IS DREIDEL EXPECTED TO LAST O(NUT S^{2}) SPINS?

Doron ZEILBERGER ^{1}

According to my Maple package DREIDEL (downloadable from my website), the expected number of spins, in a 2-player Dreidel game, that starts with each player having NUT S nuts, and terminates as soon as one of the players runs out of nuts, equals, for NUT S = 1, …, 25:

[0, 2.400000000, 8.329934935, 18.11361402, 32.14410851, 50.57079312,

73.42689226, 100.7185523, 132.4464689, 168.6106885, 209.2111984,

254.2479934, 303.7210720, 357.6304337, 415.9760785, 478.7580063,

545.9762172, 617.6307111, 693.7214880, 774.2485480, 859.2118910,

948.6115170, 1042.447426, 1140.719618, 1243.428093] .

Hence it seems very likely that the expected duration of such a Dreidel game is O(NUT S^{2}), or more precisely, a∗(NUT S)^{2} +b∗NUT S +c+o(NUT S), where a = 2.21814 …, b = −5.9804585 …, and c = 6.601118 . . .. I am offering Hanukkah gelt in the amount of $25, for settling this conjecture. I am offering an additional $25 for a (positive) confirmation of my conjecture (for which I do not have any evidence) that it takes, on the average, O(NUT S^{k}) spins to complete a k −player Dreidel game.

1 Department of Mathematics, Temple University, Philadelphia, PA 19122, USA. zeilberg@math.temple.edu

http://www.math.temple.edu/~zeilberg/ . 30 Kislev, 5760 [Dec. 9, 1999]. Supported in part by the NSF. Exclusive for the Personal Journal of Ekhad and Zeilberger http://www.math.temple.edu/~zeilberg/pj.html