Sources page biographical material

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 The Philosophy of Analysis -- unpublished collection of MSS in the BM as Add.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 Notebooks -- unpublished collection of MSS in the BM as Add.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 pościgu (Definitions for a theory of games and pursuit). Myśl 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. Jörg Bewersdorff [email of 6 Jun 1999] observes that Lasker's analysis of his Nim got very close to the idea of the Sprague-Grundy number. See: Jörg Bewersdorff; Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse und Grenzen; Vieweg, 1998, Section 2.5 Lasker-Nim: Gewinn auf verborgenem Weg, pp. 118-124.

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, Sprague-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 read Murray's History of Board Games Other than Chess, but not yet entered the material. 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.

4.B.1. TIC TAC TOE = NOUGHTS AND CROSSES
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.

Misère version: Gardner (1957); Scotts (1975);
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, 1834, 1838, 1841, 1850, 1855, 1875, 1876, 1891. [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. BMC has 1801, 1810, 1830, 1833, 1834, 1838, 1841, 1875, 1876, 1898.]

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 terra cotta 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." Reprinted in 1969 and in the 1960s?

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, and this is an old Scottish term [James T. R. Ritchie; The Singing Street Scottish Children's Games, Rhymes and Sayings; (O&B, 1964); Mercat Press, Edinburgh, 2000, p. 61].) 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.

But cf Games and Sports for Young Boys, 1859, below.

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. [= 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 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. 4r 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, but no such machine was built until 1949. 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.

Games and Sports for Young Boys. Routledge, nd [1859 - BLC]. P. 70, under Rhymes and Calls: "In the game of Tit-tat-toe, which is played by very young boys with slate and pencil, this jingle is used:--

Tit, tat, toe, my first go:

Three jolly butcher boys all in a row;

Stick one up, stick one down.

Stick one on the old man's crown."

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.

The Play Room: or, In-door Games for Boys and Girls. Dick & Fitzgerald(?), 1866. [Reprinted as: How to Amuse an Evening Party. Dick & Fitzgerald, NY, 1869.] ??NX -- the 1869 was seen at Shortz's. P. 22: Tit-tat-to. Uses O and +. "This is a game that small boys enjoy, and some big ones who won't own it."

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. Looks at Ovid's game.

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."

G. E. Felton & R. H. Macmillan. Noughts and Crosses. Eureka 11 (Jan 1949) 5-9. Mentions Dudeney's work on the 3 x 3 board and asks for generalizations. Mentions pegotty = go-bang. Then studies the 4 x 4 x 4 game -- see 4.B.1.a. Adds some remarks on pegotty, citing Falkener, Lucas and Tarry.

Stanley Byard. Robots which play games. Penguin Science News 16 (Jun 1950) 65-77. On p. 73, he says D. W. Davies, of the National Physical Laboratory, 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.

P. C. Parks. Building a noughts and crosses machine. Eureka 14 (Oct 1951) 15-17. Cites Babbage, Rogers, Davies, Byard. Parks built a simple machine with wire and tin cans in 1950 at a cost of about £6. Says G. Eastell of Thetford, Norfolk, built a machine using sixty valves for the Festival of Britain.

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 misère version (the person who makes three in a row loses), three and n dimensional forms (giving L. 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,".

C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. A ticktacktoe machine, pp. 384-385. Noel Elliott gives a brief description of a relay logic machine to play the game.

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.

Barnard. 50 Observer Brain-Twisters. 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 is a draw and that if t  n + 1 - (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.

[Henry] Joseph & Lenore Scott. Quiz Bizz. Puzzles for Everyone -- Vol. 6. Ace Books (Charter Communications), NY, 1975. P. 143: Ha-ho-ha. Misère version of noughts and crosses proposed. No discussion.

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.