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\centerline{\bf Honors Homework 2} 

\centerline{Morally Due Mon Feb 19 at 10:00AM}

\begin{enumerate}

\item
(0 points)
What is your name?  Write it clearly.


\newpage

\item
(30 points) 
Consider the DUP-SPOILER game with $(\Q,\R,4)$. 
So the orderings are $\Q$, the rationals,
$\R$ the reals and the game will go for 3 rounds. 
We denote DUP by D and SPOILER by S.

\begin{enumerate} 
\item 
Assume the game goes as follows:
\begin{enumerate}
\item
Round 1: S picks $\sqrt{2}$ in $\R$. D picks 0 in $\Q$.
\item
Round 2: S picks $\frac{1}{1000}$ in $\Q$. D picks 2 in $\R$. 
\item
Round 3: S picks $\frac{1}{2000}$ in $\Q$. 
\end{enumerate}

Give a move D can make to WIN!

\item 
Assume the game goes as follows:
\begin{enumerate}
\item
Round 1: S picks $\sqrt{2}$ in $\R$. D picks 0 in $\Q$.
\item
Round 2: S picks $\frac{1}{1000}$ in $\Q$. D picks 2 in $\R$. 
\item
Round 3: S picks $\sqrt{5}$ in $\R$. 
\end{enumerate}

Give a move D can make to WIN!
\end{enumerate} 


%XXX
\newpage

\item 
(30 points) 
Consider the DUP-SPOILER game with $(\Q,\R,1000)$. 
So the orderings are $\Q$, the rationals,
$\R$ the reals and the game will go for 1000 rounds. 
We denote DUP by D and SPOILER by S.

Assume that after 999 moves:
\begin{itemize}
\item
The points in $\R$ picked are $r_1 < r_2 < \cdots < r_{999}$.
(They did not have to be picked in that order. For example,
S's first move could be what we now call $r_{10}$.)
\item
The points in $\Q$ picked are $q_1 < q_2 < \cdots < q_{999}$.
(They did not have to be picked in that order. For example,
S's first move could be what we now call $q_{10}$.)
\item 
If in round $j$ S picked $r_i$ then in that round D picked $q_i$.
\item
If in round $j$ S picked $q_i$ then in that round D picked $r_i$.
\end{itemize}


\begin{enumerate}
\item
Assume that in Round 1000 S picks a point $r\in \R$ such that $r< r_1$.
How should D respond to win?
(The kind of answer I want, and this is NOT correct, is {\it S picks a point between
$q_{10}$ and $q_{20}$}.) 
\item
Assume that in Round 1000 S picks a point $q\in\Q$ such that
$q_{10} < q < q_{11}$. 
How should D respond to win?
\item
Assume that in Round 1000 S picks a point $r$ such that
$r_{999} <r$. 
How should D respond to win?
\end{enumerate}

\newpage

\item
(40 points) 
Informally argue that, for all $n$, D wins the game $(\Q,\R,n)$. 


\end{enumerate}

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