I don't generally bother with rule 1; ordinary games (not gambling) don't fret about compensating for the weight of the pips, and my designs in general don't lead to *heavy* differences in weight among the sides. Of course, really extreme asymmetry is not acceptable either, but I don't sweat the small stuff. (That said, I also think it's

*really cool* that you do rebalance your dice like that, and it's an awesome feature to have in a die, to be able to say that the pips were resized in order to bring the center of balance back).

Rule 2 is probably a necessary condition for fairness, but not a sufficient one. It isn't just about area; you can have faces of very different shapes with the same area, and some might be long and narrow (and unlikely to be landed on) and some might be wider and more likely. The area should be the same, and so should the shape, and also the dihedral angles with other faces need to be in similar arrangement, etc. It's about how much energy it takes to roll from one face over to the next one.

Rule 3 isn't really necessary

*or* sufficient for fairness. A die whose opposite faces don't add up right isn't less fair than one that does. But it makes it nicer. Rule 4 is pretty obviously necessary, unless you're doing some strange numbering or really creative geometry (I dunno, somehow making sure that the probabilities of a particular pair of faces add up exactly to the probability of another face or something).

I made up a quick-and-dirty spreadsheet for doing chi-square tests on a die; maybe I should post it. Then with each die you can also say "... and with 100 rolls, this die scored x% on the chi-square test, indicating no reason to suspect non-fairness"