| 1 |
Introduction
- 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
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| 2 |
Counting 101
- 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
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| 3 |
Counting Sets
- Set theory and Boolean logic
- Inclusion/exclusion—Easy as PIE
- How many handshakes?
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| 4 |
Counting Subsets
- 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
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| 5 |
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
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| 6 |
Discrete Probability
- Sample spaces and events
- Probability measures
- Sampling with and without replacement
- Conditional Probabiliity
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| 7 |
More Probability
- The Bernoulli process
- The Infinite Bernoulli process
- Analyzing games
- Solving problems
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| 8 |
Graph Theory
- A whirlwind tour
- Vertices, edges, degree, paths, cycles
- Connectivity and components
- Acyclic graphs—Trees and forests
- Directed graphs
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| 9 |
More Graph Theory
- Eulerian tours
- Graph coloring
- Ramsey Theory
- Turan's Theorem
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| 10 |
Contest Problems
- A chance to tackle some real contest problems
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