Sources page biographical material

4.B.6. PHUTBALL
Winning Ways. 1982. Philosopher's football, pp. 688 691. In 1985, Guy said this was the only published description of the game.
4.B.7. BRIDG IT
This is best viewed as played on a n x n array of squares. The n(n+1) vertical edges belong to one player, say red, while the n(n+1) horizontal edges belong to black. Players alternate marking a square with a line of their colour between edges of their colour. A square cannot be marked twice. The object is to complete a path across the board. In practice, the edges are replaced by coloured dots which are joined by lines. As with Hex, there can be no ties and there must be a first person strategy.
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.

In Oct 2000, I bought a second-hand copy of a 5 x 5 version called Connections, attributed to Tom McNamara, but with no date.
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.

Deepak Shimkhada. A preliminary study of the game of Karma in India, Nepal, and Tibet. Artibus Asiae 44 (1983) 4. ??NYS - cited in Belloli et al, p. 68.

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 Gyān caupad [the d should have an underdot] or Gyān 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.
N. W. Bazely & P. J. Davis. Accuracy of Monte Carlo methods in computing finite Markov chains. J. of Res. of the Nat. Bureau of Standards -- Mathematics and Mathematical Physics 64B:4 (Oct-Dec 1960) 211-215. ??NYS -- cited by Davis & Chinn and Bewersdorff. Bewersdorff [email of 6 Jun 1999] brought these items to my attention and says it is an analysis based on absorbing Markov chains.

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.

Philip J. Davis & William G. Chinn. 3.1416 and All That. S&S, 1969, ??NYS; 2nd ed, Birkhäuser, 1985, chap. 23 (by Davis): "Mr. Milton, Mr. Bradley -- meet Andrey Andreyevich Markov", pp. 164-171. Simply describes how to set up the Markov chain transition matrix for a game with 100 cells and ending B. Doesn't give any results.

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; not in Carroll-Wakeling, Carroll-Wakeling II or Carroll-Gardner. 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.

Jörg Bewersdorff. Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse und Grenzen. Vieweg, 1998. Das Leiterspiel, pp. 67-68 & Das Leiterspiel als Markow Kette. Discusses setting up the Markov chain, citing Bazley & Davis, with the same board as in Davis & Chinn, then states that the average duration is 39.224 moves.

Jay Belloli, ed. The Universe A Convergence of Art, Music, and Science. [Catalogue for a group of exhibitions and concerts in Pasadena and San Marino, Sep 2000 - Jun 2001.] Armory Center for the Arts, Pasadena, 2001. P. 68 has a discussion of the Jain versions of the game, called 'gyanbazi', with a colour plate of a 19C example with a 9 x 9 board with three extra cells.
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 devised by Marco Meirovitz in 1973 -- see: Reddi. One of these was the English Bulls and Cows, but I haven't seen anything written on this and it doesn't appear in Bell, Falkener or Gomme. Since 1975 there have been several books on the game and a number of papers on optimal strategies. I include a few of the latter.

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

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

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.

Robert W. Irving. Towards an optimum Mastermind strategy, JRM 11:2 (1978-79) 81-87. Knuth's algorithm takes an average of 5804/1296 = 4.478 guesses. The author presents a better strategy that takes an average of 5662/1296 = 4.369 guesses, but requires six guesses in one case. A simple adaptation eliminates this, but increases the average number of guesses to 5664/1296 = 4.370. An intelligent setter will choose a pattern with a single repetition, for which the average number of guesses is 3151/720 = 4.376.

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

Merrill M. Flood. Mastermind strategy. JRM 18:3 (1985-86) 194-202. Cites five earlier papers on strategy, including Knuth and Irving. He considers it as a two-person game and considers the setter's strategy. He has several further papers in JRM developing his ideas.

Antonio M. Lopez, Jr. A PROLOG Mastermind program. JRM 23:2 (1991) 81-93. Cites Knuth, Irving, Flood and two 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.

Jörg Bewersdorff. Glück, Logik und Bluff Mathematik im Spiel -- Methoden, Ergebnisse und Grenzen. Vieweg, 1998. Section 2.15 Mastermind: Auf Nummer sicher, pp. 227-234 & Section 3.13 Mastermind: Farbcodes und Minimax, pp. 316-319. Surveys the work on finding optimal strategies. Then studies Mastermind as a two-person game. Finds the minimax strategy for the 3² game and describes Flood's approach.
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 -- these 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.

R. H. Macmillan. Wari. Eureka 13 (Oct 1950) 12. 2 x 6 board with each cup having four to start. Says it is played on the Gold Coast.

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.

Ulrich Schädler. Mancala in Roman Asia Minor? Board Games Studies International Journal for the Study of Board Games 1 (1998) 10-25. Notes that mancala could have been played on a flat board of two parallel rows of squares, i.e. something like a 2 x n chessboard, but that archaeologists have tended to view such patterns as boards for race games, etc. Describes 52 examples from Asia Minor. Some general discussion of Greek and Roman games.

John Romein & Henri E. Bal (Vrije Universiteit, Amsterdam). New computer cluster solves 3500-year old game. Posted on www.alphagalileo.org on 29 Aug 2002. They show that Awari is a tie game. They determined all 889,063,398,406 possible positions and stored them in a 778 GByte database. They then used a 144 processor cluster to analyse the graph, which 'only' took 51 hours.
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.
4.B.15. SVOYI KOSIRI
Anonymous [R. S. & J. M. B[rew ?]]. Svoyi kosiri is an easy game. Eureka 16 (Oct 1953) 8 12. This is an intriguing game of pure strategy commonly played in Russia and introduced to Cambridge by Besicovitch. It translates roughly as 'One's own trumps'. There are two players and the hands are exposed, with one's spades and clubs being the same as the other's hearts and diamonds. At Cambridge, the cards below 6 are removed, leaving 36 cards in the deck. The article doesn't explain how trumps are chosen, but if one has spades as trumps, then the other has hearts as trumps! Players alternate playing to a central discard pile. A player can take the pile and start a new pile with any card, or he can 'cover' the top card and then play any card on that. 'Covering' is done by playing a higher card of the same suit or one of the player's own trumps -- if this cannot be done, e.g. if the ace of the player's own trumps has been played, the player has to take the pile. The object is to get rid of all one's cards.
5. COMBINATORIAL RECREATIONS
7.AZ is actually combinatorial rather than arithmetical and I may shift it.
5.A. THE 15 PUZZLE, ETC.
Pictorial versions: The Premier (1880), Lemon (1890), Stein (1898), King (1927).

Double-sided versions: The Premier (1880), Brown (1891).

Relation to Magic Squares: Loyd (1896), Cremer (1880), Tissandier (1880 & 1880?), Cassell's (1881), Hutchison (1891).

Making a magic square with the Fifteen Puzzle: Dudeney (1898), Anon & Dudeney (1899), Loyd (1914), Dudeney (1917), Gordon (1988). See also:  Ollerenshaw & Bondi in 7.N.

GENERAL
Peter Hajek. 1995 report of his 1992 visit to the Museum of Money, Montevideo, Uruguay, with later pictures by Jaime Poniachik. In this Museum is a metal chest made in England in 1870 for the National State Bank of Uruguay. The front has a 7 x 7 array of metal squares with bolt heads. These have to be slid in a 12 move sequence to reveal the three keyholes for opening the chest. This opens up a whole new possible background for the 15 Puzzle -- can anyone provide details of other such sliding devices?
S&B, pp. 126 129, shows several versions of the puzzle.

L. Edward Hordern. Sliding Piece Puzzles. OUP, 1986. Chap. 2: History of the sliding block puzzle, pp. 18 30. This is the most extensive survey of the history. He concludes that Loyd did not invent the general puzzle where the 15 pieces are placed at random, which became popular in 1879(?). Loyd may have invented the 14 15 version or he may have offered the $1000 prize for it, but there is no evidence of when (1881??) or where. However, see the entries for Loyd's Tit Bits article and Dudeney's 1904 article which seem to add weight to Loyd's claims. Most of the puzzles considered here are described by Hordern and have code numbers beginning with a letter, e.g. E23, which I will give.

I contributed a note about computer techniques of solving such puzzles and hoping that programmers would attack them as computer power increased.

In 1993-1995, I produced four Sliding Block Puzzle Circulars, totalling 24 pages (since reformatted to 21), largely devoted to reporting on computer solutions of puzzles in Hordern. Since then, a large number of solution programs have appeared and many more puzzles have appeared. The best place to look is on Nick Baxter's Sliding Block Home Page: http://www.johnrausch.com/slidingblockpuzzles/index.html .

EARLY ALPHABETIC VERSIONS
Embossing Co. Puzzle labelled "No. 2 Patent Embossed puzzle of Fifteen and Magic Sixteen. Manufactured by the Embossing Co. Patented Oct 24 1865". Illustrated in S&B, p. 127. Examples are in the collections of Slocum and Hordern. Hordern, p. 25, says that searching has not turned up such a patent.

Edward F. [but drawing gives E.] Gilbert. US Patent 91,737 -- Alphabetical Instruction Puzzle. Patented 22 Jun 1869. 1p + 1p diagrams. Described by Hordern, p. 26. This is not really a puzzle -- it has the sliding block concept, but along several tracks and with many blank spaces. I recall a similar toy from c1950.

Ernest U. Kinsey. US Patent 207,124 -- Puzzle-Blocks. Applied: 22 Nov 1877; patented: 20 Aug 1878. 2pp + 1p diagrams. Described by Hordern, p. 27. 6 x 6 square sliding block puzzle with one vacant space and tongue & grooving to prevent falling out. Has letters to spell words. He suggests use of triangular and diamond shaped pieces. This seems to be the most likely origin of the Fifteen Puzzle craze.

Montgomery Ward & Co. Catalogue. 1889. Reproduced in: Joseph J. Schroeder, Jr.; The Wonderful World of Toys, Games & Dolls 1860··1930; DBI Books, Northfield, Illinois, 1977?, p. 34. Spelling Boards. Like Gilbert's idea, but a more compact layout.

LOYD
Loyd prize puzzle: One hundred pounds. Tit-Bits (14 Oct 1893) 25 & (18 Nov 1893) 111. Loyd is described as "author of "Fifteen Puzzle," ...."

Loyd. Tit Bits 31 (24 Oct 1896) 57. Loyd asserts he developed the 15 puzzle from a 4 x 4 magic square. "[The fifteen block puzzle] had such a phenomenal run some twenty years ago. ... There was one of the periodical revivals of the ancient Hindu "magic square" problem, and it occurred to me to utilize a set of movable blocks, numbered consecutively from 1 to 16, the conditions being to remove one of them and slide the others around until a magic square was formed. The "Fifteen Block Puzzle" was at once developed and became a craze.

I give it as originally promulgated in 1872 ..." and he shows it with the 15 and 14 interchanged. "The puzzle was never patented" so someone used round blocks instead of square ones. He says he would solve such puzzles by turning over the 6 and the 9. "Sphinx" [= Dudeney] says he well remembers the sensation and hopes "Mr. Loyd is duly penitent."

Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. "... the "Fifteen Puzzle" that in 1872 and 1873 was sold by millions, .... When this puzzle was brought out by its inventor, Mr. Sam Loyd, ... he thought so little of it that he did not even take any steps to protect his idea, and never derived a penny profit from it.... We have recently tried all over the metropolis to obtain a single example of the puzzle, without success." Dudeney says the puzzle came with 16 pieces and you removed the 16. He also says he recently could not find a single example in London.

Loyd. The 14 15 puzzle in puzzleland. Cyclopedia, 1914, pp. 235 & 371 (= MPSL1, prob. 21, pp. 19 20 & 128). He says he introduced it 'in the early seventies'. One problem asks to move from the wrong position to a magic square with sum = 30 (i.e. the blank is counted as 0). This is c= SLAHP, pp. 17 18 & 89.

G. G. Bain. Op. cit. in 1, 1907. Story of Loyd being unable to patent it.

Anonymous & Sam Loyd. Loyd's puzzles, op. cit. in 1, 1896. Loyd "owns up to the great sin of having invented the "15 block puzzle"", but doesn't refer to the patent story or the date.

W. P. Eaton. Loc. cit. in 1, 1911. Loyd refers to it as the 'Fifteen block' puzzle, but doesn't say he couldn't patent it.

Loyd Jr. SLAHP. 1928. Pp. 1 3 & 87. "It was in the early 80's, ... that the world disturbing "14 15 Puzzle" flashed across the horizon, and the Loyds were among its earliest victims." He gives many of the stories in the Cyclopedia and two of the same problems. He doesn't mention the patent story.
THE 15 PUZZLE
W. W. Johnson. Notes on the 15 Puzzle -- I. Amer. J. Math. 2 (1879) 397 399.

W. E. Story. Notes on the 15 Puzzle -- II. Ibid., 399 404.

J. J. Sylvester. Editorial comment. Ibid., 404.

(This issue may have been delayed to early 1880?? Johnson & Story are not terribly readable, but Sylvester is interesting, asserting that this is the first time that the parity of a permutation has become a popular concept.)

Anonymous. Untitled editorial. New York Times (23 Feb 1880) 4. "... just now the chief amusement of the New York mind, ... a mental epidemic .... In a month from now, the whole population of North America will be at it, and when the 15 puzzle crosses the seas, it is sure to become an English mania."

Anonymous. EUREKA! The Popular but Perplexing Problem Solved at Last. "THIRTEEN -- FOURTEEN -- FIFTEEN" New York Herald (28 Feb 1880) 8. ""Fifteen" is a puzzle of seeming simplicity, but is constructed with diabolical cunning. At first sight the victim feels little or no interest; but if he stops for a single moment to try it, or to look at any one else who is trying it, the mania strikes him. ... As to the last two numbers, it depends entirely upon the way in which the blocks happen to fall in the first place .... Two or three enterprising gamblers took up the puzzle and for a time made an excellent living.... The subject was brought up in the Academy of Sciences by the veteran scientist Dr. P. H. Vander Weyde", who showed it could not be solved. The Herald reporter discovered that the problem is solvable if one turns the board 90^o, i.e. runs the numbers down instead of across, and Vander Weyde was impressed. The article implies the puzzle had already been widely known for some time.

Mary T. Foote. US Patent 227,159 -- Game apparatus. Filed: 4 Mar 1880; patented: 4 May 1880. 1p + 1p diagrams. The patent is for a box with sliding numbered blocks for teaching the multiplication tables. Lines 57-63: "I am aware that it is not novel to produce a game apparatus in which blocks are to be mixed and then replaced by a series of moves; also, that it is not novel to number such blocks, as in the "game of 15," so called, where the fifteen numbers are first mixed and then moved into place."

Persifor Frazer Jr. Three methods and forty eight solutions of the Fifteen Problem. Proc. Amer. Philos. Soc. 18 (1878 1880) 505 510. Meeting of 5 Mar 1880. Rather cryptic presentation of some possible patterns. Asserts his 26 Feb article in the Bulletin (??NYS -- ??where -- Philadelphia??) was the first "solution for the 13, 15, 14 case".

J. A. Wales. 15 - 14 - 13 -- The Great Presidential Puzzle. Puck 7 (No. 158) (17 Mar 1880) back cover.

Anonymous. Editorial: "Fifteen". New York Times (22 Mar 1880) 4. "No pestilence has ever visited this or any other country which has spread with the awful celerity of what is popularly called the "Fifteen Puzzle." It is only a few months ago that it made its appearance in Boston, and it has now spread over the entire country." Asserts that an unregenerate Southern sympathiser has introduced it into the White House and thereby disrupted a meeting of President Hayes' cabinet.

Sch. [H. Schubert]. The Boss Puzzle. Hamburgischer Correspondent (= Staats- und Gelehrte Zeitung des Hamburgischen unpartheyeischen Correspondent) No. 82 (6 Apr 1880) 11, with response on 87 (11 Apr 1880) 12 (Sprechsaal). Gives a fairly careful description of odd and even permutations and shows the puzzle is solvable if and only if it is in an even permutation. The response is signed X and says that when the problem is insoluble, just turn the box by 90^o to see another side of the problem!

Gebr. Spiro, Hofliefer (Court supplier), Jungfernsteig 3(?--hard to read), Hamburg. Hamburgischer Correspondent (= Staats- und Gelehrte Zeitung des Hamburgischen unpartheyeischen Correspondent) No. 88 (13 Apr 1880) 7. Advertises Boss Puzzles: "Kaiser-Spiel 50Pf. Bismarck-Spiel 50 Pf. Spiel der 15 u. 16, 50 Pf. Spiel der 16 separat, 15 Pf. System und Lösung, 20 Pf."

G. W. Warren. Letter: Clew to the Fifteen Puzzle. The Nation 30 (No. 774) (29 Apr 1880) 326.

Anon. Shavings. The London Figaro (1 May 1880) 12. "The "15 Puzzle," which has for some months past been making a sensation in New York equal to that aroused by "H. M. S. Pinafore" last year, has at length reached this country, and bids fair to become the rage here also." (Complete item!)

George Augustus Sala. Echoes of the Week. Illustrated London News 76 (No. 2138) (22 May 1880) 491.

Mary T. Foote. US Patent 227,159 -- Game Apparatus. Applied: 4 Mar 1880; patented: 4 May 1880. 1p + 1p diagrams. Described in Hordern, p. 27. 3 x 12 puzzles based on multiplication tables. Refers to the "game of 15" and Kinsey.

Arthur Black. ?? Brighton Herald (22 May 1880). ??NYS -- mentioned by Black in a letter to Knowledge 1 (2 Dec 1881) 100.

Anonymous. Our latest gift to England. From the London Figaro. New York Times (11 Jun 1880) 2(?). ??page

The Premier. First (?) double sided version, with pictures of Gladstone and Beaconsfield, apparently produced for the 1880 UK election. Described in Hordern, pp. 32 33 & plate I.

Ahrens. MUS II 227. 1918. Story of Reichstag being distracted in 1880.

P. G. Tait. Note on the Theory of the "15 Puzzle". Proc. Roy. Soc. Edin. 10 (1880) 664 665. Brief but valid analysis. Mentions Johnson & Story. First mention of the possibility of a 3D version.

T. P. Kirkman. Question 6489 and Note on the solution of the 15 puzzle in question 6489. Mathematical Questions with their Solutions from the Educational Times 34 (1880) 113 114 & 35 (1881) 29 30. The question considers the n x n problem. The note is rather cryptic. (No use??)

Messrs. Cremer (210 Regent St. and 27 New Bond St., London). Brilliant Melancholia. Albrecht Durer's Game of the Thirty Four and "Boss" Game of the Fifteen. 1880. Small booklet, 16pp + covers, apparently instructions to fit in a box with pieces numbered 1 to 16 to be used for making magic squares as well as for the 15 puzzle. Explains that only half the positions of the 15 puzzle are obtainable and describes them by examples. (Photo in The Hordern Collection of Hoffmann Puzzles, p. 74, and in Hordern, op. cit. above, plate IV.) Possibly written by "Cavendish" (Henry Jones).

H. Schubert. Theoretische Entscheidung über das Boss Puzzle Spiel. 2nd ed., Hamburg, 1880. ??NYS (MUS, II, p. 227)

Gaston Tissandier. Les carrés magiques -- à propos du "Taquin," jeu mathématique. La Nature 8 (No. 371) (10 Jul 1880) 81 82. Simple description of the puzzle called 'Taquin' which came from America and has had a very great success for several weeks. Says it had 16 squares and was usable as a sliding piece puzzle or a magic square puzzle. Cites Frénicle's 880 magic squares of order 4.

Anon. & C. Henry. Gaz. Anecdotique Littéraire, Artistique et Bibliographique. (Pub. by G. d'Heylli, Paris) Year 5, t. II, 1880, pp. 58 59 & 87 92. ??NYS

Piarron de Mondésir. Le dernier mot du taquin. La Nature 8 (No. 382) (25 Sep 1880) 284 285. Simple description of parity decision for the 15 puzzle. Says 'la Presse illustrée' offered 500 francs for achieving the standard pattern from a random pattern, but it was impossible, or rather it was possible in only half the cases.

Jasper W. Snowdon. The "Fifteen" Puzzle. Leisure Hour 29 (1880) 493 495.

Gwen White. Antique Toys. Batsford, London, 1971; reprinted by Chancellor Press, London, nd [1982?]. On p. 118, she says: "The French game of Taquin was played in 1880, in which 15 pieces had to be moved into 16 compartments in as few moves as possible; the word 'taquin' means 'a teaser'." She gives no references.

Tissandier. Récréations Scientifiques. 1880?

2nd ed., 1881 -- unlabelled section, pp. 143-153. As: Le taquin et les carrés magiques; seen in 1883 ed., ??NX; 1888: pp. 208-215. Adapted from the 1880 La Nature articles of Tissandier and de Mondésir. 1881 says it came from America -- 'récemment une nouvelle apparition', but this is dropped in 1888 -- otherwise the two versions are the same.

Translated in Popular Scientific Recreations, nd [c1890], pp. 731 735. Text says "Mathematical games, ..., have recently obtained a new addition .... ... from America, ...." The references to contemporary reactions are deleted and the translation is confused. E.g. the newspaper is now just "a French paper" and the English says the problem is impossible in nine cases out of ten!

Lucas. Récréations scientifiques sur l'arithmétique et sur la géométrie de situation. Sixième récréation: Sur le jeu du taquin ou du casse tête américain. Revue scientifique de France et de l'étranger (3) 27 (1881) 783 788. c= Le jeu du taquin, RM1, 1882, pp. 189 211. Revue says that Sylvester told him that it was invented 18 months ago by an American deaf mute. RM1 says "vers la fin de 1878". Cf Schubert, 1895.

Cassell's. 1881. Pp. 96 97: American puzzles "15" and "34". = Manson, pp. 246-248. Says "articles ... have appeared in many periodicals, but no one has ... publish[ed] a solution." Then sketches the parity concept and its application.

Richard A. Proctor. The fifteen puzzle. Gentlemen's Magazine 250 (No. 1801) (1881) 30 45.

"Boss". Letter: The fifteen puzzle. Knowledge 1 (11 Nov 1881) 37-38, item 13. This magazine was edited by Proctor. The letter starts: "I am told that in a magazine article which appeared some time ago, you have attempted to show that there are positions in the Fifteen Puzzle from which the won position can never be obtained." I suspect the letter was produced by Proctor. The response is signed Ed. and begins: "I thought the Fifteen Puzzle was dead, and hoped I had had some share in killing the time-absorbing monster." Notes that many people get to the position starting blank, 1, 2, 3 and view this as a win. Sketches parity argument and suggests "Boss" work on the 3 x 3 or 3 x 2 or even the 2 x 2 version.

Editorial comment. The fifteen puzzle. Knowledge 1 (25 Nov 1881) 79. "I supposed every one knew the Fifteen Puzzle." Proceeds to explain, obviously in response to readers who didn't know it.

Arthur Black. Letter: The fifteen puzzle. Knowledge 1 (2 Dec 1881) 100, item 80. Sketches a proof which he says he published in the Brighton Herald of 22 May 1880.

"Yawnups". Letter: The fifteen puzzle. Knowledge 1 (30 Dec 1881) 185. Solution from the 15-14 position obtained by turning the box. Editorial comment says the solution uses 102 moves and the editor gets an easy solution in 57 moves. Adds that a 60 move solution has been received.

Arthur Black. Letter: The fifteen puzzle. Knowledge 1 (13 Jan 1882) 230. Finds a solution from the 15-14 position in 39 moves by turning the box and asserts no shorter solution is possible. Says he also gave this in the Brighton Herald in May 1880. An addition says J. Watson has provided a similar solution, which takes 38 moves??

A. B. Letter: The fifteen puzzle. Knowledge 2 (20 Oct 1882) 345, item 598. Finds a box-turning solution in 39 moves.

C. J. Malmsten. Göteborg Handl 1882, p. 75. ??NYS -- cited by Ahrens in his Encyklopadie article, op. cit. in 3.B, 1904.

Anonymous. Enquire Within upon Everything. Houlston and Sons, London. This was a popular book with editions almost every year -- I don't know when the following material was added. Section 2591: Boss; or the Fifteen Puzzle, p. 363. Place the pieces 'indifferently' in the box. Half the positions are unsolvable. Cites Cavendish for the solution by turning the box 90^o but notes this only works with round pieces. Goes on to The thirty-four puzzle, citing Dürer. I found this material in the 66th ed., 862nd thousand, of 1883, but I didn't find the material in the 86th ed of 1892.

Letters received and short answers. Knowledge 4 (16 Nov 1883) 310. 'Impossible'.

P. G. Tait. Listing's Topologie. Philosophical Mag. (5) 17 (No. 103) (Jan 1884) 30 46 & plate opp. p. 80. Section 11, p. 39. Simple but cryptic solution.

Letters received and short answers. Letter from W. S. B. asks how to solve the problem when the last row has 13, 14, 15 [sic!]; Answer by Ed. points out the misprint and says the easiest solution is to remove the 15 and put it after the 14, or to invert the 6 and 9. Knowledge 6 (No. 159) (14 Nov 1884) 412 & 6 (No. 160) (21 Nov 1884) 429.

Don Lemon. Everybody's Pocket Cyclopedia .... Saxon & Co., London, (1888), revised 8th ed., 1890. P. 137: The fifteen puzzle. Brief description, with pieces placed randomly in the box -- "to get the last three into order is often a puzzle indeed".

John D. Champlin & Arthur E. Bostwick. The Young Folk's Cyclopedia of Games and Sports. 1890. ??NYS Cited in Rohrbough; Brain Resters and Testers; c1935; Fifteen Puzzle, p. 20. Describes idea of parity of number of exchanges. [Another reference provided more details of Champlin & Bostwick.]

Lemon. 1890. A trick puzzle, no. 202, pp. 31 & 105 (= Sphinx, no. 422, pp. 60 & 112). 15 puzzle with lines on the pieces to arrange as "a representation of a president with only one eye". The solution is a spelling of the word 'president'. Attributed to Golden Days -- ??. After The Premier puzzle of c1880, this is the second suggestion of using a picture and the first publication of the idea that I have seen.

G. A. Hutchison, ed. Indoor Games and Recreations. The Boy's Own Bookshelf. (1888); New ed., Religious Tract Society, London, 1891. (See M. Adams; Indoor Games for a much revised version, but which doesn't contain this material.) Chap. 19: The American Puzzles., pp. 240 241. "These puzzles, known as the 'Thirty four Game' and the 'Fifteen Game,' on their introduction amongst us some years ago ...." "The '15' puzzle would appear to have been, on its coming to England a few years ago, strictly a new introduction ...." He sketches the parity concept. [NOTE. I have seen a reference to the editor as Hutchinson, but the book definitely omits the first n.]

Daniel V. Brown. US Patent 471,941 -- Puzzle. Applied: 23 Apr 1891; patented: 29 Mar 1892, 2pp + 1p diagrams. Double-sided 16 block puzzle to spell George Washington on one side and Benjamin Harrison on the other. No sliding involved.

Berkeley & Rowland. Card Tricks and Puzzles. 1892. American fifteen puzzle, pp. 105-107. "The Fifteen Puzzle was introduced by a shrewd American some ten years ago, ...." Refers to Tait's 1880 paper. Says half the positions are impossible, but solves them by turning the box 90^o or by inverting the 6 and the 9.

Hoffmann. 1893. Chap IV, no. 69: The "Fifteen" or "Boss" puzzles, pp. 161 162 & 217 218 = Hoffmann-Hordern, pp. 142-144, with photo of five early examples, two or three of which also are thirty-four puzzles. (Hordern Collection, p. 74, has a photo of a version by Cremer, cf above.) "This, like a good many of the best puzzles, hails from America, where, some years ago, it had an extraordinary vogue, which a little later spread to this country, the British public growing nearly as excited over the mystic "Fifteen" as they did at a later date over the less innocent "Missing Word" competitions." He distinguishes between the ordinary Fifteen where one puts the pieces in at random, and the Boss or Master puzzle which has the 14 and 15 reversed. "Notwithstanding the enormous amount of energy that has been expended over the "Fifteen" Puzzle, no absolute rule for its solution has yet been discovered and it appears to be now generally agreed by mathematicians that out of the vast number of haphazard positions ... about half admit [of solution]. To test whether ... the following rule has been suggested." He then says to count the parity of the number of transpositions.

Hoffmann. 1893. Chap. IV, no. 70: The peg away puzzle, pp. 163 & 218 = Hoffmann Hordern, p. 145. This is a 3 x 3 version of the Fifteen puzzle, made by Perry & Co. Start with a random pattern and get to standard form. "The possibility of success in solving this puzzle appears to be governed by precisely the same rule as the "Fifteen" Puzzle." Hoffmann-Hordern has no photo of this -- do any examples exist??

H. Schubert. Zwölf Geduldspiele. Dümmler, Berlin, 1895. [Taken from his columns in Naturwissenschaftlichen Wochenschrift, 1891-1894.] Chap. VII: Boss-Puzzle oder Fünfzehner-Spiel, pp. 75-94?? Pp. 75-77 sketches the history, saying it was called "Jeu du Taquin" (Neck-Spiel) in France and was popular in 1879-1880 in Germany. Cites Johnson & Story and his own 1880 booklet. Gives the story of a deaf and dumb American inventing it in Dec 1878, saying "Sylvester communicated this at the annual meeting of the Association Française pour l'Avancement des Sciences at Reims". Cf Lucas, 1881. [There is a second edition, Teubner??, Leipzig, 1899, ??NYS. However this material is almost identical to the beginning of Chap. 15 in Schubert's Mathematische Mussestunden, 3rd ed., Göschen, Leipzig, 1909, vol. 2. The later version omits only some of the Hamburg details of 1879-1880. Hence the 2nd ed. of Zwölf Geduldspiele is probably very close to these versions.]

Dudeney. Problem 49: The Victoria Cross puzzle. Tit Bits 32 (4 & 25 Sep 1897) 421 & 475. = AM, 1917, prob. 218, pp. 60 & 194. B7. 3 x 3 board with letters Victoria going clockwise around the edges, leaving the middle empty, and starting with V in a corner. Slide to get Victoria starting at an edge cell, in the fewest moves. Does it in 18 moves, by interchanging the i's and says there are 6 such solutions.

Dudeney. Problem 65: The Spanish dungeon. Tit Bits 33 (1 Jan & 5 Feb 1898) 257 & 355. = AM, 1917, prob. 403, pp. 122-123 & 244. B14. Convert 15 Puzzle, with pieces in correct order, into a magic square. Does it in 37 moves.

Conrad F. Stein. US Design 29,649 -- Design for a Game-Board. Applied: 29 Sep 1898; patented: 8 Nov 1898 as No. 692,242. 1p + 1p diagrams. This appears to be a 3 x 4 puzzle with a picture of a city with a Spanish flag on a tower. Apparently the object is to move an American flag to the tower.

Anon. & Dudeney. A chat with the Puzzle King. The Captain 2 (Dec? 1899) 314-320; 2:6 (Mar 1900) 598-599 & 3:1 (Apr 1900) 89. The eight fat boys. 3 x 3 square with pieces: 1 2 3; 4 X 5; 6 7 8 to be shifted into a magic square. Two solutions in 19 moves. Cf Dudeney, 1917.

Addison Coe. US Patent 785,665 -- Puzzle or Game Apparatus. Applied: 17 Nov 1904; patented: 1 Mar 1905. 4pp + 3pp diagrams. Gives a 3 x 5 flat version and a 3 dimensional version -- cf 5.A.2.

Dudeney. AM. 1917.

Prob. 401: Eight jolly gaol birds, pp. 122 & 243. E23. Same as 'The eight fat boys' (see Anon. & Dudeney, 1899) with the additional condition that one person refuses to move, which occurs in one of the two previous solutions.

Prob. 403: The Spanish dungeon, pp. 122-123 & 244. = Tit-Bits prob. 65 (1898). B14.

Prob. 404: The Siberian dungeons, pp. 123 & 244. B16. 2 x 8 array with prisoners 1, 2, ..., 8 in top row and 9, 10, ..., 16 in bottom row. Two extra rows of 4 above the right hand end (i.e. above 5, 6, 7, 8) are empty. Slide the prisoners into a magic square. Gives a solution in 14 moves, due to G. Wotherspoon, which they feel is minimal. This allows long moves -- e.g. the first move moves 8 up two and left 3.

"H. E. Licks" [pseud. of Mansfield Merriman]. Recreations in Mathematics. Van Nostrand, NY, 1917. Art. 28, pp. 20 21. 'About the year 1880 ... invented in 1878 by a deaf and dumb man....'

[From sometime in the 1980s, I suspected the author's name was a pseudonym. On pp. 132-138, he discusses the Diaphote Hoax, from a Pennsylvania daily newspaper of 10 Feb 1880, which features the following people: H. E. Licks, M. E. Kannick, A. D. A. Biatic, L. M. Niscate. The diaphote was essentially a television. He says this report was picked up by the New York Times and the New York World. An email from Col. George L. Sicherman on 5 Jun 2000 agrees that the name is false and suggested that the author was "the eminent statistician Mansfield Merriman" who wrote the article on The Cattle Problem of Archimedes in Popular Science Monthly (Nov 1905), which is abridged on pp. 33-39 of the book, but omitting the author's name. Sichermann added that Merriman was one of the authors of Pillsbury's List. William Hartston says this was an extraordinary list of some 30 words which Pillsbury, who did memory feats, was able to commit to memory quite rapidly. Sicherman continued to investigate Merriman and got Prof. Andri Lange interested and Lange corresponded with a James A. McLennan, author of a history of the physics department at Lehigh University where Merriman had been. McLennan found Merriman's obituary from the American Society of Civil Engineers which states that Merriman used H. E. Licks as a pseudonym. [Email from Sicherman on 25 Feb 2002.]]

Stephen Leacock. Model Memoirs and Other Sketches from Simple to Serious. John Lane, The Bodley Head, 1939, p. 300. "But this puzzle stuff, as I say, is as old as human thought. As soon as mankind began to have brains they must have loved to exercise them for exercise' sake. The 'jig-saw' puzzles come from China where they had them four thousand years ago. So did the famous 'sixteen puzzle' (fifteen movable squares and one empty space) over which we racked our brains in the middle eighties."

G. Kowalewski. Boss Puzzle und verwandte Spiele. K. F. Kohler Verlag, Leipzig, 1921 (reprinted 1939). Gives solution of general polygonal versions, i.e. on a graph with a Hamilton circuit and one or more diagonals.

Hummerston. Fun, Mirth & Mystery. 1924.
1 2

9 Push, pp. 22 & 25. This is played on the board

3 10 11 4 shown at the left with its orthogonal lines, like

12 13 3, 10, 11, 4, and its diagonal lines, like

5 14 15 6 1, 9, 11, 13, 6. 10, 15 and 11, 14 are not

16 connected, so this is an octagram. Take 16

7 8 numbered counters and place them at random on

the board and remove counter 16. Move the pieces

to their correct locations. He asserts that 'unlike the original ["Sixteen" Puzzle],

no position can be set up in "Push" that cannot be solved'.

The six bulls puzzle, Puzzle no. 34, pp. 90 & 177. This uses the 2 x 3 + 1 0    

board shown at the right, where the 0 is the blank space. Exchange 1 2 3

3 and 6 and 4 and 5. He does it in 20 moves. [This is Hordern's 4 5 6

B3, first known from 1977 under the name Bull Pen, but is a variant of