# Visual Math

> 73 free interactive explainers you can run, step through and break. Each page starts with a plain-English explanation, then shows the real math and runnable code, with live labs that compute every number they show.

Pages state when a model is simplified ("toy model", "idealized") and label typical or approximate real-world figures as such. Weather, physics and sound pages show imperial units by default with a metric toggle.

## Math & science

- [000 N-Queens](https://visualmath.dev/n-queens/): Place N queens so none attack: backtracking, pruning and the search space, step by step.
- [001 Golden ratio](https://visualmath.dev/golden-ratio/): φ in Fibonacci ratios, sunflower seeds, continued fractions and the pentagram.
- [002 Graphs](https://visualmath.dev/graphs/): Shortest paths, minimum spanning trees and the bridges of Königsberg.
- [003 Number theory](https://visualmath.dev/number-theory/): The sieve of Eratosthenes, the Collatz conjecture and Euclid’s algorithm.
- [004 Dynamic programming](https://visualmath.dev/dynamic-programming/): Edit distance, the 0/1 knapsack and Pascal’s triangle as tables that fill themselves.
- [005 Recursion & fractals](https://visualmath.dev/recursion-fractals/): The Tower of Hanoi, self-similar shapes and a zoomable Mandelbrot set.
- [006 Probability](https://visualmath.dev/probability/): Monty Hall, Buffon’s needle and the Galton board.
- [007 Bachet’s weights](https://visualmath.dev/bachet/): Weigh 1 to 40 with four weights: balanced ternary and why 1, 3, 9, 27 can’t be beaten.
- [027 Pi](https://visualmath.dev/pi/): Squeeze π between polygons, throw darts at it, race infinite sums to it, find it hiding in primes and bell curves, then explore a million of its digits.
- [028 Rubik's cube](https://visualmath.dev/rubiks-cube/): Turn a cube, count its 43 quintillion positions, treat moves as permutations with orders, and solve the 2×2 perfectly by searching every position.

## Coding problems

- [008 FizzBuzz](https://visualmath.dev/fizzbuzz/): Branch order, modulo clocks and custom rule sets.
- [009 Palindromes](https://visualmath.dev/palindromes/): Two pointers, expand around center and reverse-and-add.
- [010 Fibonacci](https://visualmath.dev/fibonacci/): The exploding call tree, memoization, iteration and fast doubling.
- [011 Two Sum](https://visualmath.dev/two-sum/): Every pair versus one pass with a hash map.
- [012 Roman numerals](https://visualmath.dev/roman-numerals/): Read with one look-ahead, write greedily, explore 1 to 3999.
- [013 Rotate an array](https://visualmath.dev/rotate-array/): Shifting, three reversals and cyclic replacement in gcd(n, k) cycles.
- [014 Missing number](https://visualmath.dev/missing-number/): Gauss’s sum, XOR cancellation and other routes compared.
- [015 Validate a BST](https://visualmath.dev/validate-bst/): Range bounds, in-order traversal and insertion order.
- [016 Sudoku solver](https://visualmath.dev/sudoku/): Backtracking, smarter cell order and constraint propagation.
- [017 Lowest common ancestor](https://visualmath.dev/lowest-common-ancestor/): Recursive LCA, the BST split point and binary lifting.
- [018 Serialize a tree](https://visualmath.dev/serialize-tree/): Encode a tree as a string, rebuild it, and compare formats.
- [019 Topological sort](https://visualmath.dev/topological-sort/): Kahn’s algorithm, DFS finishing order, cycles and counting orders.
- [020 Regex matching](https://visualmath.dev/regex-matching/): The DP table, memoized recursion and catastrophic backtracking.
- [021 Alien dictionary](https://visualmath.dev/alien-dictionary/): Recover an unknown alphabet with a topological sort.

## Game theory

- [022 Prisoner's dilemma](https://visualmath.dev/prisoners-dilemma/): Dominant strategies, the Nash equilibrium nobody wants, and what changes when the game repeats.
- [023 Guess ⅔ of the average](https://visualmath.dev/beauty-contest/): Iterated reasoning drives the equilibrium to 0. Real crowds stop a few levels short.
- [024 Centipede game](https://visualmath.dev/centipede-game/): Backward induction says take the money at once. People pass anyway, and a little doubt explains why.
- [025 Nim](https://visualmath.dev/nim/): P and N positions, the nim-sum, Sprague–Grundy values and the golden ratio in Wythoff’s game.
- [026 Mixed strategies](https://visualmath.dev/mixed-strategies/): Rock-paper-scissors and penalty kicks: randomize so your opponent is indifferent.

## Data & statistics

- [D00 Polls & margins of error](https://visualmath.dev/polls/): Why a thousand people can speak for millions, what ±3% really means, and the bias no margin can fix.
- [D01 Misleading charts](https://visualmath.dev/misleading-charts/): Truncated axes, stretched pictograms, cherry-picked windows and dual axes, with the lie factor measured.
- [D02 Simpson's paradox](https://visualmath.dev/simpsons-paradox/): Better in every group and worse overall: weighted averages, lurking variables, and when to split the data.
- [D03 Correlation vs causation](https://visualmath.dev/correlation/): Pearson's r, confounders that fake a link, chance correlations, and why randomized experiments settle it.
- [D04 Medical tests & Bayes](https://visualmath.dev/medical-tests/): Why a 99% accurate test can be wrong most of the time, Bayes' theorem drawn, and the ROC trade-off.

## Weather & sky

- [W00 Hurricanes](https://visualmath.dev/hurricanes/): The Coriolis effect, angular momentum, the cube law of wind power and forecast cones.
- [W01 Tornadoes](https://visualmath.dev/tornadoes/): The funnel as a pressure map, vorticity tilting, damage asymmetry and the radar couplet.
- [W02 Highs & lows](https://visualmath.dev/pressure-systems/): Pressure gradient force, Coriolis and friction: why wind circles highs and lows, and how fronts lift air.
- [W03 El Niño](https://visualmath.dev/el-nino/): The Pacific seesaw, Kelvin waves crossing the ocean, and the delayed oscillator behind ENSO.
- [W04 Global circulation](https://visualmath.dev/global-circulation/): Hadley, Ferrel and polar cells, hydrostatic balance, and why the Hadley cell stops near 30°.
- [W05 Thunderstorms](https://visualmath.dev/thunderstorms/): Parcel theory and CAPE, latent heat as the engine, and how cold pools organize storm lines.
- [W06 Weather models](https://visualmath.dev/weather-models/): The primitive equations, grids and the CFL limit, and why chaos caps forecast range.
- [W07 Lightning](https://visualmath.dev/lightning/): Charging a cloud, the stepped leader as a fractal, and counting the seconds to thunder.
- [W08 Rainbows](https://visualmath.dev/rainbows/): Rays inside a raindrop, the 42° minimum deviation, and why the colors split.
- [W09 Sky color](https://visualmath.dev/sky-color/): Rayleigh’s 1/λ⁴ law, sunset path lengths, and why clouds are white.
- [W10 Tides](https://visualmath.dev/tides/): Gravity gradients raise two bulges; the Sun and Moon beat into spring and neap tides.
- [W11 Jet streams](https://visualmath.dev/jet-streams/): Thermal wind builds the jet; Rossby waves bend it into the ridges and troughs that steer your weather.
- [W12 Sea breezes](https://visualmath.dev/sea-breeze/): Land heats faster than water, the pressure tilts, and an afternoon breeze blows in from the sea.
- [W13 Lake-effect snow](https://visualmath.dev/lake-effect/): Cold air over a warm lake: instability, fetch and wind direction decide where the snow bands land.
- [W14 Monsoons](https://visualmath.dev/monsoons/): A continent-sized sea breeze: seasonal heating, winds that turn at the equator, and mountains that wring out the rain.

## Everyday physics

- [E00 Bridges & arches](https://visualmath.dev/bridges/): The hanging chain's catenary, arches in pure compression, and the forces inside trusses and suspension bridges.
- [E01 Why bikes stay up](https://visualmath.dev/bicycles/): Steering into the fall, trail and the self-steering front wheel, and what gyroscopes do and don't do.
- [E02 Traffic jams](https://visualmath.dev/traffic/): Phantom jams from one tap of the brakes, the flow–density diagram, and why zipper merging works.
- [E03 Levers, pulleys & gears](https://visualmath.dev/levers-pulleys/): Torque balance, block and tackle, gear ratios and bike gears: trading force for distance.
- [E04 Why planes fly](https://visualmath.dev/flight/): Wings turn air downward, the lift equation and stall, and the four forces that set how far a plane can glide.
- [E05 How sailboats sail upwind](https://visualmath.dev/sailing-upwind/): Apparent wind, sails and keels as wings, the no-go zone, and the zigzag that beats a straight line into the wind.
- [E06 Hull speed & planing](https://visualmath.dev/hull-speed/): A boat makes its own bow wave and gets trapped in it. Why longer boats are faster, and how planing hulls break free.
- [E07 Why sailboats don't tip over](https://visualmath.dev/sailboat-stability/): Weight down through the center of gravity, buoyancy up through the center of buoyancy, and the twist between them that rights the boat.
- [E08 Navigation at sea](https://visualmath.dev/navigation/): Why a nautical mile is a minute of latitude, fixing your position from bearings, and steering at an angle to cross a current.
- [E09 Tides for sailors](https://visualmath.dev/tides-for-sailors/): Read a tide curve, check the rule of twelfths, find when there's enough water to cross a bar, and time a passage to ride the tidal stream.
- [E10 Tennis physics](https://visualmath.dev/tennis/): Why topspin dips, where the racket's two sweet spots are, how each court changes the bounce, and how little time a 130 mph serve leaves you.

## Sound & music

- [S00 Waves & harmonics](https://visualmath.dev/waves-harmonics/): Standing waves on a string, why instruments sound different, and the beats you hear when two notes almost match.
- [S01 Scales & tuning](https://visualmath.dev/musical-scales/): Why simple ratios sound sweet, the Pythagorean comma, and the twelfth root of two behind every piano.
- [S02 The Fourier transform](https://visualmath.dev/fourier/): Wind a signal around a circle, split any drawn wave into frequencies, and redraw pictures with spinning circles.
- [S03 Doppler & sonic booms](https://visualmath.dev/doppler/): The falling pitch of a passing siren, the Mach cone of a sonic boom, and radar guns that measure speed.
- [S04 How MP3s shrink audio](https://visualmath.dev/audio-compression/): Sampling and aliasing, bit depth and noise, the sounds you can't hear, and transform coding.

## How computers work

- [C00 Bits & bytes](https://visualmath.dev/bits-bytes/): How a computer stores everything as switches: binary, hex, signed numbers, size prefixes, UTF-8 text and floating point.
- [C01 The CPU](https://visualmath.dev/cpu/): Logic gates that add, a toy processor you can step through fetch, decode and execute, and how pipelines and cores go faster.
- [C02 Memory & RAM](https://visualmath.dev/memory/): Numbered boxes, little-endian bytes and pointers, the memory hierarchy at human scale, and why caches reward neighbors.
- [C03 How the internet moves data](https://visualmath.dev/internet/): Packets and checksums, shortest-path routing, reliable delivery over a lossy network, and how far light gets in fiber.
- [C04 Encryption & hashing](https://visualmath.dev/encryption/): SHA-256 and the avalanche effect, agreeing on a secret in public with Diffie–Hellman, and RSA built from number theory.

## AI & machine learning

- [C05 How machines learn](https://visualmath.dev/machine-learning/): Learning as rolling downhill: gradient descent, neurons and backpropagation, and the line between memorizing and learning.
- [C06 Embeddings & vector search](https://visualmath.dev/embeddings/): Meaning as coordinates: cosine similarity, semantic search, and how vector databases find neighbors without checking everything.
- [C07 How a language model writes](https://visualmath.dev/llm/): Tokens, next-word probabilities, temperature and sampling, attention, and the loop that turns them into text.

## Optional

- [Sitemap](https://visualmath.dev/sitemap.xml)
- [Home](https://visualmath.dev/)