
\documentstyle[11pt,twoside,amstex,psfig]{article}

\addtolength{\topmargin}{-1in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9in}

\raggedbottom

\input{macros-course}
\iffalse
\setlength{\topmargin}{-0.50in}
\setlength{\textheight}{8.5in}
\setlength{\oddsidemargin}{0.0in}
\setlength{\evensidemargin}{-0.0in}
\setlength{\textwidth}{6.5in}
\addtolength{\parskip}{2pt}
\addtolength{\itemsep}{0.1in}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{claim}{Claim}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newcommand{\qed}{\mbox{\ \ \ }\rule{6pt}{7pt} \bigskip}
\fi
%\newtheorem{fact}{Fact}
\begin{document}

\handout{teamprobs6}{Problems for Recitation 6}{October 16, 1998}

\begin{problems}

\problem

{\textbf Life and Death.}
In ancient Mars, cannibalism was an essential way of life. 
The Marons and the Batons
were two species that lived for several thousand years in 
close contact.
Every year, first all the Marons sustained themselves by first
consuming Batons: two Marons ate one Baton between them.
Then they reproduced: one-year-old Marons 
doubled in number
while Marons that were two or older tripled in number.
Then the surviving Batons consumed one Maron each
and then all the Batons reproduced to quadruple in number. This cycle
repeated every year. After 1 year there were two 1-year old Marons and
two 1-year old Batons. After 2 years there were two 1-year olds and
two 2-year olds in each of the two species. How many of each 
will there be of each species at the end of $n$ years?

\problem

{\textbf Stepping up. } Alice the ``elegant'' climbs up a flight
of stairs one or two stairs in one stride. Bob the ``elongated beast'' on the
other hand clambers up 2, 4, 6 or any even number of stairs in one stride. 
Who can climb a flight of $n$ stairs in more ways? (Assume $n$ is even. 
A {\em way} of climbing a flight of $n$ stairs is the sequence of strides
taken to get to the top. E.g. to climb up 4 stairs, Alice can go
1,1,1,1 or 1,1,2, or 1,2,1, or 2,1,1 or 2,2 i.e., in 5 different ways in all.)  

\end{problems}
\end{document}
