Puzzles
Challenging questions whose solutions have interesting twists and broader lessons
The hardest problem on the hardest testA challenging question of geometry and probability on the Putnam. The shift in perspective used for the solution here carries with it broader loessons for problem-solving.
The unexpectedly hard windmill question (2011 IMO, Q2)Problem 2 from the 2011 IMO, a seemingly easy question that turned out to be ridiculously difficult
Five puzzles for thinking outside the boxA series of geometry puzzles with clever solutions, with a discussion about the role of higher dimensions in math.
Olympiad level counting (Generating functions)Generating functions, as applied to a hard puzzle used for IMO training.
This pattern breaks, but for a good reason | Moser's circle problemAn apparent pattern that breaks, and the reason behind it.
The impossible chessboard puzzleAn information puzzle with an interesting twist
Alice, Bob, and the average shadow of a cubeA tale of two problem solvers, looking for the average area of a cube's shadow
Darts in Higher Dimensions [Numberphile]A game of darts yields an interesting connection to higher dimensional geometry.
The three utilities puzzle with math/science YouTubersA classic puzzle in graph theory, the "Utilities problem", a description of why it is unsolvable on a plane, and how it becomes solvable on surfaces with a different topology.
Euler's Formula and Graph DualityA very clever proof of Euler's characteristic formula using spanning trees.