Labs

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Primary

Coin puzzles: transforming shapes
19 rooms · 19 challenges
Slide coins to turn one figure into another.
Primary Logic Puzzles
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Trios: logic with cards and attributes
22 rooms · 22 challenges
Learn to identify valid trios step by step.
Primary Combinatorics
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Drawing without lifting the pencil: Euler's walk
23 rooms · 35 challenges
Drawing without lifting the pencil: the magic rule of Eulerian graphs.
Primary Graphs
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Line symmetry and mandalas
22 rooms · 22 challenges
Explore axial symmetry and create mandalas
Primary Symmetry Geometry
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Mazes
20 rooms · 20 challenges
Walk through famous mazes around the world and learn the tricks to never get lost: right-hand rule, Tarry and Trémaux.
Primary Mazes Graphs Algorithms
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12 25 17 14 18 18 6 30 7 24 5 13 11 23 42 🐰 🥕
Mazes of multiples and operations
16 rooms · 16 challenges
Cross mazes stepping only on multiples of a number, and later by solving operations along the way.
Primary Mazes Operations
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Polyominoes
23 rooms · 40 challenges
Dominoes, trominoes, tetrominoes and pentominoes: cover shapes, prove things with colorings, count ways to fit pieces, and design your own challenge for a classmate.
Primary Combinatorics Logic Polyominoes
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Logic sequences and visual patterns
21 rooms · 51 challenges
Work out the next steps of a visual series and check it yourself.
Primary Logic Patterns
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Colouring maps: the four colour theorem
18 rooms · 58 challenges
Paint squares, triangles, numbers and countries by the rules: no two neighbours alike, two of a kind in every square, the product of two colours gets the third. Colour maps with just four colours and discover how a good colouring proves that some things are impossible.
Primary Logic Combinatorics
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Odds and evens: what never changes
11 rooms · 39 challenges
Turn cups over two at a time, rub numbers off the board, make a playing card flip across a grid and swap tiles around. Some challenges work and some do not: discover what never changes —whether a number is even or odd— and use it to prove that something is impossible. At the end, learn the heads-or-tails magic trick.
Primary Invariants Logic
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Secondary

? + 3 = 15
Linear equations: the hidden number
15 rooms · 24 challenges
Guess the number I am thinking of. Learn to read linear equations and undo operations to solve them.
Secondary Equations Algebra
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Powers, exponentials and logarithms
23 rooms · 82 challenges
From counting factors to graphing curves: the properties of powers, the exponential function that always ends up winning, and the logarithm as its inverse question.
Secondary Powers Functions
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x 1 corte
Polynomial and rational functions
19 rooms · 56 challenges
From lines to hyperbolas: graphs, domain, range and how each parameter changes them.
Secondary Functions
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la edad de mi padre es... x + 2 🧒 👨 x x + 2
Word problems: from the story to the equation
13 rooms · 33 challenges
Learn to translate a word problem into an equation. Variables, expressions and the key equality that leads to the answer.
Secondary Equations Word problems
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x y
Linear functions
12 rooms · 29 challenges
Lines y = m·x + n. Slope, intercept and graph recognition.
Secondary Functions
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Lines that make curves: envelopes and string art
15 rooms · 15 challenges
When many lines touch each other, they trace parabolas, hyperbolas, cardioids, astroids…
Secondary Curves
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Brachistochrone: the fastest curve
9 rooms · 9 challenges
Discover why the fastest curve is not the shortest.
Secondary Curves
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How epidemics spread: the SIR model and cellular automata
17 rooms · 20 challenges
How a disease spreads: from the simple SIR model to real COVID data. Along the way, cellular automata, vaccines, lockdowns and social networks.
Secondary Biology Graphs Percolation Cellular automata
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↻
Phantom traffic jams: the mathematics of traffic
13 rooms · 15 challenges
A traffic jam can appear with no crash, roadworks or traffic light. Find out why.
Secondary Chaos Modelling Cellular automata Upper secondary
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A B C D E F G H D E F G H I J K
Cipher zoo: classical cryptography
17 rooms · 20 challenges
Six historical ciphers for hiding messages: from the reversed Hebrew alphabet to the American telegraphers' zigzag. Ends with 3 camp cryptographer missions.
Secondary Cryptography
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Climate models: pressure, wind and hurricanes
8 rooms · 10 challenges
Pressure, wind, rain and hurricanes: control the Iberian Peninsula's weather with real physics.
Secondary Climate Geography Modelling
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Spreading on networks: rumours, viruses and fires
14 rooms · 17 challenges
How something spreads — a rumour, a virus, a wildfire — through a network or a map: links, hubs, and the famous threshold.
Secondary Graphs Percolation Cellular automata
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Order from chaos: patterns that form by themselves
23 rooms · 28 challenges
Four completely different mechanisms of the same principle: with only simple local rules, random noise spontaneously reorganizes into patterns. Spirals, flocks, sync, fractals.
Secondary Chaos Cellular automata Modelling
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The catenary
8 rooms · 8 challenges
The hanging chain, bridges and Gaudí's arches.
Secondary Curves Architecture
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The mathematics of ageing
8 rooms · 10 challenges
Gompertz, cells and mitochondria: the mathematics of ageing.
Secondary Aging Biology Modelling
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5 10 ?
Invariants: how to prove something is impossible
19 rooms · 22 challenges
Can you get from here to there? If not — what stops you?
Secondary Invariants Modular arithmetic
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O 2 … 7 … 12
Probability: coins and dice
21 rooms · 77 challenges
What do you bet on? Toss real coins and dice, bet before each experiment and win or lose Ᵽ points. Mark the winning cells on the two-dice square, count paths until you reach Pascal’s triangle and find out why 5 heads out of 10 is much more likely than 50 out of 100. At the end, a coin game where the computer almost always beats you… and its trick.
Secondary Probability Combinatorics
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High school

Function analysis
21 rooms · 105 challenges
From function families to their full graphical analysis: domain, extrema, limits and discontinuities, with zoom to see them up close.
High school Calculus Functions
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x y
The derivative
32 rooms · 85 challenges
From visual intuition to calculation: slopes, limits, numerical differentiation and rules.
High school Calculus Functions
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Circular Trigonometry
25 rooms · 55 challenges
Sine and cosine are born from a point going round a circle. From the unit circle to sound waves, ellipses, the double angle and polar coordinates — and along the way, how a computer draws a circle.
High school Trigonometry Functions Geometry
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4D
The fourth dimension
17 rooms · 24 challenges
Regular polyhedra exist in 4D: tesseract, hexadecachoron, icositetrachoron… Rotate their 3D shadows to imagine what they look like.
High school Geometry Fourth dimension
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Train a neural network by playing
7 rooms · 7 challenges
Design a spaceship's sensors and train it by playing: your moves become training examples for a neural network that learns to dodge bombs and collect stars.
High school AI Neural networks Upper secondary
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conejos zorros
Population models: predators and prey
17 rooms · 27 challenges
Foxes, rabbits and equations that pulse. From individual agents to three-species chaos: why populations oscillate and what they share with the three-body problem.
High school Differential equations Biology Modelling
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★
Reinforcement learning: an AI that learns to play
20 rooms · 25 challenges
How does a machine learn to play with nobody teaching it the strategy? Boxes of beads that reward and punish moves discover perfect play on their own — from the stone game to Hexapawn to a teamwork game with no single solution.
High school AI
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1 0 1 1 0 1 H · log₂
Information theory: Shannon and error-correcting codes
22 rooms · 28 challenges
How information is measured, encoded and protected from noise. Hartley, Shannon, Hamming.
High school Information Codes
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Axelrod's tournament
16 rooms · 18 challenges
Why does cooperation emerge among selfish players? From the one-shot prisoner's dilemma to a tournament of strategies, to the evolution that rewards the best scorers. Rebuild Axelrod's experiment and discover why winning every duel is not winning the tournament.
High school Game theory AI
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Non-Euclidean geometries: the sphere and the hyperbolic plane
24 rooms · 28 challenges
The geometry you learned is not the only one: it depends on the shape of space. Measure triangles on the plane, the sphere and the hyperbolic world to discover three different geometries.
High school Geometry
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function setup () {'{}'} createCanvas background (30) } function draw () { fill ( 255 ,0,0) circle (mouseX, mouseY, 30) }
Create with p5.js
35 rooms · 65 challenges
Learn to program animations and interactive images with p5.js. From your first shape to your own functions, all with live code and result.
High school Programming
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Chaos theory and the butterfly effect
10 rooms · 11 challenges
Deterministic systems you still cannot predict.
High school Chaos Physics
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f : A → B
Functions: concept, domain and range
16 rooms · 61 challenges
What is (and isn't) a function, injective/surjective/bijective, and how to find the domain of a real function.
High school Functions
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Infinity: Hilbert's hotel and the sizes of infinity
24 rooms · 71 challenges
Sheep, impossible hotels and a vase that empties itself: a journey through infinity, from the classic paradoxes to the different sizes of infinity and the strange Cantor set.
High school Logic Set theory
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Turing patterns
12 rooms · 15 challenges
Turing and Gray-Scott reaction-diffusion. Spots, stripes and mazes emerge from two substances that diffuse and react. Design the pattern of your own animal.
High school Modelling Biology
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OR
Logic: logic gates and truth tables
24 rooms · 43 challenges
Circuits, truth and lies, sets: learn the logic operators by building them.
High school Logic
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x y
Linear programming: graphical method and simplex
17 rooms · 62 challenges
Getting the most out of limited resources: from the word problem to the model, from the model to the picture, and from the picture to the simplex method and the dual problem.
High school Optimization Linear algebra
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University

☀️ ☁️ ☔ 🐸
Markov chains
20 rooms · 29 challenges
Predicting the weather, Bitcoin, your next word, and which web pages matter — all with one idea: tomorrow depends only on today.
University Probability Modelling
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y
Neural Networks
29 rooms · 34 challenges
Build, train and watch artificial neurons: weights, sigmoid, gradient descent, backprop. No manual calculation — everything is automatic, you design.
University Neural networks AI
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A B C D E D E F G H +3 RSA
RSA Cryptography
27 rooms · 27 challenges
Learn the RSA algorithm step by step: Caesar cipher, modular arithmetic and Euler's φ function.
University Cryptography Modular arithmetic
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x y
Function interpolation
29 rooms · 53 challenges
Approximate unknown values from known data: from the tangent line to Lagrange, Newton, Taylor and cubic splines.
University Calculus Functions Numerical analysis
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x y
Definite integrals and areas
18 rooms · 49 challenges
From counting grid squares to Riemann sums and the definite integral, with applications to areas, arc length and volumes.
University Calculus
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1
The simplex method
24 rooms · 89 challenges
Solving linear optimization problems without drawing them: standard form, the simplex tableau, walking the vertices and the dual problem.
University Optimization Linear algebra
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2ε 2δ
Continuity: ε, δ and topology
21 rooms · 52 challenges
The ε-δ definition played as a duel, metrics that change the verdict, sequences, and the moment continuity stops needing distances.
University Topology Calculus Functions
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Heuristic optimization
35 rooms · 98 challenges
Looking for the minimum of a function you cannot see and can only query point by point: from pure randomness to annealing, swarms and genetics, all compared on the same budget.
University Optimization Algorithms
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P Q P + Q
Elliptic curves
26 rooms · 76 challenges
A line meets the curve y² = x³ + ax + b in three points: from two of them you build the third. Drag points, add with chords and tangents, multiply a point by n and hunt rational points like Diophantus and Bachet. Discover torsion, the points that come back to the start, and Mazur’s theorem, which allows only 15 shapes. Finally, the same curve in modular arithmetic: multiplying is lightning fast and undoing the multiplication is almost impossible. The cryptography that protects the internet rests on that.
University Number theory Group theory Curves Algebra Cryptography
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a x⁵ − x + a
Galois theory through complex analysis
27 rooms · 124 challenges
Move a coefficient of x⁵ − x + a around a loop and the five roots come back to their places… but in a different order. With this idea of V. I. Arnold one proves that the quintic has no formula in radicals: loops, permutations, commutators and the group A₅, which never lets itself be reduced. Special rooms to pick your own polynomial, see its branch points, draw your own loops and compute its monodromy group. Follows lecture 5 of “Mathematical Omnibus” (Fuchs and Tabachnikov) and the book by V. B. Alekseev.
University Group theory Equations Topology Algebra
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Ant colonies: biological optimisation
35 rooms · 95 challenges
There are problems we cannot solve, and the vans go out every morning all the same. Here there is no optimizer: there are dumb agents that write the solution on the ground — ants, slime mould, bees and termites against the travelling salesman problem and its relatives.
University Optimization Algorithms Graphs Biology
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Matchings in graphs: Hall, König and Gale–Shapley
24 rooms · 97 challenges
Pairing people up, cameras on streets, club delegates, translators and company internships: the same problem five times, and each time a certificate proving it cannot be done better. Augmenting paths, König, Hall, the Hungarian method and Gale–Shapley. The rooms on concordant pairs are based on the worksheet by Vania Yakovlev (Кроссворд Тьюринга).
University Graphs Optimization Algorithms Combinatorics
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e Δ₃
Visual group theory
29 rooms · 184 challenges
A phone going back into its case, a triangle in its hole, a spinning tetrahedron: a group is the set of actions that leave an object in place. Cayley diagrams, coloured tables, cycle graphs, permutations, the symmetries of the tetrahedron, cube and icosahedron, the 2×2×2 Rubik’s cube and a zoo of small groups. Based on “Visual Group Theory” by Nathan Carter and “Abel’s Theorem in Problems and Solutions” by V. B. Alekseev (lectures by V. I. Arnold).
University Group theory Algebra Symmetry
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Z₄ Z₂
Homomorphisms and automorphisms
12 rooms · 66 challenges
Maps that respect the arrows: each element lights up and an arrow takes it to its image, and contradictions show up in red. Kernel and image, the first isomorphism theorem with the collapsing table, pairs of isomorphic groups, products, automorphisms of Zₙ and of Z₂×Z₂, homomorphisms from S₄ to S₃ and Arnold’s cube. Follows “Cosets and quotients”.
University Group theory Algebra Symmetry
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x₀
Shrinking down to the solution
16 rooms · 27 challenges
An equation that pencil and paper cannot crack, and the question of what it costs to hunt it down: bisection, golden section, fixed point and Newton compared in the only honest unit, evaluations of the function.
University Topology Numerical analysis Calculus Algorithms
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Braids: the group you weave
23 rooms · 154 challenges
Weave colourful braids with the crossings σᵢ, add them by stacking one under the other and discover that they form an infinite non-abelian group: the Artin group Bₙ. Simplify words with the braid relations, decide whether a tangle is trivial, find the Garside normal form, the kernel onto Sₙ and the centre. Finally, close the braids to get knots and links, untangle them with Reidemeister moves and tell them apart with tricolourings (homomorphisms to Δ₃) and the Jones polynomial.
University Group theory Algebra Knots
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Topological spaces: open sets, bases and comparing
16 rooms · 84 challenges
A topology is a list of sets called open. Build them on a three-point set by drawing blobs, find out what is missing when a family is not closed under unions or intersections, and collect the nine different types. Then, on the line and in the plane: the standard, Sorgenfrey, right-ray, cofinite, discrete and lexicographic topologies, their bases, and how to compare them by fitting bricks of one inside the other.
University Topology
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