Few words on logic and some proof techniques
- Correct and false proofs of Pythagoras theorem, proof by example (not a proof), proof by exhaustion (not always possible), Pythagorean triples
- Logic of implication, what a proof really is, direct proof, contrapositive, proof by contradiction

Counting and sets
- Counting/overcounting, equalities with combinatorial arguments
- More examples of counting/overcounting with sets
- Subsets, ordered/non-ordered subsets, binomial coefficients
- Pascal triangle, binomial theorem
- Selection with repetition

More proof techniques
- Inclusion/exclusion
- proof by pigeon-hole
- Proof by induction

Sequences and recurrences
- Sequences, recurrences, Fibonacci numbers
- Solving recurrences
- Traveling in Manhattan and Catalan numbers

Probability
- A little bit of probability, exclusive events, independent events
- law of large numbers

Number theory
- Divisibility and primes, prime factorization
- Uniqueness of prime factorization, prime number theorem
- Fermat's little theorem and primality testing, pseudoprimes, and charmicheal numbers
- GCD and Euclid's algorithm
- Congruence and modular arithmetic

Graph theory
- Introduction to graphs
- Trees, tree encoding and counting trees
- Minimum Spanning Tree
- Hamiltonian cycles, Traveling salesman, Eulerian cycles
- Planar graphs, Euler's formula, coloring