Sources page biographical material

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, called Plato?, of this in 1970.

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

Arthur Stone (letter to me of 9 Aug 1985) says '3 and 4 dimensional forms of tic-tac-toe produced by Brooks, Smith, Tutte and myself', but it's not quite clear if they invented these. Tutte became expert on the 4³ board and thought it was a first person game. They only played the 5⁴ game once - it took a long time.

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, with tactics and problems.

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 L. 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 L. 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 p_r(k) of r-sections of a kⁿ board by means of a recurrence p_r(k) = [p_r-1(k+2) - p_r-1(k)]/2r which generalises L. 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
David Fielker sent some pages from a Danish book on games, but the TP is not present in his copies, so we don't have details. This says that Hein introduced the game in a lecture to students at the Institute for Theoretical Physics (now the Niels Bohr Institute) in Copenhagen in 1942. After its appearance in Politiken, specially printed pads for playing the game were sold, and a game board was marketed in the US as Hex in 1952.

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.

Anatole Beck, Michael N. Bleicher & Donald W. Crowe. Excursions into Mathematics. Worth Publishers, NY, 1969. Chap. 5: Games (by Beck), Section 3: The game of Hex, pp. 327-339 (with photo of Hein on p. 328). Says it has been attributed to Hein and Nash. At Yale in 1952, they played on a 14 x 14 board. Shows it is a first player win, invoking the Jordan Curve Theorem

David Gale. The game of Hex and the Brouwer fixed-point theorem. AMM 86:10 (Dec 1979) 818-827. Shows that the non-existence of ties (Hex Theorem) is equivalent to the Brouwer Fixed-Point Theorem in two and in n dimensions. Says the use of the Jordan Curve Theorem is unnecessary.

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.

Cameron Browne. Hex Strategy: Making the Right Connections. A. K. Peters, Natick, Massachusetts, 2000.
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 élèves 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.

C. Ganse. The dot game. Ladies' Home Journal (Jun 1903) 41. Describes the game and states that one who makes a box gets to go again.

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.

Heinrich Voggenreiter. Deutsches Spielbuch Sechster Teil: Heimspiele. Ludwig Voggenreiter, Potsdam, 1930. Pp. 84-85: Die Käsekiste. Describes a version for two or more players. The first player must start at a corner and players must always connect to previously drawn lines. A player who completes 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.
4.B.5. OVID'S GAME AND NINE MEN'S MORRIS
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 follow 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 Ars 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, he shows a doubly crossed 3 x 3 board, but it 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 says no Roman name for this game has survived. He discusses various known artifacts for all the game, 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 [the H should have a dot under it], 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 Kitāb al Aghāni, 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 Men'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. [= Libro de Acedrex, Dados e Tablas of Alfonso El Sabio, generally known as the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete reproduction in 194 Phototropic Plates. 2 vols., Karl W. Horseman, 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 Castile (Castilla). 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 Kitāb al Aghāni, 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 XIII^e 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 Westmorland. 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 Men's 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 centre. They briefly describe Six and Nine Men's Morris.

Ralph Gasser & J. Nievergelt. Es ist entscheiden: Das Muehle-Spiel ist unentscheiden. Informatik Spektrum 17 (1994) 314-317. ??NYS -- cited by Jörg Bewersdorff [email of 6 Jun 1999].

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. Solving Nine Men's Morris. IN: Games of No Chance; ed. by Richard Nowakowski; CUP, 1996, pp. 101-113. ??NYS -- cited by Bewersdorff [loc. cit.] and 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 search 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. Bewersdorff [email of 11 Jun 1999] clarifies this by observing that draw here means a game that continues forever -- one cannot come to a stalemate where neither side can move.