Hi Ed, your square is a candidate for the most interesting magic square of order 4. It additionally has the longest Anderson Graph (a closed path from entry 1 to 2 to 3 to ... to 16 to 1) of all classical magic 4x4 squares (together with the square Frenicle # 361.) The length of the graph is 43.942968 units. For general orders we can say that all magic and semi-magic squares are balanced. (Obvious) And all centrally symmetric magic squares remain balanced if their entries were squared. (Not difficult to prove) Question: Are there non-symmetric magic squares of orders greater than 4 with this property? In this context Peter Lolys work should be mentioned. He showed that the moment of inertia is equal for all classical magic squares of the same order. See: http://home.cc.umanitoba.ca/~loly/MathGaz.pdf And: https://oeis.org/A126275 Walter