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\begin{document}

\noindent
{\large\bf Honors Calculus Challenge Problem \#5, \#6 \& \#7: Induction}

\bigskip

\noindent
{\bf Dotville.} The people of Dotville have the following
peculiarity: each person has a 
red or blue dot on his
forehead.
These people are perfectly logical
and therefore are able to live in harmony
guided by only one law: whoever learns the
color of his dot on a certain day must
commit suicide at midnight of that day!

Life in Dotville was quite peaceful until the day
a stranger came to town. After looking about
the stranger (truthfully)
remarked, ``Someone has
a red dot on his forehead!''
Then he left. What were the consequences
of the stranger's remark??

\bigskip

\noindent
{\bf Handshakes.} A couple invite several other couples
to their home for a party. As people arrive,
some people shake hands with
others. No one shakes hands with himself,
no one shakes hands with his spouse and
no one shakes hands with another person
more than once. Other than this there are
no restrictions on how the handshakes
take place. In particular, someone might
not shake hands with anyone while another person
might shake hands with everyone other than
his spouse or himself.

At the end of the party the host asks everyone
(even his spouse)
how many hands they shook. He receives
a different answer from each person.
How many hands did the hostess shake??

\bigskip

\noindent
{\bf Hell's Librarian.}
After dying you find that because of your naughty
lifestyle you have ended up in Hell.
Your punishment is to be Hell's librarian.
You have a large cart of books to reshelve
and each time you put one book back on the shelf
the library demon comes along and adds a large pile of 
extra books to your cart! The longer you
work the more books you have to reshelve!

Looking into the books on your cart you
find that each page of each book
has a positive integer on it, apparently
chosen at random.
You remind Satan that you once did a good
deed while living and ask if it is worth
a minor reduction in your workload:
you ask that when you replace a book on
a shelf the library demon is only allowed to 
give you new books with fewer pages
or with the same number of pages
but a smaller number on the last page.
The demon is still permitted to give
you any (finite) number of books,
so Satan agrees.
Will you ever complete the job
of reshelving the books on your cart??

\end{document}
