Recognising several parameters at once is a challenge for primary-school students — and often for secondary students too! This lab uses cards with 3–4 different attributes, similar to those in the game Set. It combines simple combinatorics exercises with attention practice. No prior knowledge is needed: the lab builds everything from scratch. It is a great warm-up before playing our Trios game, and it is highly recommended for developing patience and focus.

Did you know that a mathematical mirror can create flowers, stars, and mandalas? This lab takes primary-school students from their very first reflection of a point all the way to drawing their own mandala. It is a gentle lab with beautiful animations and visual effects. We recommend pairing it with our Tank Battle game.

Can you draw a figure without lifting your pencil and without retracing any line? This question, which sounds like a game, hides a mathematical secret almost 300 years old. This lab is designed for primary-school students and takes them from tracing a simple house all the way to solving the famous Königsberg bridges problem that Leonhard Euler unravelled in 1736.

Mazes have always had an irresistible pull — they are a natural challenge with an element of adventure! This lab is designed for primary-school students (though perfectly suited for secondary school too) and takes them on a journey that starts by walking through famous real-world mazes and ends with students designing their own. Along the way they discover that not getting lost is not a matter of luck: there are methods that always work.

What if mazes had numbers on the floor? In this lab a rabbit must cross grids full of digits to reach its carrot, but it can only step on squares that are multiples of the current number. Then the adventure gets harder: the path is no longer a grid but a series of branching operations where each choice adds, subtracts, or multiplies your score. Only one exact route leads to the goal.

Working with coins or counters develops spatial perception, planning and an understanding of rules — and it's fun! In this lab the student has to transform one configuration of coins into another using different procedures: jumps, slides and flipping coins, across 3 different types of grid.

A polyomino is a shape made by joining equal squares along full sides. The lab follows the series by size: dominoes (2 squares), trominoes (3), tetrominoes (4) and pentominoes (5), across 23 rooms. The student drags pieces onto a board (they can be rotated and flipped), paints cells to discover shapes, and answers multiple-choice questions or types numbers. There are no bets or simulated series: everything is solved by handling the pieces.

In this lab the student never picks between options to continue a series: they build it with their own hands. They switch on lamps, paint cells, lay matchsticks, or remove cubes to construct the next step of a visual sequence, and the lab itself checks whether what was built is correct. When they get it right, the earlier steps are hidden and they are asked for one more step based only on the one they just made: getting it right once can be luck, getting it right twice in a row is not.

A primary-school lab (ages 8-12), very visual: everything is done by painting squares, triangles, honeycomb cells, numbers and countries with a click. Each room sets a colouring rule and asks the student to follow it, to count how many ways there are, or to decide whether it can be done at all.

A primary-school lab (ages 8-12) on parity invariants. Everything is hands-on: turning cups over, rubbing numbers off a board, making a playing card somersault, swapping tiles and, at the end, a magic trick. Every block has challenges that work and one that does not; students find out by pressing “It is impossible”, and an animated explanation shows them why.

What are functions? They may be the most important concept in mathematics, and yet few students ever truly understand them. Through real-life examples like exam scores and metaphors like a postman delivering letters, we introduce the ideas of domain, range, and value. Students will learn that a function is a transformation, explore domain and range visually, learn to graph affine functions, parabolas and hyperbolas by following a step-by-step algorithm, and meet the main families taught in secondary school and upper secondary: affine, quadratic, cubic and rational.

Studying functions by hand is tedious — drawing a single graph can take fifteen minutes. The goal of this lab is to help students experiment with the parameters of the linear equation so they can visualise them directly. Why does a line go up, go down, or cross the axis exactly there? In 12 rooms we explore the full anatomy of y = m·x + n, from watching a line take shape in real time to deducing its equation from just two points. It is designed for secondary-school students encountering linear functions rigorously for the first time.

Why can a curved slide be faster than a straight one, even when it is longer? This question obsessed Galileo, the Bernoulli brothers, and Newton. The lab starts with that broken intuition — the straight line is not the fastest — and guides secondary-school students to discover the exact answer: the cycloid. Along the way a second equally beautiful surprise appears: the tautochrone property.

What curve does a chain hanging from two points form? At first glance it looks like a parabola — Galileo thought so too — but Huygens proved him wrong at the age of 17. This secondary-school lab traces that historical mistake, uncovers the true curve (the catenary), and follows it through bridges, arches, and minimal surfaces.

Why do we age at such different rates? A mouse barely makes it to 3 years, while its naked mole-rat cousin is potentially immortal. This secondary-school lab invites students to explore the mathematics behind biological ageing: from survival curves to cellular automata, through feedback loops and oxidative stress.

Why does it pour with rain in Galicia while Almería bakes in the heat? What turns a Mediterranean low into the feared DANA? This secondary-school lab puts students at the controls of the Iberian Peninsula's weather: they drag pressure centres, adjust sea temperature, and watch real atmospheric physics react to their decisions. They can even create their own hurricane (a very realistic simulation in room 3)!

You've just walked into a rather peculiar game room… In front of you are machines that convert one kind of object into another. How do you reach your goal? Is it always possible?

"I thought of a number, added 3, and got 15. What number did I think of?" With that hook, this secondary-school lab turns a guessing game into a first real encounter with linear equations in one unknown. Across 15 rooms of gradually increasing difficulty, students learn through riddles that grow into equations.

Word problems are the bridge between mathematics and real life, but many secondary-school students don't know where to start. This lab trains exactly that skill: reading a problem, choosing a variable, translating each sentence into an algebraic expression, and building the key equality that leads to the solution.

This is one of the most visually striking labs because it generates beautiful, colourful images. Designed for secondary-school students, it starts from the simplest possible question — "What happens if I connect points on two segments?" — and leads students, room by room, to discover parabolas, hyperbolas, cardioids, astroids, and more.

What do a rumour, a virus, and a wildfire have in common? How can we model their behaviour? We'll see that graphs and cellular automata create plausible mathematical models for studying and predicting the behaviour of spreading phenomena. And we'll end by making strategic decisions to save a village from fire!

We all have secrets we want to protect. How? By encrypting our messages.

How does a virus spread from one person to thousands? This lab uses COVID-19 as its through-line to help secondary-school students build, step by step, the real mathematical models used by epidemiologists. It starts with a grid of people infecting each other at random and ends with three crisis-management missions — flu A, Ebola, and COVID — where students decide when to lock down, how many vaccines to distribute, and who should get them.

How can a school of fish move in perfect unison with no one leading the way? Or how does an ant colony organize itself without anyone giving orders? This secondary-school lab explores eight families of emergent-rule phenomena — from fireflies to cellular automata and fractals — to show that complex order can arise from ridiculously simple local rules applied repeatedly across thousands of agents. This lab is worth highlighting in particular for its first room, where fish magically begin to form shoals; the third room, where fireflies synchronize; and room 23, where ants find food and start carrying it back to the nest. Each simulation is followed by an explanatory room where students must apply the rules locally themselves.

Why does traffic sometimes stop dead and then flow again a while later, with no crash, roadworks or traffic light? The answer isn't an external cause: the jam organises itself. This secondary-school lab (with extensions for upper secondary) follows the phenomenon from the famous Sugiyama experiment to the mathematical models that explain it, showing that a jam can emerge purely from too many cars and the small human imperfections of driving.

From counting factors to graphing curves. The lab, with 23 rooms, starts with the properties of powers, continues with the exponential function and ends with the logarithm as its inverse question. It is split into three blocks: powers (rooms 1-11), the exponential function (12-16) and the logarithm (17-23).

In this lab the student does not read about probability: they play it. They toss real coins and dice (with physics and all), bet before each experiment and then check whether their decision was a good one. Each student starts with 10 Ᵽ; a won bet adds 5 Ᵽ and a lost bet takes away 3 Ᵽ. The bet is settled with a long series (normally 100 rounds), so a good decision pays off almost every time; in one room, the Grand Duke’s, the series is deliberately short and misleading, and the lab says so at the end. The Ᵽ points make choosing with care matter: betting by gut feeling and betting after counting have different consequences, and the student sees it in their purse.

What is (and isn't) a function? Before working with formulas and specific families, it's worth understanding the idea of a correspondence itself: one element of the domain, exactly one image. Students will learn to tell functions apart from "non-functions" using finite sets and real graphs, meet the injective, surjective and bijective properties (including how to count them), and learn to find the domain of a real function, both from its graph and from its formula.

Can a fully deterministic system be unpredictable? In 1961, meteorologist Edward Lorenz discovered that the answer is yes: tiny differences in initial conditions produce radically different trajectories. This lab takes upper-secondary and university students to explore that phenomenon first-hand, moving from the Lorenz attractor to the double pendulum and population dynamics.

What is information? How much does it "weigh"? Which codes are better? Can we correct errors in the transmission of information? This upper-secondary/university-level lab walks through information theory from its foundations to error-correcting codes, following in the footsteps of Hartley, Shannon, and Hamming. It consists of 22 rooms with a total estimated duration of about half an hour. We recommend using it alongside our game Oráculo Mentiroso.

Why does a leopard have spots and a zebra have stripes? It is not random: it is mathematics. This upper-secondary lab explores Turing's reaction-diffusion model (implemented with the Gray-Scott equations) and shows that two substances diffusing and interacting at different speeds are enough to generate, from scratch, the patterns that cover animal skin. Throughout the rooms students learn to create patterns by tuning the physical reaction parameters, and finish by painting the silhouettes of real animals.

The derivative is the most important idea in calculus, and it is almost always introduced all at once as a hard-to-digest limit. This lab builds it the other way round: first the intuition —"how fast does this function rise?"— and only then the formula. Everything happens on a 1:1 grid, so the student can estimate slopes by eye, counting squares, before computing anything. Across 32 rooms it walks the full path, from tapping where a curve rises fastest to differentiating with the chain rule, understanding how a computer approximates a derivative, and solving optimisation problems. It is aimed at High school and University.

How can a machine learn to play without anyone teaching it the strategy? This lab for lower and upper secondary (ESO and Bachillerato) answers that question with the most beautiful and concrete idea in artificial intelligence: reinforcement learning. We rebuild, step by step, Donald Michie’s matchbox machine (MENACE) and Martin Gardner’s Hexapawn robot: drawers full of counters that, by rewarding the moves that lead to a win and penalising those that lead to a loss, end up discovering the perfect move on their own. At the end we take a leap: a game where there is no fixed optimal move because the best choice depends on what the other player does.

Why does cooperation emerge among selfish individuals who only seek their own benefit? This upper-secondary lab rebuilds Robert Axelrod's famous experiment (1980): a tournament where different strategies face each other playing the prisoner's dilemma over and over. From the one-shot dilemma we move to the repeated game, to the everyone-vs-everyone tournament and, finally, to evolutionary models —including spatial ones— that show how cooperation can emerge… or collapse. The conclusion is surprising: the winning strategy is not the cleverest or the most aggressive, but a very simple one based on reciprocity.

The geometry you learned in school isn't the only one possible: it depends on the shape of the space you live in. How would an inhabitant discover this if they could never leave their world to see it "from outside"? This upper-secondary lab answers with a recurring character — the angle-counting ant, who can only measure angles and distances — and travels through three complete geometries (flat, spherical, hyperbolic) across 24 rooms, using real 2D and 3D renders (Three.js) instead of formulas alone.

What would a world with four spatial dimensions look like? Starting from a question that seems impossible, this Bachillerato lab builds the answer step by step, using the same trick a 2D being would use to imagine 3D space: dimensional analogy and projecting shadows.

A short course in creative programming with p5.js, the JavaScript library for drawing and animating on a canvas. The student starts by watching an animated drawing that takes only a few lines of code and, room by room, learns the pieces used to write something like it: setup() and draw(), shapes, colours, variables, conditions, loops, text, custom functions and arrays. It ends with two complete sketches, a particle system and a recursive fractal tree.

How does an autonomous spaceship "see" its surroundings, and how does it learn to dodge danger without anyone hand-coding explicit rules? This upper-secondary lab tackles supervised learning from the opposite end of the "Neural Networks" lab: instead of designing weights by hand, the student designs an agent's perception and then trains it by playing. Across 7 short rooms, the arc runs from "what information does an agent need?" to watching a neural network navigate on its own, thanks to examples the student generated themselves.

Foxes, rabbits and equations that pulse. Why do predator-prey populations oscillate instead of settling down? And why did fishing *less* during World War I lead to *more* sharks? This upper-secondary lab builds the Lotka-Volterra model from the ground up: starting from individual agents that move and reproduce in a meadow, moving through differential equations, all the way to deterministic chaos in three-species systems. Across 17 rooms split into four blocks, students go from watching a simulation to designing and perturbing their own dynamical system.

Sine and cosine are defined from a point that turns on the circle of radius 1: cos θ and sin θ are its coordinates (x, y). The lab starts there and, over 25 rooms, reaches the right triangle, the notable angles, the graph as a function, frequency and amplitude, sound, the ellipse, the double angle and polar coordinates. It is a Bachillerato lab, with drawings the student moves and challenges that are solved by calculating.

Recognize a function by its graph, sketch it by hand following a fixed method, and read off the graph what happens to a function near a tricky point or as x goes to infinity. Aimed at High school and University, it assumes the student already knows the derivative (the `derivada` lab) and uses it here as just another tool, not as the object of study.

Are there more whole numbers than even numbers? Can a full infinite hotel take in infinitely many more guests? Can a set have length zero and still contain as many points as the whole line? This Bachillerato/university lab, outside the curriculum, goes through set theory and Georg Cantor’s infinite cardinals: from the idea of a bijection (the same one shepherds used to count sheep with stones) to the strange Cantor set, by way of three classic paradoxes of infinity (Zeno, divergent series, the Ross-Littlewood vase) and Hilbert’s hotel.

A 24-room lab in which the student learns the logical operators by building them. Almost every room uses the same board: switches, bulbs and logic gates that can be dragged, placed without overlapping and connected by tapping two ends of different pieces (tapping again disconnects). Each operator is presented in three ways at once: the circuit (no current is 0, current is 1), true or false statements (1 is truth and 0 is a lie) and, for conjunction and disjunction, sets. The formal name and the symbol of each operator (∧, ∨, ¬, →, ↔, ⊕) are stressed throughout.

How do you get the most out of resources that will not stretch to everything? A workshop making tables and chairs with limited carpentry, upholstery and assembly hours runs through the lab, which covers the graphical method in 17 rooms: setting up the model, drawing the feasible region and finding the optimal vertex, both when maximizing and when minimizing. The simplex method and the dual problem are covered in a separate lab, *The simplex method*. Aimed at upper secondary.

How can someone send you a secret message using a key that everyone can see? That paradox is RSA — the algorithm that still protects internet communications today. This upper-secondary/university lab walks the full journey: from the simple Caesar cipher all the way to generating public and private keys, building up every piece of modular arithmetic and number theory that makes the magic work.

Neural networks are the engine behind voice recognition, machine translation, and language models. This lab guides upper-secondary/university students from a single artificial neuron to a multilayer network capable of learning on its own, building every piece with pencil and calculator before watching it work on screen. Across 29 rooms in eight sections, the journey goes from "what is a neuron?" all the way to training a real network with backpropagation.

Can a single mathematical idea predict the weather, your next WhatsApp message, which web pages matter, and whether Bitcoin will keep falling tomorrow? Yes — and that idea is the Markov chain: what happens tomorrow depends only on what happens today. This upper-secondary lab covers 20 rooms, moving from concrete intuition (your own mood, Barcelona's weather) all the way to real-world applications like PageRank and financial time series.

When we only know a handful of values of a function — a table of measurements, a few points from an experiment — how do we estimate what happens between them, or near them? This is a classically dry topic when presented as pure formulas, so this lab builds it the other way round: first the visual intuition — guessing by eye, dragging points, watching curves move live — and only afterwards the symbolic machinery. Across 29 rooms full of sliders, drags and animations, it walks the full path from the tangent line to the cubic splines used by every design program. Aimed at University level (Calculus).

How much is the area under a curve worth? Before any formula, this lab starts by counting grid squares by eye. From there it moves to Riemann sums — the rectangles that approximate that area under different criteria — and watches how, as their number grows, two different bounds converge to the same value: the definite integral. With that idea already built, the lab opens up to its applications: signed area, area between curves, and two less obvious uses of the same "slice and add" idea — the length of a curve and the volume of a solid of revolution. Designed for University (Calculus).

And what about problems with so many variables that you cannot draw them any more? This lab picks up where Linear optimization ends: there the optimum was found by looking at the feasible region, and here we build the machinery that finds it without seeing it — the simplex method — and then turn the whole problem inside out to reach shadow prices and the dual problem. The same workshop of tables and chairs runs through the first 23 rooms, and always ends at €36; the last room applies everything to a complete three-product problem, like those in exams. Aimed at University level.

How do you look for the lowest point of a landscape you cannot see? The optimizer does not have the formula: it can only point at a spot and ask how much the function is worth there, and each question costs — in a real problem it can be an eight-hour simulation. This lab goes through seven ways of spending those questions, from throwing random darts to swarms and evolving populations, and makes them compete on the same budget over very different terrains. Second of the Optimization trilogy, after *Linear optimization*. Aimed at university level.

There are problems we cannot solve, and the vans go out every morning all the same. This laboratory starts with the travelling salesman problem — how many routes there are, why we cannot even check that one is the best, and how the error can be bounded — and then looks at five creatures that solve versions of it without any individual knowing the problem: ants, termites, a slime mould, bees and the workers of an ant nest sharing out their work. Everything is measured with a fixed budget, against a bound and against classical methods, and the result is reported even when it does not flatter the creatures. Third in the Optimization trilogy, after *Linear optimization* and *Heuristic optimization*. Designed for University; blocks A and B work at upper secondary level.

A university lab on the second half of the continuity and convergence topic: from the ε-δ definition to uniform convergence, passing through the topological characterisation. The material is the usual content of a first topology course; what the lab contributes is that each definition is manipulated before it is written down.

A university lab on the numerical solution of equations, built around a single question that no room ever abandons: I want to solve this equation and I want to do it cheaply.

A university lab in Optimisation and Discrete Mathematics about a single problem that turns up in five disguises: pairing up students who get along, putting cameras on the streets of a neighbourhood, choosing a different delegate for each club, sharing documents among translators and assigning students to internships. Blocks A, B and F work from upper secondary school.

First lab of the group theory series, for upper secondary school or first-year university. It is followed by “Cosets and quotients”, “Homomorphisms and automorphisms” and “Centre, commutators and solvable groups”.

Third group theory lab, for first-year university. It follows “Cosets and quotients” and studies maps between groups that respect the structure: what is lost, what is kept, and when two groups are really the same.

Last group theory lab, for first-year university. It follows “Groups you can see”, “Cosets and quotients”, “Homomorphisms and automorphisms” and “Centre, commutators and solvable groups”, and uses what is learnt there (permutations, commutators, solvable groups) to prove that there is no formula in radicals for the general equation of degree five.

A university-level lab (first years) or advanced upper-secondary lab, about the curve *y*² = *x*³ + *ax* + *b*. It is very visual: almost everything happens by dragging points on the curve and watching how the line through them moves.

A group theory lab for first-year university students. Braids are a group you can touch: you weave them from crossings, add them by stacking one under the other and undo them with their mirror image. The group is infinite, non-abelian and has a presentation with only two kinds of relations, so it revisits almost everything in the course (generators, presentations, homomorphisms, kernels, quotients, conjugation, centre, abelianisation) with a fresh example. At the end, closing the braids produces knots, and the invariants that tell them apart turn out to be homomorphisms.

Topology lab for second-year undergraduates. A topology is a list of sets called open sets, and nothing more: the lab has the student build it by hand before using it. First on a set of three points (●, ■, ★) drawn on a triangle, where each open set is a potato that surrounds its points; then on the line and the plane, with the topologies that appear in every course: standard, Sorgenfrey, right rays, cofinite, discrete and lexicographic.