21-110: List of potential topics

This is a list of interesting topics in recreational mathematics; it can serve as a rough road map of places we might go this semester. We won’t have time to look at everything here, and we may investigate things I didn’t think of when I compiled this list. If something on this list seems especially interesting to you, let me know!

1. Problem-solving techniques
   • Strategic coloring: checkerboards, mouse in a maze, etc.
   • Working systematically
   • Backtracking
   • Solving problems with algebra
   • Finding a formula for a sequence of numbers
   • What to do when you’re stuck
   • Making conjectures, turning conjectures into theorems
   • Extending problems and generalizing results
2. Puzzles and logic
   • Chessboard problems
   • Magic squares and magic hexagons
   • Knights and knaves, truth-tellers and liars
   • The game of SET
   • Logical paradoxes
   • Conway’s Game of Life
   • Sudoku, kakuro, Ken-Ken
   • Mazes
3. Shapes and geometry
   • Polyominoes
   • Symmetry and tilings
   • Dissection problems
   • Origami
   • Fractals
   • Math in art
   • The golden ratio
   • Three-dimensional shapes: Platonic solids, Archimedean solids, Keplerian solids, Euler’s formula
   • Higher-dimensional geometry (the fourth dimension and beyond)
   • Topology: four-color theorem, spheres and toruses, Möbius strip, Klein bottle, knot theory
4. Numbers and number theory
   • Palindromes and repunits
   • Fibonacci numbers, Lucas numbers, and recursion
   • Divisibility
   • Prime numbers and the sieve of Eratosthenes
   • Prime factorization and the fundamental theorem of arithmetic
   • Special primes: Fermat primes, Mersenne primes, Sophie Germain primes
   • Diophantine equations
   • Pythagorean triples, Fermat’s Last Theorem, Euler bricks
   • Triangular numbers, square numbers, tetrahedral numbers, etc.
   • Ulam’s spiral
   • Modular arithmetic and finite fields
   • Bases and cryptarithmetic
   • Unsolved conjectures in number theory:
     • Twin primes conjecture
     • Existence of a perfect cuboid
     • Goldbach’s conjecture
     • Gilbreath’s conjecture
     • Hailstone (Collatz) conjecture
     • Smallest Sierpinski number
     • Infinitude of palindromic primes
     • Normality of π
     • Existence of an odd perfect number
   • Unimaginably large numbers
   • Irrationality of √2
   • History and approximations of π
5. Set theory and combinatorics
   • Principle of inclusion–exclusion
   • Infinite sets, Hilbert’s Hotel
   • How to count without counting
   • Permutations and combinations
   • Pascal’s triangle
   • Latin squares
6. Graph theory
   • The Königsberg bridge problem
   • Eulerian and Hamiltonian paths
   • Knight’s tour
   • Bridge-crossing and river-crossing problems
   • Weighted graphs and spanning trees
   • The traveling salesman problem
   • Graph coloring, scheduling with conflict graphs
   • Word graphs
7. Probability
   • Conditional probability and the Monty Hall problem
   • Random sampling, sampling bias, extrapolation from a sample
   • Statistics
   • Expected value, fair games
   • How many times do we need to shuffle a deck of cards to make it “perfectly” random?
8. Game theory
   • Nim
   • Tower of Hanoi
   • Rock, paper, scissors
   • Tic-tac-toe
   • Prisoners’ dilemma, travelers’ dilemma, pirate game, three-person duel
   • Auctions
   • Nontransitive dice
   • Conway’s angel problem
   • Games on graphs: cops and robbers, sprouts
   • Voting and apportionment methods
9. Codes and cryptography
   • Check digits
   • Substitution ciphers
   • Data compression
   • Error-correcting codes
10. Philosophy of mathematics
    • The nature of mathematical proof
    • Is mathematics discovered or invented?
    • Foundational issues of mathematics

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