Mathematics

Bits of maths that turned out to be more interesting than they first looked.

Geometry

• Pythagorean tiling (Article) — the two-square floor pattern that quietly contains a proof of the Pythagorean theorem.
• A Geometric View of the Square Root Algorithm (Article) — A. Grzesina rebuilds the pencil-and-paper square root method as areas: squaring 249 becomes a square cut into seven regions, and extracting √64,009 becomes a matter of subtracting those rectangles and squares back off, one digit at a time, until 253 falls out.

Game theory

• The Secret Garden of Rock Paper Scissors (Article) — how allowing ties and lopsided odds turns RPS into a whole family of balanced games with real strategy.

Sequences

• Integer sequence (Article) — the entry point to the On-Line Encyclopedia of Integer Sequences, where every named sequence has an ID (Fibonacci is A000045). The idea worth carrying away is the counting argument: computable sequences are countable while integer sequences are uncountable, so almost every sequence that exists can never be generated by any algorithm — and “definable” turns out to be no better a rescue, since there’s no absolute way to define what definable means.

Problems

• Project Euler (Website) — 997 archived problems meant to be solved by writing code, published steadily since 2001. The archive shows how many people have cracked each one, which doubles as a difficulty gauge: 860,391 solvers on problem 1, and the numbers fall away sharply after that.

Combinatorics

• Combinatorics and Graph Theory (Book) — David Guichard’s free online textbook covering counting and generating functions through graph theory, from Euler circuits and coloring to Pólya–Redfield counting.

Category theory

• Category theory: online lecture notes, etc. (Reference) — Peter Smith’s annotated shelf of free and legitimately available category theory resources, ordered by difficulty: the opening chapters of Goldblatt’s Topoi as the gentlest way in, then Leinster’s Basic Category Theory, then Riehl’s Category Theory in Context, followed by selected lecture notes, videos, and readings for philosophers.
• Introducing Category Theory (Book) — Smith’s own 510-page introduction, free as a PDF and deliberately slow-paced: Part I on what lives inside a category (products, quotients, limits, exponentials), Part II on functors, natural transformations, the Yoneda lemma, and adjunctions, Part III on elementary toposes.

Statistics

• Monte Carlo method (Article) — Stanisław Ulam dreamed it up at Los Alamos after wondering about the odds in a solitaire game, then reached for randomness to crack a neutron-diffusion problem ordinary math couldn’t touch. It’s named for the casino, on Nicholas Metropolis’s suggestion, because Ulam’s uncle used to borrow money from relatives to gamble there.

Seeing Theory (Website) — a visual introduction to probability and statistics in six interactive D3 chapters, built by Daniel Kunin and collaborators at Brown — every concept is something you drag, roll, or resample rather than read.

• Basic Probability (Interactive) — chance events, expectation, and variance, demonstrated with coins and dice you actually throw.
• Probability Distributions (Interactive) — discrete and continuous random variables, ending with a hands-on central limit theorem.
• Bayesian Inference (Interactive) — watch a prior turn into a posterior as the evidence arrives.
• Regression Analysis (Interactive) — least squares, correlation, and ANOVA, with a line you fit by hand.

Magazines

Plus Magazine (Website) — Cambridge's free online maths magazine, published by the Millennium Mathematics Project since 1997 — articles, podcasts, videos, and puzzles running from the maths of everyday life to current research, indexed by topic from Fermat's last theorem to epidemiology and maths in music.

• Maths in a minute (Series) — one mathematical idea per post, explained in about the time it takes to read it.
• Puzzles (Interactive) — the magazine’s puzzle archive, solutions published after the fact.
• Collections (Series) — themed bundles of articles, such as the set marking a century of quantum mechanics.
• The DAG behind the data (Article) — how directed acyclic graphs let you tell cause from mere correlation, and why adjusting for the wrong variable (a mediator or a collider) quietly biases your result instead of clearing it up.