Fix coprime s, t > 1. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the finitely many simultaneous (s, t)cores have average size 1 24(s−1)(t−1)(s+t+1), and that the subset of self-conjugate cores has the same average (first shown by Chen–Huang–Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse...