Lecture Notes

WEEK # TOPICS
1

Introduction (PDF)

  • Platonic solids—Counting faces, edges, and vertices
  • Planar graphs, duality
  • Euler's formula for planar graphs—A constructive proof
  • Non-existence of a sixth platonic solid
  • Proving non-planarity by counting
2

Counting 101 (PDF)

  • First Law of Counting—Multiplying the possibilities
  • Shepard's Law—To count the sheep, count the feet
  • Counting by cases—Break it down and add it up
  • Counting by subtraction—Cases to exclude
3

Counting Sets (PDF)

  • Set theory and Boolean logic
  • Inclusion/exclusion—Easy as PIE
  • How many handshakes?
4-5

Counting Subsets (PDF)

  • Binomial coefficients
  • The wonders of Pascal's triangle
  • Counting by block walking
  • Counting by committee
  • The most useful combinatorial identity known to man—"The Hockey Stick"
  • The n days of Christmas 

Problem Solving

  • Applying what we know—Examples of counting
  • How to recognize an apparently unfamiliar problem
  • What to do when you are lost in the forest—How to get unstuck
  • More examples—kids and candy, flower arrangements
6

Discrete Probability (PDF)

  • Sample spaces and events
  • Probability measures
  • Sampling with and without replacement
  • Conditional Probabiliity
7

More Probability (PDF)

  • The Bernoulli process
  • The Infinite Bernoulli process
  • Analyzing games
  • Solving problems
8

Graph Theory (PDF)

  • A whirlwind tour
  • Vertices, edges, degree, paths, cycles
  • Connectivity and components
  • Acyclic graphs—Trees and forests
  • Directed graphs
9

More Graph Theory (PDF)

  • Eulerian tours
  • Graph coloring
  • Ramsey Theory
  • Turan's Theorem
10

Contest Problems

  • A chance to tackle some real contest problems

 

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