In this paper, we consider dice having high symmetry and balance. Dice are often used to generate random numbers in board games and gambling. Even though we commonly use cube-shaped dice, there are dice of other regular polyhedra and nonregular polyhedra that have high symmetry. In this study, we consider highly symmetrical and well-balanced arrangement of dice based on a convex isohedron, under the condition that duplication of numbers and inclusion of unused faces are not allowed. It is well known that the sums of the numbers on the opposite sides of a cube die are all equal. Bosch et al. extended this constraint and proposed two new dice number symmetries, “numerically balanced vertices” and “numerically balanced faces,” in a paper published in 2017. The constraint of numerically balanced vertices means that the sum of the numbers on the faces gathered at any one vertex is almost equal (the difference is at most one), and the constraint of numerically balanced faces means that the sum of the numbers on the faces adjacent to any one face is almost equal. They called dice that have arrangements satisfying all of these three constraints, including the opposite number convention, “numerically balanced dice,” and showed that there exists an icosahedral numerically balanced die. In this study, we are trying to apply these constraints to all regular polyhedra, all Catalan’s polyhedra, and other twelve convex isohedra. As a result, first we found that any of these polyhedra never forms a die satisfying all of the three constraints. Next restricting the conditions two or one, we try to find dice satisfying the conditions, and clarified some facts.