form is common in Germany, but unknown in England and the US.  In addition, the diagonals are often drawn, producing a 'doubly crossed square'.  The squares are sometime drawn as circles giving a 'crossed circle' and a 'doubly crossed circle', though it is hard to identify the corners in a crossed circle.  The  3 x 3  array of dots sometimes occurs.  The standard  #  pattern is sometimes surrounded by a square producing a '3 x 3  chessboard'.

          Fiske 129 says the English play with  O  and  +,  while the Swedes play with  O  and  1.  My experience is that English and Americans play with  O  and  X.  One English friend said that where she grew up, it was called 'Exeter's Nose' as a deliberate corruption of 'Xs and Os'.

          The first clear references to the standard game of Noughts and Crosses are Babbage (1820) and the items discussed under Tic-tac-toe below.  Further clear references are:  Cassell's, Berg, A wrangler ..., Dudeney, White and everything entered below after White.


Murray mentions Morris, which he generally calls Merels, many times.  Besides the many specific references mentioned below and in 4.B.5, he shows, on p. 614, under Nine Holes and Three Men's Morris, a number of  3 x 3  diagrams. 

                    Kurna, Egypt, (-14C) _ a double crossed square and a double crossed circle _ see Parker below.

                    Ptolemaic Egypt (in the BM, no. 14315) _ a square with  #  drawn inside.  See below where I describe this, from a recent exhibition, as just a  #  board.

                    Ceylon _ a doubly crossed square _ see Parker below.

                    Rome and Pompeii _ doubly crossed circles. 

          Under Nine Holes, he says a piece can be moved to any vacant point;  under Three Men's Morris, he says a man can only be moved along a marked line to an adjacent point, i.e. horizontally, vertically or along a main diagonal.

                    Under Nine Holes, he shows the  #  board for English Noughts and Crosses.  He specifically notes that the pieces do not move.  His only other mention of this board is for a Swedish game called Tripp, Trapp, Trull, but he does not state that the pieces do not move.  He gives no other examples of the  #  board nor of non‑moving pieces.

                    He also mentions Five (or Six) Men's Morris, of which little is known.  On p. 133, he mentions a  3 x 3  "board of nine points used for a game essentially identical with the 'three men's merels', which has existed in China from at least the time of the Liang dynasty (A.D. 502‑557).  The 'Swei shu' (first half of the 7th c.) gives the names of twenty books on this game."

H. Parker.  Ancient Ceylon.  ??, London, 1909;  Asian Educational Services, New Delhi, 1981.  Nerenchi keliya, pp. 577‑580 & 644.  There is a crossed square with small holes at the intersections at the Temple of Kurna, Upper Egypt, ‑14C.  [Rohrbough, loc. cit. in 4.B.5, says this temple was started by Ramses I and completed by Seti in -1336/-1333, citing J. Royal Asiatic Soc. (1783) 17.]  On p. 644, he shows 34 mason's diagrams from Kurna, which include  #,  #  in a circle, crossed square with small holes at the intersections, doubly crossed square, doubly crossed circle.  He cites Bell, Arch. Survey of Ceylon, Third Progress Report, p. 5 note, for for a doubly crossed square in Ceylon, c1C, but Noughts and Crosses is not found in the interior of Ceylon.  The doubly crossed square was used in 18C Ireland.  On pp. 643-665, he discusses appearances of the crossed square and doubly crossed circle as designs or characters and claims they have mystic significance.  On p. 662, he lists many early appearances of the  #  pattern.

Murray 440, note 63, includes a reference to Soutendam; Keurboek van Delft; Delft, c1425, f. 78 (or p. 78?); who says games of subtlety are allowed, e.g. ... ticktacken.  There is no indication if this may be our game and the OED indicates that such names were used for backgammon back to 1558.  The OED doesn't cite:  W. Shakespeare; Measure for Measure, c1604.  Act I, scene ii, line 180 (or 196):  "foolishly lost at a game of ticktack".  Later it was more common as Tric-trac. 

Murray 746 notes a Welsh game Gwyddbwyll mentioned in the Mabinogion (14C).  The name is cognate with the Irish Fidchell and may be a Three Men's Morris, but the game was already forgotten by the 15C.


                    STANDARD SOURCES ON GAMES


Joseph Strutt.  The Sports and Pastimes of the People of England.  (With title starting:  Glig‑Gamena Angel-Ðeod., or the Sports ...; J. White, London, 1791, 1801, 1810).  A new edition, with a copious index, by William Hone.  Tegg, London, 1830, 1831, 1833.  [The 1830 ed. has a preface, omitted in 1833, stating that the 1810 ed. is the same as the 1801 ed. and that Hone has only changed it by adding the Index and incorporating some footnotes into the text.]  [Hall, BCB 263-266 are:  1801,  1810,  1830,  1831.  Toole Stott 647-656 are:  1791;  1801;  1810;  1828-1830 in 10 monthly parts with Index by Hone; 1830;  1830;  1833;  1838;  1841;  1876, an expanded ed, ed by Hone.  Heyl 300-302 gives  1830;  1838;  1850.  Toole Stott 653 says the sheets were remaindered to Hone, who omitted the first 8pp and issued it in  1833,  1834,  1838,  1841.  I have seen an 1855 ed.  C&B list 1801, 1810, 1830, 1903.]

                    Strutt-Cox.  The Sports and Pastimes of the People of England.  By Joseph Strutt.  1801.  A new edition, much enlarged and corrected by J. Charles Cox.  Methuen, 1903.  The Preface sketches Strutt's life and says this is based on the 'original' 1801 in quarto, with separate plates which were often hand coloured, but not consistently, while the 1810 reissue had them all done in a terracotta shade.  Hone reissued it in octavo in 1830 with the plates replaced by woodcuts in the text and this was reissued in 1837, 1841 and 1875.  (From above we see that there were other reissues.)  "Mr. Strutt has been left for the most part to speak in his own characteristic fashion ....  A few obvious mistakes and rash conclusions have been corrected, ... certain unimportant omissions have been made.  ...  Nearly a third of the book is new." 

J. T. Micklethwaite.  On the indoor games of school boys in the middle ages.  Archaeological Journal 49 (Dec 1892) 319-328.  Describes various  3 x 3  boards and games on them, including Nine Holes and "tick, tack, toe; or oughts and crosses, which I suppose still survives wherever slate and pencil are used as implements of education", Three Men's Morris and also Nine Men's Morris, Fox and Geese, etc.

Alice B. Gomme.  The Traditional Games of England, Scotland, and Ireland.  2 vols., David Nutt, London, 1894 & 1898.  Reprinted in one vol., Thames & Hudson, London, 1984.

Willard Fiske.  Chess in Iceland and in Icelandic Literature with Historical Notes on Other Table-Games.  The Florentine Typographical Society, Florence, 1905.  Esp. pp. 97-156 of the Stray Notes.  P. 122 lists a number of works on ancient games.


          These and the OED have several entries on Noughts and Crosses and Tic‑tac‑toe and many on related games, which are summarised below.  Gomme often cites or quotes Strutt.  The OED often gives the same quotes as Gomme.  Gomme's references are highly abbreviated but full details of the sources can usually be found in the OED.

          (Nine Men's) Morris, where Morris is spelled about 30 different ways, e.g. Marl, Merelles, Mill, Miracles, Morals, and Nine Men's may be given as, e.g. Nine‑peg, Nine Penny, Nine Pin.  Also known as Peg Morris and Shepherd's Mill.  Gomme I 80 & 414‑419 and Strutt 317‑318 (c= Strutt-Cox 256-258 & plate opp. 246, which adds reference to Micklethwaite) are the main entries.  See 4.B.5 for material more specifically on this game.

          Nine Holes, also known as Bubble‑justice, Bumble‑puppy, Crates, and possibly Troll‑madam, Troule‑in‑Madame.  Gomme I 413‑414 and Strutt 274‑275 & 384 (c= Strutt‑Cox 222-223 & 304) are the main entries.  Twelve Holes is similar [Gomme II 321 gives a quote from 1611].  There seem to be cases where Nine Men's Morris was used in referring to Nine Holes [Gomme I 414‑419].  There are two forms of the game:  one form has holes in an upright board that one must roll a ball or marble through;  the other form has holes in the ground, usually in a  3  x  3  array, that one must roll balls into.  Unfortunately, none of the references implies that one has to get three in a row _ see Every Little Boys Book for a version where this is certainly not the case.  There are references going back to 1572 for Crates (but mentioning eleven holes) [Gomme I 81  &  II 309] and 1573 [OED] for Nine Holes.  Botermans et al.; The World of Games; op. cit. in 4.B.5; 1989; p. 213, shows a 17C engraving by Ménian showing Le Jeu de Troumadame as having a board with holes in it, held vertically on a table and one must roll marbles through the holes.  They say it is nowadays known as 'bridge'.

          Three Men's Morris.  This is less common, but occurs in several variant spellings corresponding to the variants of Nine Men's Morris, including, e.g. Three‑penny Morris, Tremerel.  The game is played on a  3  x  3  board and each player has three men.  After making three plays each, consisting of setting men on the cells, further play consists of picking up one of your own men and placing it on a vacant cell, with the object of getting three in a row.  There are several versions of this game, depending on which cells one may play to, but the descriptions given rarely make this clear.  [Gomme I 414‑419] quotes from F. Douce; Illustrations of Shakespeare and of Ancient Manners; 1807, i.184.  "In the French merelles each party had three counters only, which were to be placed in a line to win the game.  It appears to have been the tremerel mentioned in an old fabliau.  See Le Grand, Fabliaux et Contes, ii.208.  Dr. Hyde thinks the morris, or merrils, was known during the time that the Normans continued in possession of England, and that the name was afterwards corrupted into three men's morals, or nine men's morals."  [Hyde.  Hist. Nederluddi [sic], p. 202.]  In practice, the board is often or usually drawn as a crossed square.  If one can move along all winning lines, then it would be natural to draw a doubly crossed square.  See under Alfonso MS (1283) in 4.B.5 for versions called marro, tres en raya and riga di tre.  Again, much of the material on this game is in 4.B.5.

          Five‑penny Morris.  None of the references make it clear, but this seems to be (a form of) Three Men's Morris.  Gomme I 122 and the OED [under Morrell] quote:  W. Hawkins; Apollo Shroving (a play of 1627), act III, scene iv, pp. 48-49. 

          "..., Ovid hath honour'd my exercises.  He describes in verse our boyes play.

          Twise three stones, set in a crossed square where he wins the game 

          That can set his three along in a row, 

          And that is fippeny morrell I trow." 

Most of the references (and myself) are perplexed by the reference to five, though the fact that one has at most five moves in Tic‑tac‑toe might have something to do with it??  Since Three Men's Morris is less well known, some writers have assumed Five‑penny Morris was Nine Men's Morris and others have called all such games by the same name.  A few lines later, Hawkins has: "I challenge him at all games from blowpoint upward to football, and so on to mumchance, and ticketacke.  ... rather than sit out, I will give Apollo three of the nine at Ticketacke, ..."

          Corsicrown [Gomme I 80] seems to be a version of Three Men's Morris, but using seven of the nine cells, omitting two opposite side cells.  Gomme quotes from J. Mactaggart; The Scottish Gallovidian Encyclopedia; (1871 or possibly 1824?):  "each has three men .... there are seven points for these men to move about on, six on the edges of the square and one at the centre."

          Tic‑tac‑toe.  The earliest clearly described versions are given in Babbage (with no name given), c1820, and Gomme I 311, under Kit‑cat‑cannio, where she quotes from:  Edward Moor; Suffolk Words and Phrases; 1823 (This word does not occur in the OED).  Gomme also gives entries for Noughts and Crosses [I 420‑421] and Tip‑tap‑toe [II 295‑296] with variants Tick‑tack‑toe and Tit‑tat‑toe.  In 1842-1865, Babbage uses Tit Tat To and slight variants.  Under Tip‑tap‑toe, Gomme says the players make squares and crosses and that a tie game is a score for Old Nick or Old Tom.  (When I was young, we called it Cat's Game.)  She quotes regional glossaries for Tip‑tap‑toe (1877), Tit‑tat‑toe (1866 & 1888), Tick‑tack‑toe (1892).  The OED entry for Oughts and Crosses seems to be this game and gives an 1861 quote.  Von der Lasa cites a 1838-39 Swedish book for Tripp, Trapp, Trull.  Van der Linde (1874, op. cit. in 5.F.1) gives Tik, Tak, Tol as the Dutch name. 

          Tit‑tat‑toe [Gomme II 296‑298].  This is a game using a slate marked with a circle and numbered sectors.  The player closes his eyes and taps three times with a pencil and tries to land on a good sector.  Gomme gives the verse:

                    Tit, tat, toe, my first go,

                    Three jolly butcher boys all in a row

                    Stick one up, stick one down,

                    Stick one in the old man's ground.

          The OED entries under Tick‑tack, Tip‑tap and Tit give a number of variant spellings and several quotations, which are often clearly to this game, but are sometimes unclear.  Also some forms seem to refer to backgammon.

          In her 'Memoir on the study of children's games' [Gomme II 472‑473], Gomme gives a somewhat Victorian explanation of the origin of Old Nick as the winner of a tie game as stemming from "the primitive custom of assigning a certain proportion of the crops or pieces of land to the devil, or other earth spirit."


Franco Agostini & Nicola Alberto De Carlo.  Intelligence Games.  (As:  Giochi della Intelligenza; Mondadori, Milan, 1985.)  Simon & Schuster, NY, 1987.  P. 81 says examples of boards were discovered in the lowest level of Troy and in the Bronze Age tombs in Co. Wicklow, Ireland.  Their description is a bit vague but indicates that the Italian version of Tic-tac-toe is actually Three Men's Morris.

Anonymous.  Play the game.  Guardian Education section (21 Sep 1993) 18-19.  Shows a stone board with the  #  incised on it 'from Bet Shamesh, Israel, 2000 BC'.  This might be the same as the first board below??

A small exhibition of board games organized by Irving Finkel at the British Museum, 1991, displayed the following.

Stone slab with the usual  #  Tick-Tac-Toe board incised on it, but really  a  4 x 3  board.  With nine stone men.  From Giza, >-850.  BM items EA 14315 & 14309, donated by W. M. Flinders Petrie.  Now on display in Room 63, Case C.

Stone Nine Holes board from the Temple of Artemis, Ephesus, 2C-4C.  Item BM GR 1873.5.5.150.  This is a  3  x  3  array of depressions.  Now on display in Room 69, Case 9.

Robbie Bell & Michael Cornelius.  Board Games Round the World.  CUP, 1988.  P. 6 states that the crossed square board has been found at Kurna (c-1400) and at the Ptolemaic temple at Komombo (c-300).  They state that Three Men's Morris is the game mentioned by Ovid in Ars Amatoria.  They say that it was known to the Chinese at the time of Confucius (c-500) under the name of Yih, but is now known as  Luk tsut k'i.  They also say the game is also known as Nine Holes _ which seems wrong to me.

The Spanish Treatise on Chess-Play written by order of King Alfonso the Sage in the year 1283.  [Generally known as the Alfonso MS.]  MS in Royal Library of the Escorial (j.T.6. fol).  Complete reproduction in 194 Phototypic Plates.  2 vols.,  Karl W. Hirsemann, Leipzig, 1913.  (There was also an edition by Arnald Steiger, Geneva, 1941.)

See 4.B.5 for more details of this work.  Vol. 2, f. 93v, p. CLXXXVI, shows a doubly crossed square board.  ??NX _ need to study text.

Pieter Bruegel (the Elder).  Children's Games.  Painting dated 1560 at the Kunsthistorisches Museum, Vienna.  In the right background, children are playing a game involving throwing balls into holes in the ground, but the holes appear to be in a straight line.

Anonymous.  Games of the 16th Century.  1950.  Op. cit. in 4.A.3.  P. 134 describes nine-holes, quoting an unknown poet of 1611: "To play at loggats, Nine-holes, or Ten-pinnes".  The author doesn't specify what positions the balls are to be rolled into.  P. 152 describes Troll-my-dames or Troule-in-madame: "they may have in the end of a bench eleven holes made, into which to troll pummets, or bowls of lead, ...."

William Wordsworth.  The Prelude, Book 1.  Completed 1805, published 1850.  Lines 509‑513.

                    At evening, when with pencil, and smooth slate

                    In square divisions parcelled out and all

                    With crosses and with cyphers scribbled o'er,

                    We schemed and puzzled, head opposed to head

                    In strife too humble to be named in verse.

          It is not clear if this is referring to Noughts and Crosses.

Charles Babbage.  The Philosophy of Analysis _ unpublished collection of MSS in the BM as Add. MS 37202, c1820.  ??NX.  F. 4.r is part of the Table of Contents.  It shows Noughts and Crosses games played on the  #  board and on a  4 x 4  board adjacent to entry 4: The Mill.  Ff. 124-146 are:  Essay 10 _ Of questions requiring the invention of new modes of analysis.  On f. 128.r, he refers to a game in which "the relative positions of three of the marks is the object of inquiry."  Though the reference is incomplete, a Noughts and Crosses game is drawn on the facing page, f. 127.v.  Ff. 134-144 are:  Essay 10 Part 5.  At the top of f. 134.r, he has added a note:  "This is probably my earliest Note on Games of Skill.  I do not recollect the date.  3 March 1865".  The Essay begins:  "Amongst the simplest of those games requiring any degree of skill which amuse our early years is one which is played at in the following manner."  He then describes the game in detail and makes some simple analysis, but he never uses a name for it. 

Charles Babbage.  Notebooks _ unpublished collection of MSS in the BM as Add. MS 37205.  ??NX.  On f. 304, he starts on analysis of games.  Ff. 310-383 are almost entirely devoted to Tit-Tat-To, with some general discussions.  Most of this material comprises a few sheets of working, carefully dated, sometimes amended and with the date of the amendment.  A number of sheets describe parts of the automaton that he was planning to build which would play the game.  The sheets are not always in strict chronological order.

                    F. 310.r is the first discussion of the game, called Tit Tat To, dated 17 Sep 1842.  On F. 312.r, 20 Sep 1843, he says he has "Reduced the 3024 cases D to 199 which include many Duplicates by Symmetry."  F. 321.r, 10 Sep 1860, is the beginning of a summary of his work on games of skill in general.  He refers to Tit-tat-too.  F. 322.r continues and he says:  "I have found no game of skill more simple that that which children often play and which they call Tit‑tat-to."  F. 324-333, Oct 1844, studies "General laws for all games of Skill between two players" and draws flow charts showing the basic recursive analysis of a game tree (ff. 325.v & 325.r).  On f. 332, he counts the number of positions as  9! + 8! + ... + 1!  =  409,113.  F. 333 has an idea of the tree structure of a game.  On ff. 337-338, 8 Sep 1848, he has Tit-tat too.  On ff. 347.r-347.v, he suggests Nine Men's Morris boards in triangular and pentagonal shapes and does various counting on the different shapes.  On ff. 348-349, 26 Oct 1859, he uses Tit-Tat-To.

John M. Dubbey.  The Mathematical Work of Charles Babbage.  CUP, 1978, pp. 96‑97 & 125‑130.  He discusses the above Babbage material.  On p. 127, Dubbey has:  "After a surprisingly lengthy explanation of the rules, he attempts a mathematical formulation.  The basic problem is one that appears not to have been previously considered in the history of mathematics."  Babbage represents the game using roots of unity.  Dubbey, on p. 129, says:  "This analysis ... must count as the first recorded stochastic process in the history of mathematics."  However, it is really a deterministic two-person game.

Baron Tassilo von Heydebrand und von der Lasa.  Ueber die griechischen und römischen Spiele, welche einige ähnlichkeit mit dem Schach hatten.  Deutsche Schachzeitung (1863) 162-172, 198-199, 225-234, 257-264.  ??NYS _ described on Fiske 121-122 & 137, who says van der Linde I 40-47 copies much of it.  Von der Lasa asserts that the Parva Tabella of Ovid is Kleine Mühle (Three Men's Morris).  He says the game is called Tripp, Trapp, Trull in the Swedish book Hand-Bibliothek för Sällkapsnöjen, of 1839, vol. II, p. 65 (or 57) _ ??NYS.  Van der Linde says that the Dutch name is Tik, Tak, Tol.  Fiske notes that both of these refer to Noughts and Crosses, but it is unclear if von der Lasa or van der Linde recognised the difference between Three Men's Morris and Noughts and Crosses.

C. Babbage.  Passages from the Life of a Philosopher.  1864.  Chapter XXXIV _ section on Games of Skill, pp. 465‑471.  (= pp. 152‑156 in:  Charles Babbage and His Calculating Engines, Dover, 1961.)  Partial analysis.  He calls it tit‑tat‑to.

Anonymous.  Every Little Boy's Book  A Complete Cyclopædia of in and outdoor games with and without toys, domestic pets, conjuring, shows, riddles, etc.  With two hundred and fifty illustrations.  Routledge, London, nd.  HPL gives c1850, but the text is clearly derived from Every Boy's Book, whose first edition was 1856.  But the main part of the text considered here is not in the 1856 ed of Every Boy's Book (with J. G. Wood as unnamed editor), but is in the 8th ed of 1868 (published for Christmas 1867), which was the first seriously revised edition, with Edmund Routledge as editor.  So this may be c1868.  This is the first published use of the term Noughts and Crosses found so far _ the OED's 1861 quote is to Oughts and Crosses..

                    Pp. 46-47: Slate games: Noughts and crosses.  "This is a capital game, and one which every school-boy truly enjoys."  Though the example shown is a draw, there is no mention of the fact that the game should always be a tie.

                    Pp. 85-86: Nine-holes.  This has nine holes in a row and each player has a hole.  The ball is rolled to them and the person in whose hole it lands must run and pick up the ball and try to hit one of the others who are running away.  So this has nothing to do with our games or other forms of Nine Holes.

                    P. 106: Nine-holes or Bridge-board.  This has nine holes in an upright board and the object is get one's marbles through the holes.  (This material is in the 1856 ed. of Every Boy's Book.)

Correspondent to Notes and Queries (1875)  ??NYS _ quoted by Strutt-Cox 257.  Describes a game called Three Mans' Marriage [sic] in Derbyshire which seems to be Noughts and Crosses played on a crossed square board.  Pieces are not described as moving, but in the next description of a Nine Men's Morris, they are specifically described as moving.  However, the use of a crossed square board may indicate that diagonals were not considered.

Cassell's.  1881.  Slate Games: Noughts and Crosses, or Tit‑Tat‑To, p. 84, with cross reference under Tit-Tat-To, p. 87.  = Manson, 1911, pp. 202-203 & 208.

Albert Norman.  Ungdomens Bok [Book for Youth] (in Swedish).  2nd ed., Stockholm, 1883.  Vol. I, p. 162++.  ??NYS _ quoted and described in Fiske 134-136.  Description of Tripp, Trapp, Trull, with winning cry:  "Tripp, trapp, trull, min qvarn är full."  (Qvarn = mill.)

Lucas.  RM2, 1883.  Pp. 73-99.  Analysis of Three Men's Morris, on a board with the main diagonals drawn, with moves of only one square along a winning line.  He shows this is a first person game.  If the first player is not permitted to play in the centre, then it is a tie game.  No mention of Tic-Tac-Toe.

Albert Ellery Berg, ed.  The Universal Self‑Instructor.  Thomas Kelly, NY, 1883.  Tit‑tat‑to, p. 379.  Brief description.

Mark Twain.  The Adventures of Huckleberry Finn.  1884.  Chap. XXXIV, about half-way through.  "It's as simple as tit-tat-toe, three-in-a-row, ..., Huck Finn."

"A wrangler and late master at Harrow school."  The science of naughts and crosses.  Boy's Own Paper 10:  (No. 498) (28 Jul 1888) 702‑703;  (No. 499) (4 Aug 1888) 717;  (No. 500) (11 Aug 1888) 735;  (No. 501) (18 Aug 1888) 743.  Exhaustive analysis, including odds of second player making a correct response to each opening.  For first move in:  middle, side, corner, the odds of a correct response are:  1/2,  1/2,  1/8.  He implies that the analysis is not widely known.

"Tom Wilson".  Illustred Spelbok (in Swedish).  nd [late 1880s??].  ??NYS _ described by Fiske 136-137.  This gives Tripp, Trapp, Trull as a Three Men's Morris game on the crossed square, with moves "according to one way of playing, to whatever points they please, but according to another, only to the nearest point along the lines on which the pieces stand.  This last method is always employed when the board has, in addition to the right lines, or lines joining the middles of the exterior lines, also diagonals connecting the angles".  He then describes a drawn version using the  #  board and  0  and  +  (or  1  and  2  in the North) which seems to be genuinely Noughts and Crosses.  Fiske says the book seems to be based on an early edition of the Encyclopédie des Jeux or a similar book, so it is uncertain how much the above represents the current Swedish game.  Fiske was unable to determine the author's real name, though he was still living in Stockholm at the time.

Il Libro del Giuochi.  Florence, 1894.  ??NYS _ described in Fiske, pp. 109-110.  Gives doubly crossed square board and mentions a Three Men's Morris game.

T. de Moulidars.  Grande Encyclopédie des Jeux.  Montgredien or Librairie Illustree, Paris, nd.  ??NYS _ Fiske 115 (in 1905) says it appeared 'not very long ago' and that Gelli seems to be based on it.  Fiske quotes the clear description of Three Men's Morris as Marelle Simple, using a doubly crossed square, saying that pieces move to adjoining cells, following a line, and that the first player should win if he plays in the centre.  Fiske notes that Noughts and Crosses is not mentioned.

J. Gelli. Come Posso Divertirmi?  Milan, 1900.  ??NYS _ described in Fiske 107.  Fiske quotes the description of Three Men's Morris as Mulinello Semplice, essentially a translation from Moulidars.

Dudeney.  CP.  1907.  Prob. 109: Noughts and crosses, pp. 156 & 248.  (c= MP, prob. 202: Noughts and crosses, pp. 89 & 175‑176.  = 536, prob. 471: Tic tac toe, pp. 185 & 390‑392.  Asserts the game is a tie, but gives only a sketchy analysis.  MP gives a reasonably exhaustive analysis.

A. C. White.  Tit‑tat‑toe.  British Chess Magazine (Jul 1919) 217‑220.  Attempt at a complete analysis, but has a mistake.  See Gardner, SA (Mar 1957)  = 1st Book, chap. 4.

D'Arcy Wentworth Thompson.  Science and the Classics.  OUP, 1940.  Section V  Games and Playthings, pp. 148-165.  On p. 160, he quotes Ovid and says it is Noughts and Crosses, or in Ireland, Tip-top-castle.

The Home Book of Quizzes, Games and Jokes.  Blue Ribbon Books, NY, 1941.  This is excerpted from several books _ this material is most likely taken from:  Clement Wood & Gloria Goddard; Complete Book of Games; same publisher, nd [late 1930s].  P. 150: Tit-tat-toe, noughts and crosses.  Brief description of the game on the  #  board.  "To win requires great ingenuity."

Stanley Byard.  Robots which play games.  Penguin Science News 16 (Jun 1950) 65-77.  On p. 73, he says D. W. Davies has built, and exhibited to the Royal Society in May 1949, an electro-mechanical noughts and crosses machine.  A photo of the machine is plate 16.  He also mentions Babbage's interest in such a machine and an 1874 paper to the US National Academy by a Dr. Rogers _ ??NYS.

Gardner.  Ticktacktoe.  SA (Mar 1957)  c= 1st Book, chap. 4.  Quotes Wordsworth, discusses Three Men's Morris (citing Ovid) and its variants (including versions on  4 x 4  and  5 x 5  boards), the misere version (the person who makes three in a row loses), three and  n  dimensional forms (giving Moser's result on the number of winning lines on a  kn  board), go-moku, Babbage's proposed machine, A. C. White's article.  Addendum mentions the Opies' assertion that the name comes from the rhyme starting  "Tit, tat, toe,  My first go,".

Donald Michie.  Trial and error.  Penguin Science Survey 2 (1961) 129-145.  ??NYS.  Describes his matchbox and bead learning machine, MENACE (Matchbox educable noughts and crosses engine), for the game.

Gardner.  A matchbox game-learning machine.  SA (Mar 1962)  c= Unexpected, chap. 8.  Describes Michie's MENACE.  Says it used 300 matchboxes.  Gardner adapts it to Hexapawn, which is much simpler, requiring only 24 matchboxes.  Discusses other games playable by 'computers'.  Addendum discusses results sent in by readers including other games.

D. St. P. Barnard.  Fifty Observer Brain‑Twisters.  Faber, 1962.  (US ed.:  A Book of Mathematical and Reasoning Problems: Fifty Brain Twisters;  Van Nostrand, 1962.)  Prob. 34: Noughts and crosses, pp. 39‑40, 64 & 93‑94.  Asserts there are  1884  final winning positions.  He doesn't consider equivalence by symmetry and he allows either player to start.

Donald Michie  &  R. A. Chambers.  Boxes: an experiment in adaptive control.  Machine Intelligence 2 (1968) 136-152.  Discusses MENACE, with photo of the pile of boxes.  Says there are 288 boxes, but doesn't explain exactly how he found them.  Chambers has implemented MENACE as a general game-learning computer program using adaptive control techniques designed by Michie.  Results for various games are given. 

S. Sivasankaranarayana Pillai.  A pastime common among South Indian school children.  In:  P. K. Srinivasan, ed.; Ramanujan Memorial Volumes:  1: Ramanujan _ Letters and Reminiscences;  2: Ramanujan _ An Inspiration;  Muthialpet High School, Number Friends Society, Old Boys' Committee, Madras, 1968.  Vol. 2, pp. 81-85.  [Taken from Mathematics Student, but no date or details given _ ??]  Shows ordinary tic-tac-toe is a draw and considers trying to get  t  in a row on an  n x n  board.  Shows that  n = t ³ 3  is a draw and that if  t ³ n + 1 - Ö(n/6),  then the game is a draw.

L. A. Graham.  The Surprise Attack in Mathematical Problems.  Dover, 1968.  Tic-tac-toe for gamblers, prob. 8, pp. 19-22.  Proposed by F. E. Clark, solutions by Robert A. Harrington & Robert E. Corby.  Find the probability of the first player winning if the game is played at random.  Two detailed analyses shows that the probabilities for  first player,  second player,  tie  are  (737,  363,  160)/1260.

Gyles Brandreth.  Pencil and Paper Games and Puzzles.  Carousel, 1976.  Noughts and Crosses, pp. 11-12.  Asserts "It's been played all around the world for hundreds, if not thousands, of years ...."  I've included it as a typical example of popular belief about the game.  = Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989, Tic-Tac-Toe, pp. 11‑12.

Winning Ways.  1982.  Pp. 667-680.  Complete and careful analysis, including various uncommon traps.  Several equivalent games.  Discusses extensions of board size and dimension.

Sheila Anne Barry.  The World's Best Travel Games.  Sterling, NY, 1987.  Tic-tac-toe squared, pp. 88-89.  Get  3  in a row on the  4 x 4  board.  Also considers Tic-tac-toe-toe _ get  4  in a row on  5 x 5  board.

George Markovsky.  Numerical tic-tac-toe _ I.  JRM 22:2 (1990) 114-123.  Describes and studies two versions where the moves are numbered 1, 2, ....  One is due to Ron Graham, the other to P. H. Nygaard and Markowsky sketches the histories.

Ira Rosenholtz.  Solving some variations on a variant of tic-tac-toe using invariant subsets.  JRM 25:2 (1993) 128-135.  The basic variant is to avoid making three in a row on a  4 x 4  board.  By playing symmetrically, the second player avoids losing and 252 of the 256 centrally symmetric positions give a win for the second player.  Extends analysis to  2n x 2n,  5 x 5,  4 x 4 x 4,  etc.

Bernhard Wiezorke.  Sliding caution.  CFF 32 (Aug 1993) 24-25  &  33 (Feb 1994) 32.  This describes a sliding piece puzzle on the doubly crossed square board _ see under 5.A. 

See: Yuri I. Averbakh; Board games and real events; 1995; in 5.R.5, for a possible connection.


          4.B.1.a                   IN HIGHER DIMENSIONS


C. Planck.  Four‑fold magics.  Part 2 of chap. XIV, pp. 363‑375, of W. S. Andrews, et al.; Magic Squares and Cubes; 2nd ed., Open Court, 1917;  Dover, 1960.  On p. 370, he notes that the number of  m‑dimensional directions through a cell of the  n‑dimensional board is the  m‑th term of the binomial expansion of  ½(1+2)n.

Maurice Wilkes says he played  3-D noughts and crosses at Cambridge in the late 1930s, but the game was to get the most lines on a  3 x 3 x 3  board.  I recall seeing a commercial version of this in 1970. 

Cedric Smith says he played 3-D and 4-D versions at Cambridge in the early 1940s.

Funkenbusch & Eagle, National Mathematics Mag. (1944) ??NYR.

G. E. Felton & R. H. Macmillan.  Noughts and crosses.  Eureka 11 (1949) 5‑9.  They say they first met the  4 x 4 x 4  game at Cambridge in 1940 and they give some analysis of it.

William Funkenbusch & Edwin Eagle.  Hyper‑spacial tit‑tat‑toe or tit‑tat‑toe in four dimensions.  National Mathematics Magazine 19:3 (Dec 1944) 119‑122.  ??NYR

A. L. Rubinoff, proposer;  L. Moser, solver.  Problem E773 _ Noughts and crosses.  AMM 54 (1947) 281  &  55 (1948) 99.  Number of winning lines on a  kn  board is  {(k+2)n ‑ kn}/2.  Putting  k = 1  gives Planck's result.

L. Buxton.  Four dimensions for the fourth form.  MG 26 (1964) 38‑39.  3 x 3 x 3  and  3 x 3 x 3 x 3  games are obviously first person, but he proposes playing for most lines and with the centre blocked on the  3 x 3 x 3 x 3  board.  Suggests  3n  and  4 x 4 x 4  games.

Anon.  Puzzle page: Noughts and crosses.  MTg 33 (1965) 35.  Says practice shows that the  4 x 4 x 4  game is a draw.  [I only ever had one drawn game!]  Conjectures  nn  is first player and  (n+1)n  is a draw.

Roland Silver.  The group of automorphisms of the game of 3‑dimensional ticktacktoe.  AMM 74 (1967) 247‑254.  Finds the group of permutations of cells that preserve winning lines is generated by the rigid motions of the cube and certain 'eviscerations'.  [It is believed that this is true for the  kn  board, but I don't know of a simple proof.]

Ross Honsberger.  Mathematical Morsels.  MAA, 1978.  Prob. 13: X's and O's, p. 26.  Obtains Moser's result.

Kathleen Ollerenshaw.  Presidential Address: The magic of mathematics.  Bull. Inst. Math. Appl. 15:1 (Jan 1979) 2-12.  P. 6 discusses my rediscovery of Moser's 1948 result.

Paul Taylor.  Counting lines and planes in generalised noughts and crosses.  MG 63 (No. 424) (Jun 1979) 77-82.  Determines the number  pr(k)  of  r-sections of a  kn  board by means of a recurrence   pr(k)  =  [pr-1(k+2) - pr-1(k)]/2r   which generalises Moser's 1948 result.  He then gets an explicit sum for it.  Studies some other relationships.  This work was done while he was a sixth form student.

Oren Patashnik.  Qubic:  4 x 4 x 4  tic‑tac‑toe.  MM 53 (1980) 202‑216.  Computer assisted proof that  4 x 4 x 4  game is a first player win.

Winning Ways.  1982.  Pp. 673-679, esp. 678-679.  Discusses getting  k  in a row on a  n x n  board.  Discusses  43  game (Tic-Toc-Tac-Toe) and  kn  game.

Victor Serebriakoff.  A Mensa Puzzle Book.  Muller, London, 1982.  (Later combined with A Second Mensa Puzzle Book, 1985, Muller, London, as: The Mensa Puzzle Book, Treasure Press, London, 1991.)  Chapter 7: Conceptual conflict in multi-dimensional space, pp. 80-94 (1991: 98-112) & answers on pp. 99, 100, 106 & 131 (1991: 115, 116, 122 & 147).  He considers various higher dimensional noughts and crosses on the  33,  34  and  35  boards.  He finds that there are  49  winning lines on the  33  and he finds how to determine the number of  d-facets on an  n-cube  as the coefficients in the expansion of  (2x + 1)n.  He also considers games where one has to complete a  3 x 3  plane to win and gives a problem:  OXO three hypercube planes, p. 91 (1991: 109) & Answer 29, p. 106 (1991: 122)  which asks for the number of planes in the hypercube  34.  The answer says there are  123  of them, but in 1985 I found  154  and the general formula for the number of  d-sections of a  kn  board.  When I wrote to Serebriakoff, he responded that he could not follow the mathematics and that "I arrived at the figures ... from a simple formula published in one of Art [sic] Gardner's books which checked out as far as I could take it.  Several other mathematicians have looked through it and not disagreed."  I wrote for a reference to Gardner but never had a response.  I presented my work to the British Mathematical Colloquium at Cambridge on 2 Apr 1985 and discovered that the results were known _ I had found the explicit sum given by Taylor above, but not the recurrence.


          4.B.2. HEX


Piet Hein.  Article or column in Politiken (Copenhagen) (26 Dec 1942).  ??NYR, but the diagrams show a board of hexagons.

Gardner (1957) and others have related that the game was independently invented by John Nash at Princeton in 1948-1949.  Gardner had considerable correspondence after his article which I have examined.  The key point is that one of Niels Bohr's sons, who had known the game in Copenhagen, was a visitor at the Institute for Advanced Study at the time and showed it to friends.  I concluded that it was likely that some idea of the game had permeated to Nash who had forgotten this and later recalled and extensively developed the idea, thinking it was new to him.  I met Harold Kuhn in 1998, who was a student with Nash at the time and he has no doubt that Nash invented the idea.  In particular, Nash started with the triangular lattice, i.e. the dual of Hein's board, for some time before realising the convenience of the hexagonal lattice.  Nash came to Princeton as a graduate student in autumn 1948 and had invented the game by the spring of 1949.  Kuhn says he observed Nash developing the ideas and recognising the connections with the Jordan Curve Theorem, etc.  Kuhn also says that there was not much connection between students at Princeton and at the Institute and relates that von Neumann saw the game at Princeton and asked what it was, indicating that it was not well known at the Institute.  In view of this, it seems most likely that Nash's invention was independent, but I know from my own experience that it can be difficult to remember the sources of one's ideas _ a casual remark about a hexagonal game could have re-emerged weeks or months later when Nash was studying games, as the idea of looking at hexagonal boards in some form, from which the game would be re-invented.  Sylvester was notorious for publishing ideas which he had actually refereed or edited some years earlier, but had completely forgotten the earlier sources.  In situations like Hex, we will never know exactly what happened _ even if we were present at the time, it is difficult to know what is going on in the mind of the protagonist and the protagonist himself may not know what subconscious connections his mind is making.  Even if we could discover that Nash had been told something about a hexagonal game, we cannot tell how his mind dealt with this information and we cannot assume this was what inspired his work.  In other words, even a time machine will not settle such historical questions _ we need something that displays the conscious and the unconscious workings of a person's mind.

Parker Brothers.  Literature on Hex, 1952.  ??NYS or NYR.

Claude E. Shannon.  Computers and automata.  Proc. Institute of Radio Engineers 41 (Oct 1953) 1234‑1241.  Describes his Hex machine on p. 1237.

M. Gardner.  The game of Hex.  SA (Jul 1957) = 1st Book, chap. 8.  Description of Shannon's 8 by 7 'Hoax' machine, pp. 81‑82, and its second person strategy, p. 79.

Winning Ways.  1982.  Pp. 679-680 sketches the game and the strategy stealing argument which is attributed to Nash.

C. E. Shannon.  Photo of his Hoax machine sent to me in 1983.


          4.B.3. DOTS AND BOXES


Lucas.  Le jeu de l'École Polytechnique.  RM2, 1883, pp. 90‑91.  He gives a brief description, starting: "Depuis quelques années, les élèes de l'École Polytechnique ont imaginé un nouveaux jeu de combinaison assez original."  He clearly describes drawing the edges of the game board and that the completer of a box gets to go again.  He concludes: "Ce jeu nous a paru assez curieux pour en donner ici la description; mais, jusqu'a présent, nous ne connaissons pas encore d'observations ni de remarques assez importantes pour en dire davantage."

Lucas.  Nouveaux jeux scientifiques de M. Édouard Lucas.  La Nature 17 (1889) 301‑303.  Clearly describes a game version of La Pipopipette on p. 302, picture on p. 301, "... un nouveau jeu ... dédié aux élèves de l'école Polytechnique."  This is dots and boxes with the outer edges already drawn in.

Lucas.  L'Arithmétique Amusante.  1895.  Note III:  Les jeux scientifiques de Lucas, pp. 203‑209 _ includes his booklet:  La Pipopipette, Nouveau jeu de combinaisons, Dédié aux élèves de l'École Polytechnique, Par un Antique de la promotion de 1861, (1889), on pp. 204‑208.  On p. 207, he says the game was devised by several of his former pupils at the École Polytechnique.  On p. 37, he remarks that "Pipo est la désignation abrégée de Polytechnique, par les élèves de l'X, ...."

Robert Marquard & Georg Frieckert.  German Patent 108,830 _ Gesellschaftsspiel.  Patented 15 Jun 1899.  1p + 1p diagrams.  8 x 8  array of boxes on a board with slots for inserting edges.  No indication that the player who completes a box gets to play again.  They have some squares with values but also allow all squares to have equal value.

Loyd.  The boxer's puzzle.  Cyclopedia, 1914, pp. 104 & 352.  = MPSL1, prob. 91, pp. 88‑89 & 152‑153.  c= SLAHP: Oriental tit‑tat‑toe, pp. 28 & 92‑93.  Loyd doesn't start with the boundaries drawn.  He asserts it is 'from the East'.

Ahrens.  A&N.  1918.  Chap. XIV: Pipopipette, pp. 147‑155, describes it in more detail than Lucas does.  He says the game appeared recently.

Blyth.  Match-Stick Magic.  1921.  Boxes, pp. 84-85.  "The above game is familiar to most boys and girls ...."  No indication that the completer of a box gets to play again.

Meyer.  Big Fun Book.  1940.  Boxes, p. 661.  Brief description, somewhat vaguely stating that a player who completes a box can play again.

The Home Book of Quizzes, Games and Jokes.  Op. cit. in 4.B.1, 1941.  P. 151: Dots and squares.  Clearly says the completer gets to play again.  "The game calls for great ingenuity."

"Zodiastar".  Fun with Matches and Match Boxes.  (Cover says:  Match Tricks  From the 1880s to the 1940s.)  Universal Publications, London, nd [late 1940s?].  The game of boxes, pp. 48-49.  Starts by laying out four matches in a square and players put down matches which must touch the previous matches.  Completing a box gives another play.  No indication that matches must be on lattice lines, but perhaps this is intended.

Readers' Research Department.  RMM 2 (Apr 1961) 38‑41, 3 (Jun 1961) 51‑52, 4 (Aug 1961) 52‑55.  On pp. 40‑41 of No. 2, it says that Martin Gardner suggests seeking the best strategy.  Editor notes there are two versions of the rules _ where the one who makes a box gets an extra turn, and where he doesn't _ and that the game can be played on other arrays.  On p. 51 of No. 3, there is a symmetry analysis of the no‑extra‑turn game on a board with an odd number of squares.  On pp. 52‑54 of No. 4, there is some analysis of the extra‑turn case on a board with an odd number of boxes.

Everett V. Jackson.  Dots and cubes.  JRM 6:4 (Fall 1973) 273‑279.  Studies 3‑dimensional game where a play is a square in the cubical lattice.

Gyles Brandreth.  Pencil and Paper Games and Puzzles.  Carousel, 1976.  Worm, pp. 18-19.  This is a sort of 'anti-boxes' _ one draws segments on the lattice forming a path without any cycles _ last player wins.  = Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989, pp. 18-19.

Winning Ways.  1982.  Chap. 16: Dots-and-Boxes, pp. 507-550

David B. Lewis.  Eureka!  Perigee (Putnam), NY, 1983.  Pp. 44‑45 suggests playing on the triangular lattice.

Sheila Anne Barry.  The World's Best Travel Games.  Sterling, NY, 1987.

Eternal triangles, pp. 80-81.  Gives the game on the triangular lattice.

Snakes, pp. 81-82.  Same as Brandreth's Worm.  I think 'snake' would be a better title as only one path is drawn.


          4.B.4. SPROUTS


M. Gardner.  SA (Jul 1967) = Carnival, chap. 1.  Describes Michael Stewart Paterson and John Horton Conway's invention of the game on 21 Feb 1967 at tea time in the Department common room at Cambridge.  The idea of adding a spot was due to Paterson and they agreed the credit for the game should be 60% Paterson to 40% Conway.

Gyles Brandreth.  Pencil and Paper Games and Puzzles.  Carousel, 1976.  Sprouts, p. 13.  "... actually born in Cambridge about ten years ago."  c= Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989, p. 13: "... was invented about ten years ago."

Winning Ways.  1982.  Sprouts, pp. 564-570 & 573.  Says the game was "introduced by M. S. Paterson and J. H. Conway some time ago".  Also describes Brussels Sprouts and Stars-and-Stripes.  An answer for Brussels Sprouts and some references are on p. 573.

Sheila Anne Barry.  The World's Best Travel Games.  Sterling, NY, 1987.  Sprouts, pp. 95-97.

Karl-Heinz Koch.  Pencil & Paper Games.  (As: Spiele mit Papier und Bleistift, no details); translated by Elisabeth E. Reinersmann.  Sterling, NY, 1992.  Sprouts, pp. 36-37, says it was invented by J. H. Conway & M. S. Paterson on 21 Feb 1976 [sic _ misprint of 1967] during their five o'clock tea hour.




          See also 4.B.1 for historical material.

          The classic Nine Men's Morris board consists of three concentric squares with their midpoints joined by four lines.  The corners are sometimes also joined by another four diagonal lines, but this seems to be used with twelve men per side and is sometimes called Twelve Men's Morris _ see 1891 below.  Fiske 108 says this is common in America but infrequent in Europe, though on 127 he says both forms were known in England before 1600, and both were carried to the US, though the Nine form is probably older.


Murray 615 discusses Nine Men's Morris.  He cites Kurna, Egypt (‑14C), medieval Spain (Alquerque de Nueve), the Gokstad ship and the steps of the Acropolis of Athens.  He says the board sometimes has diagonals added and then is played with 9, 11 or 12 pieces.

Dudeney.  AM. 1917.  Introduction to Moving Counter Problems, pp. 58-59.  This gives a brief survey, mentioning a number of details that I have not seen elsewhere, e.g. its occurrence in Poland and on the Amazon.  Says the board was found on a Roman tile at Silchester and on the steps of the Acropolis in Athens among other sites.

J. A. Cuddon.  The Macmillan Dictionary of Sports and Games.  Macmillan, London, 1980.  Pp. 563‑564.  Discusses the history.  Says there is a c‑1400 board cut in stone at Kurna, Egypt and similar boards were made in years 9 to 21 at Mihintale, Ceylon.  Says Ars Amatoria may be describing Three Men's Morris and Tristia may be describing a kind of Tic‑tac‑toe.  Cites numerous medieval descriptions and variants.

Claudia Zaslavsky.  Tic Tac Toe and Other Three‑in‑a‑Row Games from Ancient Egypt to the Modern Computer.  Crowell, NY, 1982.  This is really a book for children and there are no references for the historical statements.  I have found most of them elsewhere,  and the author has kindly send me a list of source books, but I have not yet tracked down the following items _ ??.

                    There is an English court record of 1699 of punishment for playing Nine Holes in church.

                    There is a Nine Men's Morris board on a stone on the temple of Seti I (presumably this is at Kurna).  There is a picture in the 13C Spanish 'Book of Games' (presumably the Alfonso MS _ see below) of children playing Alquerque de Tres (c= Three Men's Morris).  A 14C inventory of the Duc de Berry lists tables for Mérelles (=? Nine Men's Morris) (see Fiske 113-115 below) and a book by Petrarch shows two apes playing the game.


H. Parker.  Ancient Ceylon.  Loc. cit. in 4.B.1.  Nine Men's Morris board in the Temple of Kurna, Egypt, ‑14C.  [Rohrbough, below, says this temple was started by Ramses I and completed by Seti in -1336/-1333, citing J. Royal Asiatic Soc. (1783) 17.]  Two diagrams for Nine Men's Morris are cut into the great flight of steps at Mihintale, Ceylon and these are dated c1C.  He cites Bell; Arch. Survey of Ceylon, Third Progress Report, p. 5 note, for another diagram of similar age.

Jack Botermans, Tony Burrett, Pieter van Delft & Carla van Spluntern.  The World of Games.  (In Dutch, 1987);  Facts on File, NY, 1989. 

                    P. 35 describes Yih, a form of Three Men's Morris, played on a doubly crossed square with a man moving "one step along any line".  A note adds that only the French have a rule forbidding the first player to play in the centre, which makes the game more challenging and is recommended.

                    Pp. 103-107 is the beginning of a section:  Games of alignment and configuration and discusses various games, but rather vaguely and without references.  They mention Al-Qurna, Mihintale, Gokstad and some other early sites.  They say Yih was described by Confucius, was played c-500 and is "the game, that we now know as tic-tac-toe, or three men's morris."  They describe Noughts and Crosses in the usual way.  They then distinguish Tic-Tac-Toe, saying "In Britain it is generally known as three men's morris ...." and say it is the same as Yih, "which was known in ancient Egypt".  They say "Ovid mentions tic-tac-toe" in Ars Amatoria, that several Roman boards have survived and that it was very popular in 14C England with several boards for this and Nine Men's Morris cut into cloister seats.  They then describe Three-in-a-Row, which allows pieces to move one step in any direction, as a game played in Egypt.  They then describe Five or Six Men's Morris, Nine Men's Morris, Twelve Men's Morris and Nine Men's Morris with Dice, with nice 13C & 15C illustration of Nine Men's Morris.

Bell & Cornelius.  Board Games Round the World.  Op. cit. in 4.B.1.  1988.  Pp. 6-8.  They discuss the crossed square board _ see 4.B.1 _ and describe Three Men's Morris with moves only along the lines to an adjacent vacant point.  They then describe Achi, from Ghana, on the doubly crossed square with the same rules.  They then describe Six Men's Morris which was apparently popular in medieval Europe but became obsolete by c1600.

Ovid.  Ars Amatoria.  -1.  II, 203-208  &  III, 353-366.  Translated by J. H. Mozley; Loeb Classical Library, 1929, pp. 80-81 & 142-145.  Translated by B. P. Moore, 1935, used in A. D. Melville; Ovid  The Love Poems; OUP, 1990, pp. 113, 137, 229 & 241.

                    II, 203-208 are three couplets apparently referring to three games: two dice games and Ludus Latrunculorum.  Mozley's prose translation is:

                              "If she be gaming, and throwing with her hand the ivory dice, do you throw amiss and move your throws amiss; or if is the large dice you are throwing, let no forfeit folow if she lose; see that the ruinous dogs often fall to you; or if the piece be marching under the semblance of a robbers' band, let your warrior fall before his glassy foe."

          'Dogs' is the worst throw in Roman dice games.

                    Moore's verse translation of 207-208 is:

                              "And when the raiding chessmen take the field,  Your champion to his crystal foe must yield." 

          Melville's note says the original has 'bandits' and says the game is Ludus Latrunculorum.

                    III, 357-360 is probably a reference to the same game since 'robbers' occurs again, though translated as brigands by Mozley, and again it immediately follows a reference to throwing dice.  Mozley's translation of 353-366 is:

                              "I am ashamed to advise in little things, that she should know the throws of the dice, and thy powers, O flung counter.  Now let her throw three dice, and now reflect which side she may fitly join in her cunning, and which challenge,  Let her cautiously and not foolishly play the battle of the brigands, when one piece falls before his double foe and the warrior caught without his mate fights on, and the enemy retraces many a time the path he has begun.  And let smooth balls be flung into the open net, nor must any ball be moved save that which you will take out.  There is a sort of game confined by subtle method into as many lines as the slippery year has months: a small board has three counters on either side, whereon to join your pieces together is to conquer."

          Moore's translation of 357-360 is:

                              "To guide with wary skill the chessmen's fight,  When foemen twain o'erpower the single knight,  And caught without his queen the king must face  The foe and oft his eager steps retrace". 

          This is clearly not a morris game _ Mozley's note above and the next entry make it clear it is Ludus Latrunculorum, which had a number of forms.  Mozley's note on pp. 142-143 refers to Tristia II, 478 and cites a number of other references for Ludus Latrunculorum.

                    Moore's translation of 363-366 is:

                              "A game there is marked out in slender zones  As many as the fleeting year has moons;  A smaller board with three a side is manned,  And victory's his who first aligns his band." 

          Mozley's notes and Melville's notes say the first two lines refer to the Roman game of Ludus Duodecim Scriptorum _ the Twelve Line Game _ which is the ancestor of Backgammon.  Mozley says the game in the latter two lines is mentioned in Tristia, "but we have no information about it."  Melville says it is "a 'position' game, something like Nine Men's Morris" and cites R. C. Bell's article on 'Board and tile games' in the Encyclopaedia Britannica, 15th ed., Macropaedia ii.1152‑1153, ??NYS.

Ovid.  Tristia.  c10.  II, 471‑484.  Translated by A. L. Wheeler.  Loeb Classical Library, 1945, pp. 88‑91.  This mentions several games and the text parallels that of Ars Amatoria III. 

                              "Others have written of the arts of playing at dice _ this was no light sin in the eyes of our ancestors _ what is the value of the tali, with what throw one can make the highest point, avoiding the ruinous dogs; how the tessera is counted, and when the opponent is challenged, how it is fitting to throw, how to move according to the throws; how the variegated soldier steals to the attack along the straight path when the piece between two enemies is lost, and how he understands warfare by pursuit and how to recall the man before him and to retreat in safety not without escort; how a small board is provided with three men on a side and victory lies in keeping one's men abreast; and the other games _ I will not describe them all _ which are wont to waste that precious thing, our time." 

          A note says some see a reference to Ludus Duodecim Scriptorum at the beginning of this.  The next note says the next text refers to Ludus Latrunculorum, a game on a squared board with 30 men on a side, with at least two kinds of men.  The note for the last game says "This game seems to have resembled a game of draughts played with few men." and refers to Ara Amatoria and the German Mühlespiel, which he describes as 'a sort of draughts', but which is Nine Men's Morris.

R. G. Austin.  Roman board games _ I & II.  Greece and Rome  4 (No. 10) (Oct 1934) 24‑34  &  4 (No. 11) (Feb 1935) 76-82.  Claims the Ovid references are to Ludus Latrunculorum (a kind of Draughts?), Ludus Duodecim Scriptorum (later Tabula, an ancestor of Backgammon) and (Ars Amatoria.iii.365-366) a kind of Three Men's Morris.  In the last, in is not clear which rule he adopts for the later movement of pieces, but he says: "the first player is always able to force a win if he places his first man on the centre point, and this suggests that the dice may have been used to determine priority of play, although there is no evidence of this."  He discusses various known artifacts, citing several Roman  8 x 8  boards found in Britain.  He gives an informal bibliography with comments as to the value of the works.

D'Arcy Wentworth Thompson.  Science and the Classics.  OUP, 1940.  Section V  Games and Playthings, pp. 148-165.  On p. 160, he quotes Ovid, Ars Amatoria.iii.365-366 and says it is Noughts and Crosses, or in Ireland, Tip-top-castle.

The British Museum has a Nine Men's Morris board from the Temple of Artemis, Ephesus, 2C-4C.  Item BM GR 1872,8-3,44.  This was in a small exhibition of board games in 1990.  I didn't see it on display in late 1996.

Murray, p. 189.  There was an Arabic game called Qirq, which Murray identifies with Morris.  "Fourteen was a game played with small stones on a wooden board which had three rows of holes (al‑Qâbûnî)."  Abû‑Hanîfa, c750, held that Fourteen was illegal and Qirq was held illegal by writers soon afterward.  On p. 194, Murray gives a 10C passage mentioning Qirq being played at Mecca.

Fiske 255 cites the Kit_b al Agh_ni, c960, for a reference to qirkat, i.e. morris boards.

Paul B. Du Chaillu.  The Viking Age.  Two vols., John Murray, London, 1889.  Vol. II, p.168, fig. 992 _ Fragments of wood from Gokstad ship.  Shows a partial board for Nine Mens's Morris found in the Gokstad ship burial.  There is no description of this illustration and there is only a vague indication that this is 10C, but other sources say it is c900.

Gutorm Gjessing.  The Viking Ship Finds.  Revised ed., Universitets Oldsaksamling, Oslo, 1957.  P. 8:  "... there are two boards which were used for two kinds of games; on one side figures appear for use in a game which is frequently played even now (known as "Mølle")."

Thorlief Sjøvold.  The Viking Ships in Oslo.  Universitets Oldsaksamling, Oslo, 1979.  P. 54:  "... a gaming board with one antler gaming piece, ...."

In medieval Europe, the game is called Ludus Marellorum or Merellorum or just Marelli or Merelli or Merels, meaning the game of counters.  Murray 399 says the connection with Qirq is unclear.  However, medieval Spain played various games called Alquerque, which is obviously derived from Qirq.  Alquerque de Nueve seems to be Nine Men's Morris.  However, in Italy and in medieval France, Marelle or Merels could mean Alquerque (de Doze), a draughts‑like game with 12 men on a side played on a  5 x 5  board (Murray 615).  Also Marro, Marella can refer to Draughts which seems to originate in Europe somewhat before 1400.

Stewart Culin.  Korean Games, with Notes on the Corresponding Games of China and Japan.  University of Pennsylvania, Philadelphia, 1895.  Reprinted as:  Games of the Orient; Tuttle, Rutland, Vermont, 1958.  Reprinted under the original title, Dover and The Brooklyn Museum, 1991.  P. 102, section 80: Kon-tjil _ merrells.  This is the usual Nine Men's Morris.  The Chinese name is Sám-k'i (Three Chess).  "I am told by a Chinese merchant that this game was invented by Chao Kw'ang-yin (917-975), founder of the Sung dynasty."  This is the only indication of an oriental source that I have seen.

Gerhard Leopold.  Skulptierte Werkstücke in der Krypta der Wipertikirche zu Quedlinburg.  IN: Friedrich Möbius & Ernst Schubert, eds.; Skulptur des Mittelalters; Hermann Böhlaus Nachfolger, Weimar, 1987, pp. 27-43; esp. pp. 37 & 43.  Describes and gives photos of several Nine-Men's-Morris boards carved on a pillar of the crypt of the Wipertikirche, Quedlinburg, Sachsen-Anhalt, probably from the 10/11 C.

Richard de Fournivall.  De Vetula.  13C.  This describes various games, including Merels.  Indeed the French title is:  Ci parle du gieu des Merelles ....  ??NYS _ cited by Murray, pp. 439, 507, 520, 628.  Murray 620 cites several MSS and publications of the text.

"Bonus Socius" [Nicolas de Nicolaï?].  This is a collection of chess problems, compiled c1275, which exists in many manuscript forms and languages.  See 5.F.1 for more details of these MSS.  See Murray 618‑642.  On pp. 619‑624 & 627, he mentions several MSS which include 23, 24, 25 or 28 Merels problems.  On p. 621, he cites "Merelles a Neuf" from 14C.  Fiske 104 & 110-111 discusses some MSS of this collection.

The Spanish Treatise on Chess-Play written by order of King Alfonso the Sage in the year 1283.  [Generally known as the Alfonso MS.]  MS in Royal Library of the Escorial (j.T.6. fol).  Complete reproduction in 194 Phototypic Plates.  2 vols.,  Karl W. Hirsemann, Leipzig, 1913.  (See in 4.A.1 for another ed.)  This is a collection of chess problems produced for Alfonso X, the Wise, King of Castilla (Castile).  Vol. 2, ff. 92v‑93r, pp. CLXXXIV‑CLXXXV, shows Nine Men's Morris boards.  ??NX _ need to study text.  See:  Murray 568‑573;  van der Linde I 137 & 279 ??NYS  &  Quellenstudien 73 & 277‑278, ??NYS (both cited by Fiske 98);  van der Lasa 116, ??NYS (cited by Fiske 99). 

                    Fiske 98-99 says that the MS also mentions Alquerque, Cercar de Liebre and Alquerque de Neuve (with 12 men against one).  Fiske 253-255 gives a more detailed study of the MS based on a transcript.  He also quotes a communication citing al Querque or al Kirk in Kazirmirski's Arabic dictionary and in the Kit_b al Agh_ni, c960.

                    José Brunet y Bellet.  El Ajedrez.  Barcelona, 1890.  ??NYS _ described by Fiske 98.  This has a chapter on the Alfonso MS and refers to Alquerque de Doce, saying that it is known as Tres en Raya in Castilian and Marro in Catalan (Fiske 102 says this word is no longer used in Spanish).  Brunet notes that there are five miniatures pertaining to alquerque.  Fiske says that all this information leaves us uncertain as to what the games were.  Fiske says Brunet's chapter has an appendix dealing with Carrera's 1617 discussion of 'line games' and describing Riga di Tre as the same as Marro or Tres en Raya as a form of Three Men's Morris

Murray gives many brief references to the game, which I will note here simply by his page number and the date of the item. 

                    438‑439 (12C);  446 (14C);

                    449 (c1400 _ 'un marrelier', i.e. a Merels board);

                    431 (c1430);  447 (1491);  446 (1538).

Anon.  Romance of Alexander.  1338.  (Bodleian Library, Mss Bodl. 264).  ??NYS.  Nice illustration clearly showing Nine Men's Morris board.  I. Disraeli (Amenities of Literature, vol. I, p. 86) also cites British Museum, Bib. Reg. 15, E.6 as a prose MS version with illustrations.  Prof. D. J. A. Ross tells me there is nothing in the text corresponding to the illustrations and that the Bodleian text was edited by M. R. James, c1920, ??NYS.  Illustration reproduced in:  A. C. Horth; 101 Games to Make and Play; Batsford, London, (1943;  2nd ed., 1944);  3rd ed., 1946; plate VI facing p. 44, in B&W.  Also in:  Pia Hsiao et al.; Games You Make and Play; Macdonald and Jane's, London, 1975, p. 7, in colour.

Fiske 113-115 gives a number of quotations from medieval French sources as far back as mid 14C, including an inventory of the Duc de Berry in 1416 listing two boards.  Fiske notes that the game has given rise to several French phrases.  He quotes a 1412 source calling it Ludus Sanct Mederici or Jeu Saint Marry and also mentions references in city statutes of 1404 and 1414.

MS, Montpellier, Faculty of Medicine, H279 (Fonts de Boulier, E.93).  14C.  This is a version of the Bonus Socius collection.  Described in Murray 623-624, denoted M, and in van der Linde I 301, denoted K.  Lucas, RM2, 1883, pp. 98-99 mentions it and RM4, 1894, Quatrième Récréation: Le jeu des mérelles au XIIIe siècle, pp. 67-85 discusses it extensively.  This includes 28 Merels problems which are given and analysed by Lucas.  Lucas dates the MS to the 13C.

Household accounts of Edward IV, c1470.  ??NYS _ see Murray 617.  Record of purchase of "two foxis and 46 hounds" to form two sets of "marelles".

Civis Bononiae [Citizen of Bologna].  This is a collection of chess problems compiled c1475, which exists in several MSS.  See Murray 643‑703.  It has 48 or 53 merels problems.  On p. 644, 'merelleorum' is quoted.

A Hundred Sons.  Chinese scroll of Ming period (1368-1644).  18C copy in BM.  ??NYS _ extensively reproduced and described in:  Marguerite Fawdry; Chinese Childhood; Pollock's Toy Theatres, London, 1977.  On p. 12 of Fawdry is a scene, apparently from the scroll, in which some children appear to be playing on a Twelve Men's Morris board.

Elaborate boards from Germany (c1530) and Venice (16C) survive in the National Museum, Munich and in South Kensington (Murray 757‑758).  Murray shows the first in B&W facing p. 757.

William Shakespeare.  A Midsummer Night's Dream.  c1610.  Act II, scene I, lines 98-100:  "The nine men's morris is fill'd up with mud,  And the quaint mazes in the wanton green  For lack of tread are indistinguishable."  Fiske 126 opines that the latter two lines may indicate that the board was made in the turf, though he admits that they may refer just to dancers' tracks, but to me it clearly refers to turf mazes.

J. C. Bulenger.  De Ludis Privatis ac Domesticus Veterum.  Lyons, 1627.  ??NYS  Fiske 115 & 119 quote his description of and philological note on Madrellas (Three Men's Morris).

Paul Fleming (1609-1640).  In one of his lyrics, he has Mühlen.  ??NYS _ quoted by Fiske 132, who says this is the first German mention of Morris. 

Fiske 133 gives the earliest Russian reference to Morris as 1675.

Thomas Hyde.  Historia Nerdiludii, hoc est dicere, Trunculorum; ....  (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.)  From the Sheldonian Theatre (i.e. OUP), Oxford, 1694.  Historia Triodii, pp. 202-214, is on morris games.  (Described in Fiske 118-124, who says there is further material in the Elenchus at the end of the volume _ ??NYS)  Hyde asserts that the game was well known to the Romans, though he cannot find a Roman name for it!  He cites and discusses Bulenger, but disagrees with his philology.  Gives lots of names for the game, ranging as far as Russian and Armenian.  He gives both the Nine and Twelve Men's Morris boards on p. 210, but he has not found the Twelve board in Eastern works.  On p. 211, he gives the doubly crossed square board with a title in Chinese characters, pronounced 'Che-lo', meaning 'six places', and having three white and three black men already placed along two sides.  He says the Irish name is Cashlan Gherra (Short Castle) and that the name Copped Crown is common in Cumberland and Westmoreland.  He then describes playing the Twelve Man and Nine Man games, and then he considers the game on the doubly crossed square board.  He seems to say there are different rules as to how one can move.  ??need to study the Latin in detail.  This is said to throw light on the Ovid passages.  Hyde believes the game was well known to the Romans and hence must be much older.  Fiske remarks that this is history by guesswork.

Murray 383 describes Russian chess.  He says Amelung identifies the Russian game "saki with Hölzchenspiel (?merels)".  Saki is mentioned on this page as being played at the Tsar's court, c1675.

Archiv der Spiele.  3 volumes, Berlin, 1819-1821.  Vol. 2 (1820) 21-27.  ??NYS  Described and quoted by Fiske 129-132.  This only describes the crossed square and the Nine Men's Morris boards.  It says that the Three Men's Morris on the crossed square board is a tie, i.e. continues without end, but it is not clear how the pieces are allowed to move.  Fiske says this gives the most complete explanation he knows of the rules for Nine Men's Morris.

Charles Babbage.  Notebooks _ unpublished collection of MSS in the BM as Add. MS 37205.  ??NX.  For more details, see 4.B.1.  On ff. 347.r-347.v, 8 Sep 1848, he suggests Nine Mens Morris boards in triangular and pentagonal shapes and does various counting on the different shapes.

The Family Friend (1856) 57.  Puzzle 17. _ Two and a Bushel.  Shows the standard  #  board.  "This very simple and amusing games, _ which we do not remember to have seen described in any book of games, _ is played, like draughts, by two persons with counters.  Each player must have three, ...  and the game is won when one of the players succeeds in placing his three men in a row; ...."  There is no specification of how the men move.  The word 'bushel' occurs in some old descriptions of Three Men's Morris and Nine Men's Morris as the name of the central area.

The Sociable.  1858.  Merelles: or, nine men's morris, pp. 279-280.  Brief description, notable for the use of Merelles in an English book.

Von der Lasa.  Ueber die griechischen und römischen Spiele, welche einige ähnlichkeit mit dem Schach hatten.  Deutsche Schachzeitung (1863) 162-172, 198-199, 225-234, 257‑264.  ??NYS _ described on Fiske 121-122 & 137, who says van der Linde I 40-47 copies much of it.  He asserts that the Parva Tabella of Ovid is Kleine Mühle (Three Men's Morris).  Von der Lasa says the game is called Tripp, Trapp, Trull in the Swedish book Hand-Bibliothek för Sällkapsnöjen, of 1839, vol. II, p. 65 (or 57??).  Van der Linde says that the Dutch name is Tik, Tak, Tol.  Fiske notes that both of these refer to Noughts and Crosses, but it is unclear if von der Lasa or van der Linde recognised the difference between Three Men's Morris and Noughts and Crosses.

Albert Norman.  Ungdomens Bok [Book for Youth] (in Swedish).  2nd ed., Stockholm, 1883.  Vol. I, p. 162++.  ??NYS _ quoted and described in Fiske 134-136.  Plays Nine Men's Morris on a Twelve Men's Morris board.

Webster's Dictionary.  1891.  ??NYS _ Fiske 118 quotes a definition (not clear which) which includes "twelve men's morris".  Fiske says:  "Here we have almost the only, and certainly the first mention of the game by its most common New England name, "twelve men's morris," and also the only hint we have found in print that the more complicated of the morris boards _ with the diagonal lines ... _ is used with twelve men, instead of nine, on each side."  Fiske 127 says the name only appears in American dictionaries.

Dudeney.  CP.  1907.  Prob. 110: Ovid's game, pp. 156‑157 & 248.  Says the game "is distinctly mentioned in the works of Ovid."  He gives Three Men's Morris, with moves to adjacent cells horizontally or diagonally, and says it is a first player win.

Blyth.  Match-Stick Magic.  1921.  Black versus white, pp. 79-80.  4 x 4  board with four men each.  But the men must be initially placed  WBWB  in the first row and  BWBW  in the last row.  They can move one square "in any direction" and the object is to get four in a row of your colour.

Games and Tricks _ to make the Party "Go".  Supplement to "Pearson's Weekly", Nov. 7th, no year indicated [1930s??].  A matchstick game, p. 11.  On a  4 x 4  board, place eight men,  WBWB  on the top row and  BWBW  on the bottom row.  Players alternately move one of their men by one square in any direction _ the object is to make four in a line.

Lynn Rohrbough, ed.  Ancient Games.  Handy Series, Kit N, Cooperative Recreation Service, Delaware, Ohio, (1938), 1939. 

Morris was Player [sic] 3,300 Years Ago, p. 27.  Says the temple of Kurna was started by Ramses I and completed by Seti in -1336/-1333, citing J. Royal Asiatic Soc. (1783) 17.

Three Men's Morris, p. 27.  After placing their three men, players 'then move trying to get three men in a row.'  Contributor says he played it in Cardiff more than 50 years ago.

Winning Ways.  1982.  Pp. 672-673.  Says Ovid's Game is conjectured to be Three Men's Morris.  The current version allows moves by one square orthogonally and is a first person win if the first person plays in the centre.  If the first player cannot play in the centre, it is a draw.  They use Three Men's Morris for the case with one step moves along winning lines, i.e. orthogonally or along main diagonals.  An American Indian game, Hopscotch, permits one step moves orthogonally or diagonally (along any diagonal).  A French game, Les Pendus, allows any move to a vacant cell.  All of these are draws, even allowing the first player to play in the center.  They briefly describe Six and Nine Men's Morris.

L. V. Allis.  Beating the World Champion _ The state of the art in computer game playing.  IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 155-175.  On p. 163, he states that Ralph Gasser showed that Nine Men's Morris is a draw in Oct 1993, but the only reference is to a letter from Gasser.

Ralph Gasser.  ??.  IN: Games of No Chance; ed. by Richard Nowakowski; CUP, 1997??, ??NYS - described in  William Hartston; What mathematicians get up to; The Independent Long Weekend (29 Mar 1997) 2.  Demonstrates that Nine Men's Morris is a draw.  Gasser's abstract: "We describe the combination of two serach methods used to solve Nine Men's Morris.  An improved analysis algorithm computes endgame databases comprising about 1010 states.  An 18-ply alpha-beta search the used these databases to prove that the value of the initial position is a draw.  Nine Men's Morris is the first non-trivial game to be solved that does not seem to benefit from knowledge-based methods."  I'm not sure about the last statement _  4 x 4 x 4  noughts and crosses (see 4.B.1.a) and Connect-4 were solved in 1980 and 1988, though the first was a computer aided proof and the original brute force solution of Connect-4 by James Allen in Sep 1988 was improved to a knowledge-based approach by L. V. Allis by Aug 1989.  The five-in-a-row version of Connect-4 was shown to be a first person win in 1993.


          4.B.6. PHUTBALL


Winning Ways.  1982.  Philosopher's football, pp. 688‑691.


          4.B.7. BRIDG‑IT


M. Gardner.  SA (Oct 1958) c= 2nd Book, Chap. 7.  Introduces David Gales's game, later called Bridg‑it.  Addendum in the book notes that it is identical to Shannon's 'Bird Cage' game of 1951 and that it was marketed as Bridg‑it in 1960.

M. Gardner.  SA (Jul 1961) c= New MD, Chap. 18.  Describes Oliver Gross's simple strategy for the first player to win.  Addendum in the book gives references to other solutions and mentions.

M. Gardner.  SA (Jan 1973) c= Knotted, Chap. 9.  Article says Bridg‑it was still on the market.

Winning Ways.  1982.  Pp. 680-682.  Covers Bridg-it and Shannon Switching Game.


          4.B.8. CHOMP


Fred Schuh.  Spel van delers (Game of divisors).  Nieuw Tijdschrift vor Wiskunde 39 (1951‑52) 299‑304. ??NYS _ cited by Gardner, below.

M. Gardner.  SA (Jan 1973) c= Knotted, Chap. 9.  Gives David Gale's description of his game and results on it.  Addendum in Knotted points out that it is equivalent to Schuh's game and gives other references.

David Gale.  A curious Nim-type game.  AMM 81 (1974) 876-879.  Describes the game and the basic results.  Wonders if the winning move is unique.  Considers three dimensional and infinite forms.  A note added in proof refers to Gardner's article, says two programmers have consequently found that the  8 x 10  game has two winning first moves and mentions Schuh's game.

Winning Ways.  1982.  Pp. 598-600.  Brief description with extensive table of good moves.  Cites an earlier paper of Gale and Stewart which does not deal with this game.


          4.B.9. SNAKES AND LADDERS


          I have included this because it has an interesting history and because I found a nice way to express it as a kind of Markov process or random walk, and this gives an expression for the average time the game lasts.  I then found that the paper by Daykin et al. gives most of these ideas.

          The game has two or three rules for finishing.

          A.      One finishes by going exactly to the last square, or beyond it.

          B.       One finishes by going exactly to the last square.  If one throws too much, then one stands still.

          C.      One finishes by going exactly to the last square.  If one throws too much, one must count back from the last square.  E.g., if there are 100 squares and one is at 98 and one throws 6, then one counts:  99, 100, 99, 98, 97, 96  and winds up on 96.  (I learned this from a neighbour's child but have only seen it in one place _ in the first Culin item below.)

          Games of this generic form are often called promotion games.  If one considers the game with no snakes or ladders, then it is a straightforward race game, and these date back to Egyptian and Babylonian times, if not earlier.

          In fact, the same theory applies to random walks of various sorts, e.g. random walks of pieces on a chessboard, where the ending is arrival exactly at the desired square.


In the British Museum, Room 52, Case 24 has a Babylonian ceramic board (WA 1991-7.20,I) for a kind of snakes and ladders from c-1000.  The label says this game was popular during the second and first millennia BC.

Sheng-kuan t'u [The game of promotion].  7C.  Chinese game.  This is described in:  Nagao Tatsuzo; Shina Minzoku-shi [Manners and Customs of the Chinese]; Tokyo, 1940-1942, perhaps vol. 2, p. 707, ??NYS  This is cited in:  Marguerite Fawdry; Chinese Childhood; Pollock's Toy Theatres, London, 1977, p. 183, where the game is described as "played on a board or plan representing an official career from the lowest to the highest grade, according to the imperial examination system.  It is a kind of Snakes and Ladders, played with four dice; the object of each player being to secure promotion over the others."

Thomas Hyde.  Historia Nerdiludii, hoc est dicere, Trunculorum; ....  (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.)  From the Sheldonian Theatre (i.e. OUP), Oxford, 1694.  De ludo promotionis mandarinorum,  pp. 70-101 _ ??NX.  This is a long description of  Shing quon tu,  a game on a board of 98 spaces, each of which has a specific description which Hyde gives.  There is a folding plate showing the Chinese board, but the copy in the Graves collection is too fragile to photocopy.  I did not see any date given for the game.

Stewart Culin.  Chinese Games with Dice and Dominoes.  From the Report of the U. S. National Museum for 1893, pp. 489‑537.  Pp. 502-507 describes several versions of the Japanese Sugoroku (Double Sixes) which is a generic name for games using dice to determine moves, including backgammon and simple race games, as well as Snakes and Ladders games.  One version has ending in the form C.  Then says  Shing Kún T’ò  (The Game of the Promotion of Officials)  is described by Hyde as The Game of the Promotion of the Mandarins and gives an extended description of it.  There is a legend that the game was invented when the Emperor Kienlung (1736-1796) heard a candidate playing dice and the candidate was summoned to explain.  He made up a story about the game, saying that it was a way for him and his friends to learn the different ranks of the civil service.  He was sent off to bring back the game and then made up a board overnight.  However Hyde had described the game a century before this date.  It seems that this is not really a Snakes and Ladders game as the moves are determined by the throw of the dice and the position _ there are no interconnections between cells.  But Culin notes that the game is complicated by being played for money or counters which permit bribery and rewards, etc.

Culin.  Chess and Playing Cards.  Op. cit. in 4.A.4.  1898. 

                    Pp. 820-822 & plates 24 & 25 between 821 & 822.  Says  Shing Kún T’ò  (The Game of the Promotion of Officials)  is described by Hyde as  The Game of the Promotion of the Mandarins  and refers to the above for an extended description.  Describes the Korean version:  Tjyong-Kyeng-To  (The Game of Dignitaries)  and several others from Korea and Tibet, with 108, 144, 169 and 64 squares. 

                    Pp. 840-842 & plate 28, opp. p. 841 describes  Chong ün Ch’au  (Game of the Chief of the Literati)  as 'in many respects analogous' to  Shing Kún T’ò  and the Japanese game Sugoroku (Double Sixes) _ in several versions.  Then mentions modern western versions _  Jeu de L'Oie,  Giuoco dell'Oca,  Juego de la Oca,  Snake Game.  Pp. 843-848 is a table listing 122 versions of the game in the University of Pennsylvania Museum of Archaeology and Paleontology.  These are in 11 languages, varying from 22 to 409 squares.

Bell & Cornelius.  Board Games Round the World.  Op. cit. in 4.B.1.  1988.  Snakes and Ladders and the Chinese Promotion Game, pp. 65‑67.  They describe the Hindu version of Snakes and Ladders, called  Moksha-patamu.  Then they discuss  Shing Kun t'o  (Promotion of the Mandarins),  which was played in the Ming (1368-1616) with four or more players racing on a board with 98 spaces and throwing 6 dice to see how many equal faces appeared.  They describe numerous modern variants.

Andrew Topsfield.  The Indian game of snakes and ladders.  Artibus Asiae 46:3 (1985) 203‑214 + 14 figures.  Basically a catalogue of extant Indian boards.  He says the game is called  Gy_n caupad  or  Gy_n chaupar  in Hindi.  He states that  Moksha-patamu  sounds like it is Telugu and that this name appeared in Grunfield's Games of the World (1975) with no reference to a source and that Bell has repeated this.  Game boards were drawn or painted on paper or cloth and hence were perishable.  The oldest extant version is believed to be an 84 square board of 1735, in the Museum of Indology, Jaipur.  There were Hindu, Jain, Muslim and Tibetan versions representing a kind of Pilgrim's Progress, finally arriving at God or Heaven or Nirvana.  The number of squares varies from 72 to 360. 

                    He gives many references and further details.  An Indian version of the game was described by F. E. Pargiter; An Indian game: Heaven or Hell; J. Royal Asiatic Soc. (1916) 539-542, ??NYS.  He cites the version by Ayres (and Love's reproduction of it _ see below) as the first English version.  He cites several other late 19C versions.


F. H. Ayres.  [Snakes and ladders game.]  No. 200682 Regd.  Example in the Bethnal Green Museum, Misc. 8 - 1974.  Reproduced in:  Brian Love; Play The Game; Michael Joseph, London, 1978; Snakes & Ladders 1, pp. 22-23.  This is the earliest known English version of the game, with  100  cells in a spiral and  5  snakes and  5  ladders.


D. E. Daykin, J. E. Jeacocke & D. G. Neal.  Markov chains and snakes and ladders.  MG 51 (No. 378) (Dec 1967) 313-317.  Shows that the game can be modelled as a Markov process and works out the expected length of play for one player  (47.98  moves) or two players  (27.44  moves) on a particular board with finishing rule A.

Lewis Carroll.  Board game for one.  In: Lewis Carroll's Bedside Book; ed. by Gyles Brandreth (under the pseud. Edgar Cuthwellis); Methuen, 1979, pp. 19-21.  ??look for source.  Board of 27 cells with pictures in the odd cells.  If you land on any odd cell, except the last one, you have to return to square 1.  "Sleep is almost certain to have overwhelmed the player before he reaches the final square."  Ending A is probably intended.  (The average duration of this game should be computable.)

S. C. Althoen, L. King & K. Schilling.  How long is a game of snakes and ladders?  MG 77 (No. 478) (Mar 1993) 71-76.  Similar analysis to Daykin, Jeacocke & Neal, using finishing rule B, getting  39.2  moves.  They also use a simulation to find the number of moves is about  39.1.

David Singmaster.  Letter [on Snakes and ladders].  MG 79 (No. 485) (Jul 1995) 396-397.  In response to Althoen et al.  Discusses history, other ending rules and wonders how the length depends on the number of snakes and ladders.

Irving L. Finkel.  Notes on two Tibetan dice games.  IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 24-47.  Part II: The Tibetan 'Game of Liberation', pp. 34-47, discusses promotion games with many references to the literature and describes a particular game.


          4.B.10.         MU TORERE


          This is a Maori game which can be found in several books on board games.  I have included it because it has been completely analysed.  There are eight (or  2n)  points around a central area.  Each player has four (or  n)  markers, originally placed on consecutive points.  One can move from a point to an adjacent point or to the centre, or one can move from the centre to a point, provided the position moved to is empty.  The first player who cannot move is the loser.  To prevent the game becoming trivial, it is necessary to require that the first two (or one) moves of each player involve his end pieces, though other restrictions are sometimes given.


Marcia Ascher.  Mu Torere: An analysis of a Maori game.  MM 60 (1987) 90-100.  Analyses the game with  2n  points.  For  n = 1,  there are 6 inequivalent positions (where equivalence is by rotation or reflection of the board) and play is trivially cyclic.  For  n = 2,  there are  12  inequivalent positions, but there are no winning positions.  For  n = 3,  there are  30  inequivalent positions, some of which are wins, but the game is a tie.  Obtains the number of positions for general  n.  For the traditional version with  n = 4,  there are  92  inequivalent positions, some of which are wins, but the game is a tie, though this is not at all obvious to an inexperienced player.  In 1856, it was reported that no foreigner could win against a Maori.  For  n = 5,  there are  272  inequivalent positions, but the game is a easy win for the first player _ the constraints on first moves need to be revised.  Ascher gives references to the ethnographic literature for descriptions of the game.

Marcia Ascher.  Ethnomathematics.  Brooks/Cole Publishing, Pacific Grove, California, 1991.  Sections 4.4-4.7, pp. 95-109  &  Notes 4-7, pp. 118-119.  Amplified version of her MM article.


          4.B.11.         MASTERMIND, ETC.


          There were a number of earlier guessing games of the Mastermind type before the popular version in the early 1970s _ see:  Reddi.  Since then there have been a number of papers on optimal strategies.  I include a few of these. 

          NOTATION.  If there are  h  holes and  c  choices at each hole, then I abbreviate this as  ch.


S. S. Reddi.  A game of permutations.  JRM 8:1 (1975) 8-11.  Mastermind type guessing of a permutation of  1,2,3,4  can win in  5  guesses.

Donald E. Knuth.  The computer as Master Mind.  JRM 9:1 (1976-77) 1-6.  64 can be won in  5  guesses.

A. K. Austin.  Strategies for Mastermind.  G&P 71 (Winter 1978) 14-16.  Presents Knuth's results and some other work.

A. K. Austin.  How fo You play 'Master Mind'.  MTg 71 (Jun 1975) 46-47.  How to state the rules correctly.

Merrill M. Flood.  Mastermind strategy.  JRM 18:3 (1985-86) 194-202.  Cites five earlier papers on strategy, including Knuth.

Antonio M. Lopez, Jr.  A PROLOG Mastermind program.  JRM 23:2 (1991) 81-93.  Cites Knuth, Flood and three other papers on strategy.

Kenji Koyama and Tony W. Lai.  An optimal Mastermind strategy.  JRM 25:4 (1994) 251‑256.  Using exhaustive search, they find the strategy that minimizes the expected number of guesses, getting expected number  5625/1296 = 4.340.  However, the worst case in this problem requires  6  guesses.  By a slight adjustment, they find the optimal strategy with worst case requiring  5  guesses and its expected number of guesses is  5626/1296 = 4.341.  10 references to previous work, not including all of the above.


          4.B.12.         RITHMOMACHIA  =  THE PHILOSOPHERS' GAME


          I have generally not tried to include board games in any comprehensive manner, but I have recently seen some general material on this which will be useful to anyone interested in the game.  The game is one of the older and more mathematical of board games, dating from c1000, but generally abandoned about the end of the 16C along with the Neo-Pythagorean number theory of Boethius on which the game was based.


Arno Borst.  Das mittelalterliche Zahlenkampfspiel.  Sitzungsberichten der Heidelberger Akademie der Wissenschaften, Philosophisch-historische Klasse 5 (1986) Supplemente.  Available separately:  Carl Winter, Heidelberg, 1986.  Edits the surviving manuscripts on the game.  ??NYS _ cited by Stigter & Folkerts.

Detlef Illmer, Nora Gädeke, Elisabeth Henge, Helen Pfeiffer & Monika Spicker-Beck.  Rhythmomachia.  Hugendubel, Munich, 1987.

Jurgen Stigter.  Emanuel Lasker: A Bibliography  AND  Rithmomachia, the Philosophers' Game: a reference list.  Corrected, 1988 with annotations to 1989, 1 + 15 + 16pp preprint available from the author, Molslaan 168, NL‑2611 CZ Delft, Netherlands.  Bibliography of the game.

Jurgen Stigter.  The history and rules of Rithmomachia, the Philosophers' Game.  14pp preprint available from the author, as above.

Menso Folkerts.  'Rithmimachia'.  In:  Die deutsche Litteratur des Mittelalters: Verfasserlexikon; 2nd ed., De Gruyter, Berlin, 1990; vol. 8, pp. 86-94.  Sketches history and describes the 10 oldest texts.

Menso Folkerts.  Die Rithmachia des Werinher von Tegernsee.  In:  Vestigia Mathematica, ed. by M. Folkerts & J. P. Hogendijk, Rodopi, Amsterdam, 1993, pp. 107-142.  Discusses Werinher's work (12C), preserved in one MS of c1200, and gives an edition of it.


          4.B.13.         MANCALA GAMES


          This is a very broad field and I will only mention a few early items.  Four row mancala games are played in south and east Africa.  Three row games are played in Ethiopia and adjacent parts of Somaliland.  Two row games are played everywhere else in Africa, the Middle East and south and south-east Asia.  See the standard books by R. C. Bell and Falkener for many examples.  Many general books mention the game, but I only know a few specific books on the game _ this are listed first below.

          One article says that game boards have been found in the pyramids of Khamit (-1580) and there are numerous old boards carved in rocks in several parts of Africa.

          An anonymous article, by a member of the Oware Society in London, [Wanted: skill, speed, strategy; West Africa (16-22 Sep 1996) 1486-1487] lists the following names for variants of the game: Aditoe (Volta region of Ghana), Awaoley (Côte d'Ivoire), Ayo (Nigeria), Chongkak (Johore), Choro (Sudan), Congclak (Indonesia), Dakon (Philippines), Guitihi (Kenya), Kiarabu (Zanzibar), Madji (Benin), Mancala (Egypt), Mankaleh (Syria), Mbau (Angola), Mongola (Congo), Naranji (Sri Lanka), Qai (Haiti), Ware (Burkina Faso), Wari (Timbuktu), Warri (Antigua),

Stewart Culin.  Mancala, The National Game of Africa.  IN: US National Museum Annual Report 1894, Washington, 1896, pp. 595-607.

Chief A. O. Odeleye.  Ayo  A Popular Yoruba Game.  University Press Ltd., Ibadan, Nigeria, 1979.  No history.

Laurence Russ.  Mancala Games.  Reference Publications, Algonac, Michigan, 1984.  Photocopy from Russ, 1995.

Kofi Tall.  Oware  The Abapa Version.  Kofi Tall Enterprise, Kumasi, Ghana, 1991.

Salimata Doumbia  &  J. C. Pil.  Les Jeux de Cauris.  Institut de Recherches Mathématiques, 08 BP 2030, Abidjan 08, Côte d'Ivoire, 1992.

Pascal Reysset & François Pingaud.  L'Awélé.  Le jeu des semailles africaines.  2nd ed., Chiron, Paris, 1995 (bought in Dec 1994).  Not much history.

François Pingaud.  L'awélé  jeu de strategie africain.  Bornemann, 1996.

Alexander J. de Voogt.  Mancala  Board Games.  British Museum Press, 1997.  ??NYR.

Larry (= Laurence) Russ.  The Complete Mancala Games Book  How to Play the World's Oldest Board Games.  Foreword by Alex de Voogt.  Marlowe & Co., NY, 2000.  His 1984  book is described as an earlier edition of this.


William Flinders Petrie.  Objects of Daily Use.  (1929);  Aris & Phillips, London??, 1974.  P. 55 & plate XLVII.  ??NYS _ described with plate reproduced in Bell, below.  Shows and describes a  3 x 14  board from Memphis, ancient Egypt, but with no date given, but Bell indicates that the context implies it is probably earlier than ‑1500.  Petrie calls it 'The game of forty-two and pool' because of the 42 holes and a large hole on the side, apparently for storing pieces either during play or between games.

R. C. Bell.  Games to Play.  Michael Joseph (Penguin), 1988.  Chap. 4, pp. 54-61, Mancala games.  On pp. 54-55, he shows the ancient Egyptian board from Petrie and his own photo of a  3 x 6  board cut into the roof of a temple at Deir-el-Medina, probably about ‑87.

Thomas Hyde.  Historia Nerdiludii, hoc est dicere, Trunculorum; ....  (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.)  From the Sheldonian Theatre (i.e. OUP), Oxford, 1694.  De Ludo Mancala, pp. 226-232.  Have X of part of this.

Vernon A. Eagle.  On some newly described mancala games from Yunnan province, China, and the definition of a genus in the family of mancala games.  IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 48-62.  Discusses the game in general, with many references.  Attempts a classification in general.  Describes six forms found in Yunnan.


          4.B.14.         DOMINOES, ETC.


R. C. Bell.  Games to Play.  1988.  Op. cit. in 4.B.13.  P. 136 gives some history.  The Académie Français adopted the word for both the pieces and the game in 1790 and it was generally thought that they were an 18C invention.  However, a domino was found on the Mary Rose, which sank in 1545, and a record of Henry VIII (reigned 1509-1547) losing £450 at dominoes has been found. 

Bell, p. 131, describes the modern variant Tri-Ominos which are triangular pieces with values at the corners.  They were marketed c1970 and marked  © Pressman Toy Corporation, NY.

Hexadoms are hexagonal pieces with numbers on the edges _ opposite edges have the same numbers.  These were also marketed in the early 1970s _ I have a set made by Louis Marx, Swansea, but there is no date on it.