\pb
\section{Extra problems} 

\pb
\subsection{Quarto publishing}

In Quarto publishing, four book pages are printed on each side of
the large sheets of paper actually used.  One side of a quarto sheet
is shown in the illustration.  Once the sheets are printed on both sides,
they are stacked up, and the stack is folded twice: first from top
to bottom on the dashed horizontal line, and then from side to side
on the dashed vertical line.  The tops of the pages are cut apart.
Thus every quarto page produces eight book pages; see the illustration.

\vfill

1.  How many quarto sheets are needed for a book of 1240 pages?

\vfill

2.  When the pages of this book are stacked and ready to be folded,
what does the top sheet look like?

\vfill

3.  Draw a picture that shows where on the sheet the
page numbers appear, and which are upside down.

\vfill

4.  What other page numbers on on the same quarto sheet as page 13?

\vfill

\pagebreak

\begin{figure}[h!]
\includegraphics[scale=0.7]{figures/quarto} 
\end{figure}

\pb
\subsection{A frame-up}

I painted a picture that is 4 inches wider than it is high.  If I put a 
3-inch frame all the way around my picture, the area increases by 
48 square inches.  What are the dimensions of my picture?

\pb
\subsection{Four cuts}

If you were to take a block of cheese and make a cut all the way through,
you would end up with two pieces.  If you were then to make another cut 
without moving either of those pieces, you should be able to do it in such
a way that you end up with four pieces.  What is the maximum number of
pieces you can have after three cuts (remember you can't move any of 
the pieces)?  What about after four cuts?  You should be ready to explain 
how to make the cuts so as to obtain the maximum possible number 
of pieces.

\pb
\subsection{Squares and pathways}

\bigskip

\noindent At home, work on each of the following two problems (Squares and
Paths) for half an hour each.  Write down how far you get on each, and then
choose one of the two to finish solving.

\bigskip

\begin{figure}[h!]
\includegraphics[scale=0.7]{figures/squares} 
\end{figure}
\hfill\parbox[b]{3in}{
\centerline{\Large Squares}

\bigskip

\noindent 1.  How many squares of all sizes are contained in the $5 \times 5$
grid shown at left?

\bigskip

\noindent 2.  How many squares of all sizes are contained in an $n \times n$
grid?  ($n$ can be any whole number: 1, 2, 3, 4, ...)

\vspace{0.6in}
}

\bigskip
\bigskip

\noindent\parbox[b]{2.2in}{
\centerline{\Large Paths}

\bigskip

How many paths are there that start at some A and end at Z, that contain each
letter of the alphabet exactly once, and traverse the letters in alphabetical
order?  (Diagonal moves are not allowed.)
} \hfill \parbox{4in}{
\tt

\centerline{A}
\centerline{ABA}
\centerline{ABCBA}
\centerline{ABCDCBA}
\centerline{ABCDEDCBA}
\centerline{ABCDEFEDCBA}
\centerline{ABCDEFGFEDCBA}
\centerline{ABCDEFGHGFEDCBA}
\centerline{ABCDEFGHIHGFEDCBA}
\centerline{ABCDEFGHIJIHGFEDCBA}
\centerline{ABCDEFGHIJKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPQPONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPQRQPONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPQRSRQPONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPQRSTSRQPONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPQRSTUTSRQPONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPQRSTUVUTSRQPONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPQRSTUVWVUTSRQPONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPQRSTUVWXWVUTSRQPONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPQRSTUVWXYXWVUTSRQPONMLKJIHGFEDCBA}
\centerline{ABCDEFGHIJKLMNOPQRSTUVWXYZYXWVUTSRQPONMLKJIHGFEDCBA}
} % end tt parbox

\pb
\subsection{Glacks and Glocks}

\bigskip

\noindent Read the following statement and indicate TRUE, FALSE,
or CAN'T TELL for each of the conclusions listed.

\bigskip

\begin{enumerate}

\item A {\bf Glack} is an odd number that either is less than 29 or is
a divisor of 48.

\medskip

\noindent\begin{tabular}{|l||c|c|c|}
\hline
{\sc Statement} & {\sc True?} & {\sc False?} & {\sc Can't tell?} \\
\hline
a.   All Glack numbers less than 15 are odd. & & & \\
\hline
b.   14 is a Glack. & & & \\
\hline
c.   All odd numbers less than 21 are Glacks. & & & \\
\hline
d.   16 is a Glack. & & & \\
\hline
e.   No odd number greater than 27 is a Glack. & & & \\
\hline
f.   Nine Glacks are prime. & & & \\
\hline
\end{tabular}

\vfill

\item All {\bf Glock} numbers are divisible by 2 and are multiples of 13.

\medskip

\noindent\begin{tabular}{|l||c|c|c|}
\hline
{\sc Statement} & {\sc True?} & {\sc False?} & {\sc Can't tell?} \\
\hline
a.   169 is a Glock. & & & \\
\hline
b.   No prime number is a Glock. & & & \\
\hline
c.   52 is a Glock. & & & \\
\hline
d.   All Glocks are even. & & & \\
\hline
e.   All Glocks are divisible by 26. & & & \\
\hline
f.   All numbers divisible by 26 are Glocks. & & & \\
\hline
\end{tabular}

\end{enumerate}

\renewcommand{\baselinestretch}{1}
\small\normalsize

\vfill

\pb
\subsection{Sets: extra problems}
%
\begin{enumerate}
\item In a special ed classroom where there are to be 20 children, 9 of
the children will need new desks high enough to accommodate wheelchairs, and
seven of the students in the classroom will need desks fitted with electrical
power strips for special equipment.  If eight of the children can use already
existing (lower) desks without power strips, how many of the new, higher desks
need to be fitted with power strips?
%
\item In a given class of 25 students, there are a total of 11 boys,
and a total of 7 left-handed children.  If 9 of the boys are right-handed, how
many of the girls are right-handed?
%
%\item In a class of 35 students, 20 have a last name beginning with a
%consonant.  25 students have a last name with more than one syllable.  What can
%you say about the number of students with a multisyllabic last name that begins
%with a vowel?  What about the number of students with a multisyllabic last name
%that begins with a consonant?
%
\item For each Venn diagram below, shade the numbered regions
indicated, and find a way to express the region in terms of $A$,
$B$ and $C$ using unions, intersections, and complements.
\end{enumerate}

\begin{figure}[h!]
\hspace{.5in}\includegraphics[scale=0.55]{figures/setextra4} 
\end{figure}

\pb
\subsection{Driving in Septobasiland}

\bigskip

The king of Septobasiland has proclaimed that the digits 7, 8 and 9 are never
to be used in Septobasiland upon pain of death.  As a result, the little wheels
on the odometers of all the cars in Septobasiland have just the digits 0, 1, 2,
3, 4, 5, 6 on them.  Thus, if the odometer registers 0006 and you drive one
more mile, the odometer will register 0010, and if the odometer registers 0016
and you drive one more mile, it will register 0020.

\bigskip

\begin{enumerate}

\item If the odometer registers 0066 and you drive one more mile, what
will the odometer register?

\vfill

\item If the odometer registers 0325, how many miles has the car gone?

\vfill

\item How many miles will the car have gone when the odometer turns
over to all zeroes again?

\vfill

\item After a car has gone nine hundred miles, what will its odometer
register?

\vfill

\item You and your companions go on a trip in Septobasiland, traveling
together in two cars.  The first car, which is new, starts with its odometer
reading 0000, and the second starts with the odometer reading 1435.  At the end
of the trip, the odometer of the first car registers 0324.  What will the
odometer of the second car register?

\vfill

\item Suppose again that the odometers of the two cars register 0000
and 1435 at the beginning of the trip.  But this time at the end of the trip
the odometer of the second car registers 2153.  What will the odometer of the
first car register?
\end{enumerate}

\vfill

\pb 
\subsection{What {\em are} they doing?}

For each of the following students, try to figure out how they 
are computing.  Then, try their algorithm on the given problems 
and describe why their algorithm works.

\vfill

\noindent{\bf Adrian}

\begin{tabular}{rrrr}
& 5 & 2 & 8 \\
- & 2 & 8 & 3 \\
\hline
&&&
\end{tabular}
\hspace{.5in}becomes\hspace{.5in}
\begin{tabular}{rrrr}
& 5 & 12 & 8 \\
- & 3 & 8 & 3 \\
\hline
& 2 & 4 & 5 
\end{tabular}

\begin{tabular}{rrrrr}
& 2 & 0 & 0 & 3 \\
- & & 8 & 9 & 6 \\
\hline
&&&&
\end{tabular}
\hspace{.5in}becomes\hspace{.5in}
\begin{tabular}{rrrrr}
& 2 & 10 & 10 & 3 \\
- & 1 & 9 & 10 & 6 \\
\hline
& 1 & 1 & 0 & 7 
\end{tabular}

\vfill

Now use Adrian's algorithm to compute: 

(a) \hspace{.1in} \begin{tabular}{rrrr}
& 8 & 1 & 2 \\ - & 4 & 5 & 9 \\ \hline &&&
\end{tabular}
\hspace{.5in}(b) \hspace{.1in} \begin{tabular}{rrrrr}
& 6 & 2 & 2 & 1 \\ - & 4 & 2 & 2 & 7 \\ \hline &&&&
\end{tabular}
\hspace{.5in}(c) \hspace{.1in} \begin{tabular}{rrr}
& 4 & $2_{five}$ \\ - & 1 & $3_{five}$ \\ \hline &&
\end{tabular}

\vfill 

\pb
\noindent{\bf Bill}

$27 \times 34$ becomes
\begin{figure}[h!]
\includegraphics{figures/bill} 
\end{figure}
thus, $27 \times 34 = 918$.  

\vfill

Now compute the following products using this algorithm:
$$(a) \; \; 38 \times 74 \hspace{2in} (b) \; \; 125 \times 35$$

\vfill
\vfill


\pb
\subsection{$1/3$ and $2/5$}

\bigskip

The following problems all involve the fractions $\frac13$ and $\frac25$;
however, each has different answer.  Explain what different roles
the fractions are playing (use diagrams when appropriate) and what
fraction operations are represented in the stories.

\bigskip
\bigskip

\begin{enumerate}

\item In your fourth grade class, $\frac13$ of the students voted in the
class elections.  $\frac25$ of the votes in your class were for Tammy. 
What proportion of the vote cast were for Tammy? 

\bigskip

\item In the same class, $\frac13$ of the boys get free lunch and
on $\frac25$ of the days, the only vegetable in the free lunch is catsup. 
What proportion of the lunches (at least) have catsup as the vegetable?

\bigskip

\item In the same class, $\frac13$ of the students have a TI 82 and
$\frac25$ of them have a TI 86.  Nobody has both, and nobody has any
other graphing calculator.  What proportion of the class has a 
graphing calculator?

\bigskip

\item In your class, $\frac13$ of the students remembered to turn in their
field trip permission slips.  In the other fourth grade class, $\frac25$
returned theirs.  If both classes have the same number of students, what
proportion of the fourth graders remembered to turn in their
permission slips? 

\bigskip

\item In chess club, 2 out of the 5 boys wear pocket protectors, and
1 out the 3 girls does.  What is the proportion of chess club members
that wear pocket protectors?

\end{enumerate}


%%%SECTION