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\homework{4: More Counting}{Due Friday, Week 2}

Do \textbf{one} of the \textbf{three} problems below!   Prove all of your claims.  If you've seen some of these problems before, try the ones you have not seen first.


\begin{enumerate}
 \item \begin{enumerate}
\item In class, we showed that if $A,B$ are a pair of dice with associated polynomials $A(x), B(x)$, and $D(x)$ is the polynomial associated to rolling $A$ and $B$ and summing their result, we have
\begin{align*}
A(x) \cdot B(x) = D(x).
\end{align*}
Mimicking this result, show that if $A,B,C$ are three dice with associated polynomials $A(x), B(x), C(x)$, then
\begin{align*}
A(x) \cdot B(x) \cdot C(x) = D(x),
\end{align*}
where $D(x)$ is the polynomial associated to rolling $A,B,C$ and summing the result.
\item Using this result, decide whether or not there exists a triple $A,B,C$ of nonstandard 6-sided dice such that when they are rolled and summed, they are indistinguishable from rolling and summing three standard 6-sided dice.
\end{enumerate}

\item \begin{enumerate}
\item Show that the polynomial
\begin{align*}
p(x) = 1 + x + x^2 + x^3 + x^4
\end{align*}
is irreducible over the integer polynomials: in other words, it cannot be factored into two polynomials of strictly lower degree with integer coefficients.

\item Prove that there are no pairs of nonstandard 5-dice that when rolled and summed, are indistinguishable from a pair of standard 5-dice.
\end{enumerate}

\item Consider the following sequence of ``faux-bonacci'' numbers $\{\phi_n\}_{n=0}^\infty:$
\begin{align*}
\phi_0 = 0, \phi_1 = 4, \phi_{n+1} = \phi_n + \phi_{n-1}.
\end{align*}
Find a closed form for these numbers using generating functions, in a similar method to how we derived a closed form for the Fibonacci numbers in lecture today!
\end{enumerate}










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