The properties we discuss are all actually conjectures based on extensive robust computer computation. We say they are 'magical' because in each case we have no idea why they should be true, mathematical or physical, hand-waving or rigorous. Three of the results involve the monomer-dimer problem, i.e. generalizations of the question: how many ways are there to cover a chess board with dominoes, with no overlaps, and each domino covering two squares. This problem is generalized to lattices in any dimension, and also graphs. The forth result concerns lattice dimer gases. For any of these we have that all terms in the virial series are positive! The research embodied is very recent, from just published to not yet on the web.