E201 -- Calcul de la probabilité dans le jeu de rencontre
(Calculation of the probability in the game of Rencontre)
The Game of Recontre (coincidence), also called the game of treize (thirteen), involves shuffling
13 numbered cards, then dealing them one at a time, counting aloud to 13. If the nth card is dealt when the player says the number 'n,' the dealer wins (this is known in combinatorics as a
derangement of 13 objects.). Euler calculates the probability that the dealer will win.
It should be noted that this problem was solved earlier, by P.R. de Montmort, in 1713, though his work was unknown to Euler.
Originally published in Memoires de l'academie des sciences de Berlin 7, 1753, pp. 255-270
Opera Omnia: Series 1, Volume 7, pp. 11 - 25
- Original publication: E201
- E201 can be viewed or downloaded from Digitalisierte Akademieschriften und Schriften zur Geschichte der Königlich Preußischen Akademie der Wissenschaften, which includes serial publications of the Prussian Academy of Science in the 18th and 19th Centuries.
- Richard Pulskamp has put together a fantastic page on Euler's Probability & Statistics. The page includes a translation of E201.
- Another translation, prepared independently by Geoff Burke, is available at Ed Sandifer's
Euler Project page.
- Euler published one other work on derangements, namely E738.
- Much has been written about derangements in general, and any introductory combinatorics book will address them. More more about Euler's work on derangements, along with information on even earlier work in this area (including a translation of de Montmort's paper), see the above link to Pulskamp's Probablility and Statistics page.
- E201 is discussed in Ed Sandifer's How Euler Did It September 2004 column published online by the MAA.
- The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E201 include:
- Takacs L., “On the probleme-des-menages.” Discrete Mathematics, 36 (3), pp. 289-297 (1981).
- Takacs L., “The problem of coincidences.” Archive for History of Exact Sciences, 21 (3), pp. 229-244 (1980).
- Walsh TRS, Wright EM., “K-connectedness of unlabeled graphs.” Journal of the London Mathematical Society - second series, 18 (Dec), pp. 397-402 (1978).
- Wright EM., “Arithmetical properties of Euler's recontre number.” Journal of the London Mathematical Society - second series, 4 (Apr), pp. 437-& (1972).
- Wright EM., “Graphs on unlabeled nodes with a large number of edges.” Proceedings of the London Mathematical Society, 28 (Jun), pp. 577-594 (1974).
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