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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Tue Apr 01 16:47:53 1997}
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\input{tcilatex}
\begin{document}


\section{Is Algebra the Clue?}

Algebra can be used for keeping (and telling) secrets. Company ALG needs to
protect a secret file, the key to which is a hidden number. They are asking
for a scheme so that any two employees working together could learn the
hidden number but no one working alone could find the number.\medskip

\begin{quotation}
{\small You can develop such schemes using polynomial functions, that is,
functions of the form } 
\[
f(x)=a_{n}x^{n}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}. 
\]
{\small These include, in particular,}
\end{quotation}

\begin{description}
\item  {\small Lines: }$f(x)=ax+b$\quad \FRAME{itbpF}{0.6979in}{0.4653in}{%
0.0804in}{}{}{}{\special{language "Scientific Word";type "MAPLEPLOT";width
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TRUE;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}%
{\small \ }

\item  {\small Parabolas: }$f(x)=ax^{2}+bx+c${\small \quad }\FRAME{itbpF}{%
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4;plottickdisable TRUE;numpoints 49;axesstyle "normal";xis
\TEXUX{x};var1name \TEXUX{$x$};}}

\item  {\small Cubics: }$f(x)=ax^{3}+bx^{2}+cx+d${\small \ \quad }\FRAME{%
itbpF}{0.7178in}{0.5984in}{0.0701in}{}{}{}{\special{language "Scientific
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45;theta 45;plottype 4;plottickdisable TRUE;numpoints 49;axesstyle
"normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}
\end{description}

\begin{quotation}
{\small The schemes explored in these activities are based on geometric
properties.}
\end{quotation}

\begin{description}
\item  {\small Two points determine a line and there are many lines passing
through any one given point.}

\item  {\small Three points, not on a line, determine a parabola and there
are many parabolas passing through any two given points.}

\item  {\small Four points on a cubic curve determine the cubic and there
are many cubics passing through any three such points.}
\end{description}

\begin{quotation}
{\small Since these lines and curves are determined by a polynomial of
degree one, two or three, you can use algebra to implement the schemes. The
scheme you are asked to explore is based upon finding the }$y${\small %
-intercept of a line.\bigskip }
\end{quotation}

At ALG, Inc., each employee is given the coordinates of a different point on
a line. The following table lists the points for eight employees. 
\[
\begin{array}{ccccccccc}
x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 
y & -196 & 41 & 278 & 515 & 752 & 989 & 1226 & 1463
\end{array}
\]

Suppose the employees with $x$-coordinates $2$ and $8$ put their clues
together.

\begin{itemize}
\item  Find the $y$-intercept determined by these two employees' coordinates.

\item  Choose another set of two employees. Show that they determine the
same $y$-intercept.

\item  Choose one employee. Find and graph several different solutions that
use this employee's coordinates. Verify that they cross the $y$-axis in
different places.\bigskip 
\end{itemize}

\begin{quotation}
\emph{Reminder}{\small \quad For each command on the Maple menu, place the
insertion point in the mathematical expression and then choose the
command.\medskip }
\end{quotation}

\subsection{Carol's and Darel's Solution}

\FRAME{itbpF}{0.8856in}{0.5163in}{0in}{}{}{solution.wmf}{\special{language
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The system for revealing a hidden number to any two employees working
together, but hiding the number from anyone working alone, is based upon
finding the $y$-intercept of a line. Each employee is given the coordinates
of a different point on the line. The following table lists coordinates for
a company of eight employees. 
\[
\begin{array}{ccccccccc}
x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 
y & -196 & 41 & 278 & 515 & 752 & 989 & 1226 & 1463
\end{array}
\]

\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
0in;original-width 29.375pt;original-height 30.875pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"NPEU";}}]{Margin Hint}{$\rhd $ To make this table\medskip
\par
Choose \textsf{Insert + Display}
\par
Choose \textsf{Insert + Matrix}
\par
Set \textsf{Rows} to $2$, \textsf{Columns} to $8$, and choose \textsf{OK}.
\par
Type the numbers in the input boxes.}

When the employees with $x$-coordinates $2$ and $8$ put their clues
together, their coordinates determine a unique line passing through the two
points $\left( 2,41\right) $ and $\left( 8,1463\right) $. To find the $y$%
-intercept determined by these coordinates, define the function $p(x)=ax+b$
and solve the system of equations $p(2)=41$ and $p(8)=1463$. Solving this
system gives the solution $\left\{ a=237,b=-433\right\} $ and hence the $y$%
-intercept is $-433$.

\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
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"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"NPEU";}}]{Margin Hint}{%
\ As a precaution, first choose \textsf{Maple + Define + Clear Definitions}
\par
With the insertion point in the equation $p(x)=ax+b$, choose \textsf{Maple +
Define + New Definition}
\par
Choose \textsf{Insert + Display}, and in the input box, type\textsf{\ }$%
p(2)=41$.
\par
Press \textsf{Enter} to add a line to the display, and type $p(8)=1463$ in
the input box.
\par
With the insertion point in the display 
\begin{eqnarray*}
p(2)=41 \\
p(8)=1463
\end{eqnarray*}
choose \textsf{Maple + Solve + Exact}\QTR{MarginHint}{\ }}

When the employees with $x$-coordinates $1$ and $5$ put their clues
together, their coordinates determine a unique line passing through the two
points $\left( 1,-196\right) $ and $\left( 5,752\right) $. Solving the
system $p(1)=-196$ and $p(5)=752$ gives the same solution as before, $%
\{a=237,\,b=-433\}$, and hence the same $y$-intercept of $-433$.\medskip

\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
0in;original-width 29.375pt;original-height 30.875pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"NPEU";}}]{Margin Hint}{%
Choose \textsf{Insert + Display}
\par
In the input box, type\textsf{\ }$p(1)=-196$%
\par
Press \QTR{KeyTopics}{enter}, and type $p(5)=752$%
\par
With the insertion point in the display 
\begin{eqnarray*}
p(1)=-196 \\
p(5)=752
\end{eqnarray*}
choose \textsf{Maple + Solve + Exact}}

The employee with $x$-coordinate $3$ determines a family of lines passing
through the point $\left( 3,278\right) $. Solving the equation $p(3)=278$
for $b$ gives the solution $\{b=-3a+\,278\}$.

\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
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"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"NPEU";}}]{Margin Hint}{%
With the insertion point in the equation\textsf{\ }$p(3)=278$, choose 
\textsf{Maple + Solve + Exact} (Solve for $b$.)}

Taking $a=1,2,3$ gives the three equations 
\begin{eqnarray*}
y &=&x+275 \\
y &=&2x+272 \\
y &=&3x+269
\end{eqnarray*}
with three different $y$-intercepts $275$, $272$, and $269$ shown in the
following graph.\FRAME{dtbpFU}{3in}{2.0003in}{0pt}{\Qcb{{}}}{}{}{%
\special{language "Scientific Word";type "MAPLEPLOT";width 3in;height
2.0003in;depth 0pt;display "PICT";function
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49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}

\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
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TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
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"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"NPEU";}}]{Margin Hint}{%
Choose \textsf{Insert + Math}
\par
Type the list $x+275,2x+272,3x+269$ with the three expressions separated by
red commas.
\par
With the insertion point in the list, choose \textsf{Maple + Plot2D +
Rectangular}}

An individual working alone has no clue to the hidden number, since the $y$%
-intercept for a line passing through a single point off of the $y$-axis
could be any number.\bigskip

\begin{quotation}
\emph{Remark}{\small \quad The line }$y=237x-433${\small \ is shown in the
following plot, together with the eight employees' coordinates.}
\end{quotation}

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";function \TEXUX{$\left(
1,-196,2,41,3,278,4,515,5,752,6,989,7,1226,8,1463\right) $};linecolor
"black";linestyle 1;linethickness 1;pointstyle "circle";pointplot
TRUE;function \TEXUX{$237x-433$};linecolor "blue";linestyle 1;linethickness
1;pointstyle "point";xmin "-1";xmax "9";xviewmin "-1.2";xviewmax
"9.204";yviewmin "-717.4";yviewmax "1748";rangeset"X";phi 45;theta
45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{x};var1name
\TEXUX{$x$};}}

\CustomNote[\FRAME{itbpF}{0.4367in}{0.4575in}{0in}{}{}{snbhint.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.4367in;height 0.4575in;depth
0in;original-width 29.375pt;original-height 30.875pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'Snbhint.wmf';file-properties
"NPEU";}}]{Margin Hint}{%
To obtain this result, plot the expression for the line: 
\[
237x-433 
\]
Select and drag to the frame a list of the employees' points, enclosed in
parentheses: 
\[
\left( 1,-196,2,41,3,278,4,515,5,752,6,989,7,1226,8,1463\right) 
\]
\par
Choose \textsf{Edit + Properties}, select \textsf{Plot Components} tab.
Choose \textsf{Item 2} and change the graph of employees' points to a
``point'' plot with symbol ``circle.''
\par
Change \textsf{Domain Interval} to $-1\leq x\leq 9$.
\par
Choose \textsf{OK}.}

\subsection{Activities}

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"XNPEU";}}

\begin{enumerate}
\item  Company $\mathbf{Y}$ is asking for a system that would permit any
three employees working together to learn a secret number, but that would
hide the secret number from any group of two or fewer employees.

\begin{enumerate}
\item  Describe a scheme based upon finding the $y$-intercept of a
polynomial $p(x)$ of degree $2$ and giving each employee the coordinates of
a different point that lies on the graph of this polynomial. Use the
following table of points. 
\[
\begin{array}{ccccccccc}
x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 
y & 6758 & 6397 & 6110 & 5897 & 5758 & 5693 & 5702 & 5785
\end{array}
\]

\item  Suppose three employees with $x$-coordinates $2$, $4$, and $8$ get
together to authorize access. Find the $y$-intercept determined by these
three employees' coordinates.

\item  Choose another set of three employees. Show that they determine the
same $y$-intercept.

\item  Choose a set of two employees. Find and graph several different
solutions. Verify that they cross the $y$-axis in different places.
\end{enumerate}

\item  Company $\mathbf{Z}$ is asking for a system that would permit any
four employees working together to learn a secret number, but that would
hide the secret number from any group of three or fewer employees.

\begin{enumerate}
\item  Describe a scheme, based upon finding the $y$-intercept of a
polynomial $p(x)$ of degree $3$, in which you give each employee the
coordinates of a different point that lies on the graph of this polynomial.
Use the following table of points. $
\begin{array}{ccccccccc}
x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 
y & 27807 & 32919 & 36919 & 39993 & 42327 & 44107 & 45519 & 46749
\end{array}
$

\item  Suppose four employees with $x$-coordinates $2$, $4$, $7$, and $8$
get together to authorize access. Find the $y$-intercept determined by these
four employees' coordinates.

\item  Choose another set of four employees. Show that they determine the
same $y$-intercept.

\item  Choose a set of three employees. Find and graph several different
solutions. Verify that they cross the $y$-axis in different places.
\end{enumerate}

\item  Develop a scheme for a company with ten employees plus four
supervisors that allows any three employees working together, or any two
supervisors working together, to find the secret number $52293$.\bigskip 
\end{enumerate}

\begin{quotation}
\emph{Reminder}{\small : Don't forget to check your answers by looking at
the problem several ways, including algebraically, numerically, and
geometrically.}
\end{quotation}

\ \bigskip

\noindent \hrulefill

\subsubsection{Calculus Activities\label{Activities}}

In the Extras directory on your \textsl{Scientific Notebook }CD, you'll find
a lengthy series of activities for single-variable calculus---open-ended
problems that explain and extend mathematical concepts and introduce a
variety of applications. These activities aren't placed on your hard disk
when you install the program; you must access them from the CD. Each
activity is presented with an introductory activity illustrating the
concept, an explanatory solution (with hints for using \textsl{Scientific
Notebook}), and the activities themselves.

\subsubsection{\noindent $\blacktriangleright $\quad \textbf{To access the
activities}}

\begin{enumerate}
\item  Insert your\textsl{\ Scientific Notebook }CD into your CD-ROM drive.

\item  Open \textsl{Scientific Notebook.}

\item  Click\FRAME{itbpF}{0.2759in}{0.2638in}{0.0406in}{}{}{fileopen.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
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"NPEU";}}, or from the File menu, choose Open.

\item  Select the correct drive for your CD.

\item  Select the Extras directory.

\item  Select 000\_StartActivities.tex

\item  Choose OK.\medskip 
\end{enumerate}

\noindent \hrulefill

\end{document}
