Sources page biographical material

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.