Problem 14

Consider the polynomial

in the ring $\mathbb{Z}[x]$.

1. Explain why the coefficient of the term of degree $k$ (where $2\leq k\leq 12$) in $P(x)$ gives exactly the number of ways that a pair of two dice could sum to $k$.

2. Factor $P(x)$ into a product of irreducible polynomials. Prove that the factorization, indeed, consists only of irreducible polynomials.

3. Define the polynomials

   $\displaystyle f(x)$ $\displaystyle=x^{a_{1}}+x^{a_{2}}+x^{a_{3}}+x^{a_{4}}+x^{a_{5}}+x^{a_{6}}\text{ and}$ $\displaystyle g(x)$ $\displaystyle=x^{b_{1}}+x^{b_{2}}+x^{b_{3}}+x^{b_{4}}+x^{b_{5}}+x^{b_{6}},$

   where each $a_{i}$ and $b_{i}$ are positive integers. Suppose we have $P(x)=f(x)g(x)$. Based on (ii), explain why

   $$f(x)=x^{q}(x+1)^{r}(x^{2}+x+1)^{s}(x^{2}-x+1)^{t}$$

   where $0\leq q,r,s,t\leq 2$.

4. Show $r=s=1$, and $q\neq 0$. Note that we define $0^{0}=1$ for this part only.(Hint: compute $f(1)$ in two ways. Then compute $f(0)$ in two ways.)

5. Explain why $q\neq 2$.

6. Hence, show that one possible choice of $a_{1},a_{2},\dots,a_{6}$ is 1, 2, 2, 3, 3, and 4. Find the other choice(s).

7. Deduce the labels of a pair of dice (that are not the standard labels of 1 to 6 on both dice) that will lead to the same likelihood of rolling a sum of, say, $k$ (where $2\leq k\leq 12$), as regular dice.