• How to solve it? The table from G. Polya's book "How to solve it" (in English and in Russian)
• 1. Pigeonhole principle.
• 2. Proof by contradiction.
• 3. Induction.
• 4. Extremal principle.
• 5. Invariants.
• 6. Semi-invariants.
• 7. And how to solve it?
• 8. Solving problems about problem solving
• 9. Smile!
• 10. Remainders and leftovers
• Some questions - with hints and solutions.

Resources:

• 1. 500 Questions on Pigeonhole principle on problems.ru (in Russian)
• 4. Hermit Crabs trade shells (5 min BBC video)

Geometries

• 1. Inversion.
• 2. Are there different infinities?
• 3. How to use inversions?
• 4. Spherical geometry
• 5. Hyperbolic geometry
• 6. Geometries: overview
• 7. How to see in 4 dimensions?
• 8. One day of a four-dimensional student
• 9. Geometric games
• 10. Questions around tilings

Resources:

• 1. Inversion:
  • Animation demonstrating Inversion in circles, by Malin Christersson.
  • Lipkin's Inversor, on Mathematical Etudes.
• 2. Different types of infinities:
  • Where is more?, by Sergei Tabachnikov.
• 4. Spherical Geometry:
  • Spherical Droste video, 2-minute video by Henry Segerman.
• 5. Hyperbolic Geometry:
  • Make hyperbolic tilings of images, by Malin Christersson.