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Jaclyn Anderson proved that if s and t are relatively prime positiveintegers, then there are exactly (s+t-1)!/(s!t!) partitions whose set ofhook-lengths is disjoint from the set {s,t}. Drew Armstrong conjectured (andPaul Johnson, and a bit later, Victor Wang, proved) a beautiful expression forthe average size, namely (s-1)(t-1)(s+t+1)/24 .  In the present article, we go far beyond the average, and state absolutelycertain expressions (but "officially" still conjectures) for the variance(showing in particular that it is rather large, and there is no "concentrationabout the mean"), and the third through the sixth moments. For the special caseof (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 theOEIS foundation in honor of the first provers, regarding a "soft" and "global",yet rigorous, justification of our empirical approach, and for proving anintriguing conjecture about the limiting distribution.  This version reports (thanks to Marko Thiel and Nathan Williams) that thesecond challenge mentioned above has been done by Paul Johnson (but not in twopages). A donation to the OEIS was made. The first challenge is still wideopen.

Explicit Expressions for the Variance and Higher Moments of the Size of a Simultaneous Core Partition and its Limiting Distribution