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0. (Random) Pascal's Triangle

a. Extending Pascal's Triangle
b. Finding Numerical Patterns in Pascal's Triangle
c. Defining Binomial Coefficients
d. Using Pascal's triangle to find the k^th term of (x + y)ⁿ (Binomial Theorem)
e. Using Pascal's triangle to expand (x + y)ⁿ (Binomial Expansion)
f. Using Pascal's triangle with Numbers
g. Combinatoric Problems
h. Coloring Multiples in Pascal's Triangle
i. Petal Problems in Pascal's Triangle (Pascal's Flower)

1. (Random) Combinatorics

a. Combinations
b. Permutations (And other cases where order matters)
c. Simplifying Factorial Expressions
d. Combinations with Replacement
e. Binomial Theorem
f. Multinomial Coefficients
g. Counting Subsets

2. (Random) Counting Methods

a. Counting Cases
b. Complementary Counting
c. Counting with Restrictions
d. Counting Pairs

3. (Random) Probability

a. Finding the Sample Space
d. Multiplication Rule
c. Addition Rule
d. Combining Probabilities
e. Probability Trees
f. Probability using Combinatorics
g. Probability with Replacement
h. Probability without Replacement
i. Binomial Probability
j. Expected Value
k. Pigeonhole Principle

4. (Random) Graph Theory

a. Define a graph and its parts
b. Finding Bridges, Loops, and Multiple Edges
c. Identifying if Graph has an Open or Closed Unicursal Tracing
d. Applying Euler's Graph Theory
e. Creating Eulerizations and Semi-Eulerizations
f. Solving Euler Path/Circuit Word Problems
g. Finding Hamilton Circuits and Paths
h. Using and Identifying Complete Graphs
i. Solving the Traveling Salesman Problem (Brute Force)
j. Solving the Traveling Salesman Problem (Nearest Neighbor)
k. Solving the Traveling Salesman Problem (Repetitive Nearest Neighbor)
l. Solving the Traveling Salesman Problem (Cheapest Link)
m. Using and Identifying Trees
n. Find the Number of Spanning Trees (simple cases)
o. Find Minimum Spanning Trees Using Kruskal's Algorithm
p. Find Minimum Spanning Trees Using Prim's Algorithm
q. Find Minimum Spanning Trees Using Dijkstra's Algorithm
r. The Shortest Network and Steiner Points

5. (Random) Scheduling Problems

a. Find the chromatic number of a graph
b. Find the chromatic index of a graph
c. Identifying Snarks
d. Identifying Planar Graphs
e. Coloring Graphs vs. Maps
f. Use Graph Coloring for Scheduling Problems
g. Directed Graph (Digraph) Terminology
h. Applying Euler's Graph Theory for Directed Graphs
i. Finding Hamilton Circuits and Paths for Directed Graphs
j. Facts about Schedules
k. Applying the List Processing Algorithm
l. Applying the Decreasing Time Algorithm
m. Compute the earliest Completion Time using Critical Paths
n. Applying the Critical Path Algorithm
o. Identifying Bin Packing vs. Scheduling Problems
p. Bin Packing Problems

6. (Random) Voting and Elections

a. Creating a Preference Schedule
b. Using Plurality
c. Using Borda
d. Using Plurality with Elimination
e. Using Condorcet (Pairwise Comparisons)
f. Using Extended Rankings
g. Using Plurality with Runoff
h. Fairness Criterion and Arrow's Theorem
i. Using Approval Voting
j. Using Weighted Voting Systems
k. Computing Power Index (Banzhaf)
l. Computing Shapley-Shubik Power Index

7. (Random) Fair Division

a. Fair Division: The value of things
b. Allocating Items with Adjusted Winner Procedure
c. Resolving A Dispute with Adjusted Winner Procedure
d. Divider-Chooser Method
e. Lone-Divider Method
f. Lone-Chooser Method
g. Last Diminisher Method
h. Method of Sealed Bids (Knaster Inheritance)
i. Method of Markers
j. Trimming Procedure
k. Equitable, Envy-Free, and Pareto-Optimal

8. (Random) Apportionment

a. Standard Divisors and Quotas
b. Hamilton's Method
c. Jefferson's Method
d. Adam's Method
e. Webster's Method
f. Huntington-Hill Method

9. (Random) Game Theory

a. Create a Game Matrix
b. Determine the Expected Value of the Game Matrix
c. Identifying and Using Game terms such as Zero-Sum and Minimax
d. Nash equilibrium, the Prisoner's Dilemma, and the Game of Chicken
e. Nash Equilibrium for 3x3 and 4x4 Games
f. Rules of TOM (Theory of Moves)
g. Creating a Tree Diagram in TOM (Theory of Moves) and Determining if it is Cyclic
h. Finding all NME (Nonmyopic Equilibria)
i. Larger TOM Games (Theory of Moves)

10. (Random) Matrices

a. Performing Matrix Operations (Addition/Subtraction/Scalar Multiplication)
b. Performing Matrix Multiplication
c. Finding the determinants of Matrices
d. Finding Matrix Inverses (3x3)
e. Finding Matrix Inverses (4x4)
f. Apply Adjacency Matrices
g. Apply the Leslie matrix model to population models
h. Apply the Markov chain to probability models
i. Apply the Leontief model to economic situations

11. (Random) Recursion

a. Fibonacci Numbers
b. The Golden Ratio
c. Fibonacci Numbers and Quadratic Equations
d. Gnomons and Similarity
e. General Recursive Definitions
f. Hand Shake Problem
g. The Tower of Hanoi Puzzle
h. Using Recursive Relations
i. Linear Growth
j. Exponential Growth
k. Logistic Growth (Verhulst Law of Restrained Growth)
l. Percentages of Percentages

12. (Random) Sequences, Series, and Induction

a. Sequences and Series
b. Finite Differences
c. Using Finite Differences
d. Finding Values of Difference Equations
e. Finding Equilibriums of Difference Equations
f. Basic Induction
g. Series Induction
h. Divisibility Induction

13. (Random) Chaos and Fractals

a. Applying Iteration
b. Iterative Fractals
c. The Game of Chaos
d. Evaluating Polynomials with Complex Numbers
e. Mandelbrot Set
f. Julia Set
g. Sierpinski's Triangle and Carpet
h. Von Koch Snowflake and Anti-Snowflake
i. Menger Sponge
j. Cantor Set and Cantor Dust
k. Tessellations (Tilings)

14. (Random) Coding and Encoding

a. Check Digits
b. Weighted Check Digits
c. Using Codabar
d. Postnet Bar Codes
e. Information: ZIP, UPC, and Social Security Numbers
f. Using the Soundex Code
g. Binary Codes
h. Parity Check Sums
i. Data Compression
j. Cryptograms
k. RSA Encryption

15. (Random) Number Theory

a. Clock arithmetic (mod)
b. Using The Division Algorithm
c. Finding the GCD using the Euclidean Algorithm
d. Finding Integer Solutions of Diophantine Equations
e. Using Fermat's Little Theorem
f. Using the Chinese Remainder Theorem
g. Using Parity

16. (Random) Set Theory

a. Set Notation
b. Unions, Intersections, Differences and Complements
c. Venn Diagrams
d. DeMorgan's Laws and Other Properties
e. Subsets
f. Partitions
g. Power Sets
h. Set Cardinality and Inclusion-Exclusion Principle

17. (Random) Logic

a. Truth Tables
b. Truthtellers and Liars
c. Logically Equivalent Statements
d. Conditional Statements
e. Valid/Invalid Arguments, Syllogism, Modus Pollens and Modus Tollens
f. Digital Logic Circuits
g. Boolean Algebra

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