Couple of thoughts: 1. The squares and the polygons have equal areas, or 2. The squares and the polygons have equal anglular areas based on the center of the polyhedron, or 3. The centers of the squares and the polygons are equal distance from the center of the polyheadron, or 4. The squares and the polygons are set a distance from the center such that, when they are are rest, the will have equal potential energies. I like the last one best. This means there is no 'hole' to make it more likely that a square or a polygon would trap the die. I am also concerned (and get stuck at) modeling how this thing rolls. Things like spin and surface friction are important. I suspect, however, that it would end up rolling with the edges aligned. I.e., pick an edge of a square. Its final rotation would be in a direction perpendicular to that edge. This type of rotation would require the least kinetic energy to maintain. I note that this type of rotation would pass over 2 squares and 4 polygons so the squares might need to be slightly bigger to catch an appropriate share of the die. ..................... I'd bet that the answer would lie some place close to the four thoughts mentioned above [which themselves should be fairly close together]. I think it would come down to a trial and error engineering problem. >>>>>>>>>>> I have thought about this some more. I am pretty sure the die would end up rolling the way I described. Square- polygon -polygon- Square- polygon- polygon. I then note that the angles for the Square polygon is 62 degress and the plygon-polygon is 54 degrees. I'd further assume that the moment of interia is as if a ball and a constant resitive force. I would then look at the energy necessary to get over each angle and size everything so a square lands 3 times for every 4 polygons. *Note, it won't land on the polygon's evenly, there will be a tendency to land on the second polygon Then it is off to make a grant proposal for funding because there are simply too many variable and unknowns. >>>>>>>>>> Just buy an 8-sided die and blacken one side. That should work as a 7-side die. >>>>>>>>>> Duke's method inspired me. Just take a sphere and grind down 14 points equally (to form circles), with each circle centered where the square or the polygon would be. Leave a little room between the circles. It would be close to being perfectly fair. It would also look good.