Math 28 - Winter 2013

Dartmouth College

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Course Description:

Math 28 is a course in combinatorial mathematics. Combinatorics is a branch of mathematics that studies sets (usually finite) of objects that satisfy some properties. In particular, it is concerned with "counting" the objects in a set (enumerative combinatorics), with determining when an object with a required list of properties exist, with constructing and analyzing objects meeting certain properties (as in combinatorial designs and matroid theory), with finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and with finding algebraic structures these objects may have (algebraic combinatorics).

This year the course is being taught as a "guided discovery" course. This means that the primary text for the course is a book of problems with just enough prose so that the problems will make sense. It is available at the Wheelock Books in the form of course notes. By working these problems, students will discover for themselves the main theorems of combinatorial mathematics (and their proofs) and examples of how these theorems are used. Students are encouraged to work in groups on the problems. Students are expected to work as many of the problems as they can and submit them. Certain problems will be designated each week in advance as "mandatory" problems. These problems will be graded and returned to the student.

Class time will be devoted largely to a discussion of the problems and the broad themes in combinatorial mathematics that the problems illustrate. This discussion will be split between whole class discussion, including possible student presentations of their ideas, and small group discussions.

The objectives of the course are:

  • Students will be able to solve problems in enumerative combinatorics.
  • Students will improve their oral communication skills by explaining solutions to problems.
  • Students will improve their proof-writing skills.
  • Students will be well-prepared for Computer Science courses and Mathematics courses that use combinatorics.
  • "Mathematics is not for spectators; in order to gain in understanding, confidence, and enthusiasm one has to participate." M.A. Armstrong

    Last Updated Jan. 4, 2013 by R.C. Orellana