%dreidel.tex: ``Does Dreidel Last O(NUTS^2) SPINS?
%a Plain TeX file by Doron Zeilberger (1 page)
 
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\bf
\centerline
{
IS DREIDEL EXPECTED TO LAST $O(NUTS^2)$ SPINS?
}
\rm
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\centerline{ {\it 
Doron ZEILBERGER
}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Temple University,
Philadelphia, PA 19122, USA. 
%\break
{\eighttt zeilberg@math.temple.edu} \hfill \break
{\eighttt http://www.math.temple.edu/\Tilde zeilberg/   .}
30 Kislev, 5760 [Dec. 9, 1999].  Supported in part by the NSF.
Exclusive for the Personal Journal of Ekhad and Zeilberger
{\eighttt http://www.math.temple.edu/\Tilde zeilberg/pj.html}
} 
}
According to my Maple package {\tt DREIDEL}
(downloadable from my website), the 
expected number of spins, in a 2-player Dreidel game, that
starts with each player having $NUTS$ nuts, 
and terminates as soon as one of the players runs out of nuts,
equals,
for $NUTS=1, \dots, 25$:
{\bf
 
[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 $O(NUTS^2)$, or more precisely,
$a*(NUTS)^2+b*NUTS+c+ o(NUTS)$, where
$a=2.21814\dots$, $b=-5.9804585\dots$, and $c=6.601118\dots$.
I am offering {\it 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, $O(NUTS^k)$ spins to
complete a $k-player$ Dreidel game.
 
\bye
 
 
