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I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically balanced die with more than 120 faces. Given your expertise in mathematics, I turn to you to seek clarification on this subject that I find fascinating.

I am already aware of certain elements related to the balance of a die, such as the requirement that the sum of opposite faces be equal and that the average of adjacent faces at the same vertex must be equal. I have also become acquainted with two examples of perfectly balanced dice, namely the 48-sided die (d48) based on a Disdyakis dodecahedron, and the 120-sided die (d120) based on a Disdyakis triacontahedron, both of which are considered Catalan solids.

However, the assertion that there would be no numerically balanced die beyond 120 faces seems to be related to the dual properties and symmetry of these polyhedra. I would like to deepen my understanding of this assertion and comprehend why it is impossible to create a numerically balanced die beyond this limit.

Thus, I would like to pose the following questions to you: A. Could you explain in detail why it is impossible to manufacture a numerically balanced die with more than 120 faces? B. In the event that achieving perfect balance is not possible, could you guide me on how to propose a balanced or semi-balanced numbering for a die? What mathematical considerations must be taken into account to achieve an equitable distribution of values?

I sincerely thank you in advance for taking the time to address my questions and share your knowledge on this subject. Your expertise would greatly contribute to my understanding of the geometry and mathematics associated with dice.

Best regards,

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    $\begingroup$ Do you have a source for that assertion? What do you mean by "numerically balanced" exactly? Can't you get a balanced die by generalizing the construction for the 10-sided die, that is, two equal pyramids built on a regular $n$-gon as their common base? $\endgroup$
    – Federico Poloni
    Commented Aug 15, 2023 at 10:06
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    $\begingroup$ @FedericoPoloni wired.com/2016/05/… the term is used here $\endgroup$
    – user479223
    Commented Aug 15, 2023 at 11:12
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    $\begingroup$ @user479223 Thanks! I can infer a definition, but I don't see any impossibility claims there. The article just says that they had to face physical difficulties building it, and that they brute-forced a solution to the problem for the 120-sided die. $\endgroup$
    – Federico Poloni
    Commented Aug 15, 2023 at 11:27
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    $\begingroup$ This question is formulated rather naïvely, but I think there’s a good mathematical question at the heart of it: Is there some formal sense of “numerically balanced die”, for which the d120 disdyakis triacontahedron is the largest possible such die? Sharpened up like this, it seems quite reasonably on-topic. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Aug 15, 2023 at 20:09

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