# Puzzles

Challenging questions whose solutions have interesting twists and broader lessons

The unreasonable effectiveness of complex numbers in discrete mathGenerating functions, as applied to a hard puzzle used for IMO training.May 23, 2022

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.Dec 8, 2017

The unexpectedly hard windmill question (2011 IMO, Q2)Problem 2 from the 2011 IMO, a seemingly easy question that turned out to be ridiculously difficultAug 4, 2019

Why do prime numbers make these spirals?A curious pattern in polar plots with prime numbers, together with discussion of Dirichlet's theoremOct 8, 2019

The impossible chessboard puzzleAn information puzzle with an interesting twistJul 5, 2020

Alice, Bob, and the average shadow of a cubeA tale of two problem solvers, looking for the average area of a cube's shadowDec 20, 2021

Circle Division SolutionMoser's circle problem, and its solution.May 24, 2015

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.Dec 23, 2017