MATHEMATICS 198 &C1: Mathematics Via Problem-Solving
Call # 05056
10 M W F
??? Altgeld Hall
This course introduces students to mathematical ideas, improving their ability
to understand and communicate careful arguments via practice in problem-solving
and writing proofs. We develop techniques to solve interesting problems that
may sound like "brain-teasers" (sample problems appear below). After
understanding the structure of logical statements, many problems can be solved
using elementary techniques involving functions, induction, counting models, and
equivalence relations. Later techniques include the pigeonhole principle,
recurrence relations, and limits.
Mathematics is an active process of thought; one learns mathematics best by
doing mathematics. Understanding of mathematical ideas is clarified by
communicating them to others. For these reasons, this course will run in
seminar format, with no lectures. Class time will be spent mostly on student
presentations of solutions to problems, with comments by fellow students
and by the instructor. For well-motivated students in a small group, this
format provides a more enriching experience than a lecture course about proofs,
and it encourages students to develop their talents more fully and quickly.
Skills in written communication of logical arguments are also valuable.
Students will write solutions for two problems per week in addition to the
problems discussed in class. Most of the problems and the development of
mathematical ideas useful for solving them come from the text
Mathematical Thinking: Problem-Solving and Proofs,
by John D'Angelo and Douglas West.
This course has no college-level mathematics prequisite. Furthermore, since it
introduces students to the discovery and communication of logical arguments,
students should not have taken Math 247 or 300-level mathematics courses before
or with this course. It will be good preparation for advanced courses (or
occupations) where such skills are used.
Here are examples of problems we may solve:
Does every year have a Friday the 13th?
Which positive integers are sums of consecutive smaller positive integers?
How does one count the squares of all sizes in a square grid, or the
triangles of all sizes in a triangular grid?
At a party with five married couples, no person shakes hands with his or her
spouse. No two people other than the host shake hands with the same number of
people. With how many people does the hostess shake hands?
If each resident of New York City has 100 coins in a jar, is it possible that
no two residents have the same number of coins of each type (pennies, nickels,
dimes, quarters, half-dollars)?
How can we find the greatest common divisor of two large numbers without
Why are there infinitely many prime numbers?
A dart board has regions worth a points and b points, where
a and b are positive integers with no common factors. What is the
largest point total that cannot be obtained by throwing darts at the board?
A bear's cage has two jars of jelly beans, one with x beans and the other
with y. Each jar has a button. When a jar has at least two beans,
pressing its button gives one bean to the bear and moves one bean to the
other jar. Under what conditions (on (x,y)) can the bear eat all the
beans except one?
1500 soldiers arrive in training camp. A few soldiers desert. The sergeants
divide the remaining soldiers into groups of 5 and discover that there is 1 left
over. When they divide them into groups of 7 or 11, there are 3 left over.
How many soldiers deserted?
Are there more rational numbers than integers? Are there more real numbers
than rational numbers? What does ``more'' mean for infinite sets?
Airlines A and B serve the same airports. Can A have a
better on-time performance than B at every airport but worse on-time
performance than B over the full system?
Candidates A and B in an election receive a and b
votes, respectively. If the votes are counted in a random order, what is the
probability that candidate A never trails?
Can a 6 by 6 chessboard be covered with 1 by 2 rectangles so that the
arrangement cannot be cut along any horizontal or vertical line?
How many positive integers less than 1,000,000 have no common factors with
After n students take an exam, the papers are distributed randomly for
peer grading. What is the probability that no student gets his or her own
There are n girls and n boys at a party, and some boy/girl
pairings are compatible. Under what conditions is it possible to pair off the
girls and boys using only compatible pairs?
How many regions are created by drawing all chords joining n points on a
circle, if no three chords have a common point?
Why are there only five Platonic solids (the tetrahedron, cube, octahedron,
dodecahedron, and icosahedron)?
n spaces are available for parking along a curb. Rabbits take one space,
and Cadillacs take two spaces. In how many ways can we fill the spaces? In
other words, how many lists of 1's and 2's sum to n?
What numbers have more than one decimal expansion?
For each point in a tennis game, the server has probability p of winning
the point, independently of other points. What is the probability that the
server wins the game?
Two thieves steal a circular necklace with 2m gold beads and 2n
silver beads arranged in some unknown order. Can they cut the necklace along
some diameter so that each thief gets half the beads of each color? Does every
circular wire always contains two diametrically opposite points having the same
temperature? How are these questions related?
INSTRUCTOR: Douglas West received his Ph.D. in mathematics from the
Massachusetts Institute of Technology. He then taught one year at Stanford
University and three years at Princeton University before joining the
U. of I. faculty in 1982. He has published two textbooks and numerous research
articles. His area of research is discrete mathematics, especially graph theory
and partially ordered sets. He is currently Vice Chair of the Activity Group in
Discrete Mathematics of the Society of Industrial and Applied Mathematics, and
he is an Associate Editor of the American Mathematical Monthly. He also
sings in the chorus of the Illinois Opera Theatre and is an avid squash player.
Campus Honors Program at the
University of Illinois
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