For what follows, we define a loaded die as a discrete probability distribution with six outcomes (labelled {1,…,6}), each of which has positive probability. To each such die, we associate a generating polynomial, given by
in which denotes the probability of the outcome i. If
corresponds to another such die, we note that the product
has coefficients which reflect the probabilities of certain dice sums for p and q (and this is the utility of generating polynomials). We are now ready to ask the following question:
Question: Does there exist a pair of loaded dice such that the probability of rolling any dice sum ({2,…12}) is equally likely?
Proof: Equivalently, we question the existence of dice polynomials p and q such that
To see that the answer must be no, we observe that pq must then have precisely two real roots (counting with multiplicity). On the other hand, p (resp. q) has at least two real roots (counting multiplicity), since is a real polynomial of odd degree (hence admits a real root). Thus pq has at minimum four real roots, and we have a contradiction.
An extension of this argument shows the following:
Theorem: There is no finite set of n>1 loaded dice such that the probability of any dice sum ({n,…6n}) is equally likely.
We can extend the class of loaded dice to the class of null-loaded dice by allowing the probabilities of certain die rolls to equal zero. In this case, our theorem still holds: the proof holds verbatim provided that and
remain non-zero, and if either is zero, our theorem holds trivially. Moreover, we may consider the class of null-loaded k-sided dice, and ask if there exists a collection of n>1 such dice (throughout which k may vary, in general) such that the probability of any (attainable) dice sum is equally likely. If
is fixed as our dice
vary, we have the following:
Theorem: There is no simulation of a fair (m(k-1)+1)-sided die via dice sums on a collection of m null-loaded k-sided dice.
Proof: If, to the contrary, such a collection were to exist, we may write
Equating coefficients gives . Moreover, consideration of the
coefficient gives us
It follows by the AM-GM inequality that
which presents a contradiction for .
Note: when k is even, this result admits a proof analogous to that given in the case. If we restrict the scope of our previous theorem to loaded k-sided dice, the following is immediate:
Corollary: There is no simulation of a fair die via dice sums on a collection of m loaded k-sided dice.
Matters become far more complicated when is allowed to vary with the die
. In this case, the simulation of a fair die via dice sums becomes possible:
Example: Let denote the fair 2-sided die with outcomes
, and let
denote the null-loaded 3-sided die with outcomes
and respective probabilities
. Then the dice sums for
and
have outcomes {2,3,4,5}, each of which is equally likely. (In other words, we may simulate a fair 4-sided die via dice sums of two fair 2-sided dice (as long as one die has been renumbered {1,3}.)
Underlying all of this lies the factorization of the polynomial
into scaled cyclotomic polynomials. To wit, if may be written as a product of polynomials
with non-negative coefficients such that
, then the null-loaded
-sided dice (now numbered
for convenience) associated to the polynomials
have equally probable dice sums. In particular, it is an easy exercise to show that any fair m-sided die can be simulated as a null-loaded dice sum, provided that m is composite. In what follows, we’ll prove just one theorem concerning varied
(concerning the factorization of
over
).
Theorem: Suppose that the dice sums of a collection of n null-loaded -sided dice
simulate a fair m-sided die. If the probability of each outcome of
is rational, then each non-null outcome of the die
is equally likely.
Proof: Let denote the generating polynomials for the dice
(numbered
). Then
and Gauss’s lemma gives rational constants such that
(with
). Specifically,
is the constant term of
, and we note that each coefficient of
is a non-negative integer. Let
be the
coefficient of
. If
, then the
coefficient of
exceeds 1 (by non-negativity of the coefficients), a contradiction. Hence
is 0 or 1, and our result is immediate.
Some food for thought –
Exercise: We say that a null-loaded k-sided die (labelled {0,…k-1}) is primitive if the associated polynomial is irreducible in . Prove that a fair m-sided die arises as the dice sum of a collection of primitive dice if and only if m is a prime power.