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\title{End of Course Problem Set \\
       Math 804T \\
       Experimentation, Conjecture and Reasoning \\
}
\author{Steven R. Dunbar}
\date{Fall Semester 2007}

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\maketitle

\begin{enumerate}

\item Consider the sequence of Lucas numbers $L_n$, defined in Section
  2.2, page 59, problem 10, and the sequence of Fibonacci numbers
  $F_n$.  For 2 points, can
  you conjecture what the ratio $L_n/F_n$ seems to approach
  as $n$ gets larger?  For 3 points can you provide some mathematical
  reasoning that justifies your conjecture with the methods of
  Section 2.2?  (Hint: Use the result of Problem 12, page 59 and some
  of the ideas in of Problem 29, page 61.)

\item Make a new sequence $S_n$ which is the sum of the first $n$
  Fibonacci numbers.  For 2 points, make a table of values of $S_n$
  and make a conjecture about expressing this sequence in terms of
  the Fibonacci sequence.  For 3 points, provide mathematical
  reasoning about why your conjecture is true.

\item Can you make a triangle with the following leg lengths:
 one leg is the sum of two consecutive odd integers, the other leg is
 the product of two consecutive odd integers, and the hypotenuse is
 two more than the product?  Can you make a right triangle with those
 leg lengths? For two points, provide some
 experimentation and a conjecture, for 3 points, provide some
 mathematical reasoning which justifies your conjecture.

\item The cube can be inscribed in a
  sphere, that is, a sphere which touches the 8 vertices of the cube.
  What is the radius of that sphere which has an inscribed cube of
  side length 1?  (For bonus points, not required, just for fun and
  extra points if you get it right!  What
  is the radius of a sphere which has an inscribed tetrahedron of side
  length 1?)

\item A circle passes through the three vertices of an isosceles
  triangle that has two sides of length 3 and a base of length 2.
  What is the area of the circle?  (Two points for a decimal
  approximation correct to two places, 3 points for mathematical
  reasoning that gives a mathematical answer.)

\item At a teaching council 4 math teachers, 3 English teachers and 3
  foreign language teachers are to be seated in a row.  How many
  seating arrangements are possible when teachers of the same subject
  are required to sit together?

\item A mother is holding a birthday party with several excited young
  children. She has $n$ distinctively wrapped party favors to give to
  $n$ children.  She now has a headache, so she quickly hands out each
  favor package randomly without looking to see if the recipient
  already has a package.  For $n = 3$, $4$, and $5$, find the
  probability that each child gets a package.

  %% Answer is $n!/n^n$.  For n = 3,4,5 this
  %% is approximately .22, %% and 0.09, ad 0.04.m

\item A paper bag discovered at the back of a closet shelf is found to
  contain twelve old light bulbs.  Five of them are 25-watt, six are
  burned out and three are both.  Find the probability that a bulb
  drawn at random from the bag is burned out, given that it is
  25-watt.  Also find the probability that it is not 25-watt given that
  it is burned out.  

\item Suppose that an insurance company classifies people into one of
  three classes -- good risks, average risks and bad risks.  The
  company records indicate that probabilities that good, average and
  bad risk persons will be involved in an accident over a 1-year span
  are respectively $0.05$, $0.15$ and $0.30$.  If 20\% of the
  population are good risks, 50\% are average risks and 30\% are bad
  risks, what proportion of people have accidents in a given calendar year?
  If a policy holder Steve had no accidents in 2007, what is the
  probability that Steve is a good risk?

\item Write a short (one page or less) reflection on what was your
  favorite problem in the course, and why it was your favorite.
  Explain what you learned about mathematics and problem solving from
  your favorite problem.

\end{enumerate}

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