  Jaclyn Anderson proved that if s and t are relatively prime positive
integers, then there are exactly (s+t-1)!/(s!t!) partitions whose set of
hook-lengths is disjoint from the set {s,t}. Drew Armstrong conjectured (and
Paul Johnson, and a bit later, Victor Wang, proved) a beautiful expression for
the average size, namely (s-1)(t-1)(s+t+1)/24 .
  In the present article, we go far beyond the average, and state absolutely
certain expressions (but "officially" still conjectures) for the variance
(showing in particular that it is rather large, and there is no "concentration
about the mean"), and the third through the sixth moments. For the special case
of (s,s+1)-core partitions, we go all the way to the 9th moment.
  We pose two challenges, and will be glad to donate 100 dollars each, to the
OEIS foundation in honor of the first provers, regarding a "soft" and "global",
yet rigorous, justification of our empirical approach, and for proving an
intriguing conjecture about the limiting distribution.
  This version reports (thanks to Marko Thiel and Nathan Williams) that the
second challenge mentioned above has been done by Paul Johnson (but not in two
pages). A donation to the OEIS was made. The first challenge is still wide
open.
