A loaded die is biased by placing a dense insert low inside the body of the die, shifting the centre of gravity downward so that the weighted face is statistically more likely to land underneath. Increasing the thickness of the load initially lowers the centre of gravity and increases the bias, but beyond a certain depth the load’s own centroid rises enough that the overall centre of gravity begins to rise again. The formula below calculates the load thickness that produces the minimum possible centre of gravity for a cubic die, based on the die size, wall thickness, and the relative densities of the die material and the loading material. The accompanying diagram defines the geometry and variables used in the calculation and summarises the key physical principles behind the optimisation.
