Bonus puzzles & labs

Pascal's Triangle Lab

Add the two numbers just above to build the triangle, then color odd and even numbers to reveal a fractal pattern.

Fibonacci Sequence & Spiral Lab

Watch squares sized by the Fibonacci sequence trace out a spiral, and see the ratio approach the golden ratio.

Arithmetic & Geometric Sequences Lab

Change the first term and common difference (or ratio) with sliders and see why the sum formulas work, including a compound-interest example.

Fibonacci in Nature

Adjust the angle seeds grow at and see why 137.5°, the golden angle, packs them in with no overlap.

Collatz Conjecture Lab

Halve evens, triple-plus-one odds — follow the still-unproven "3n+1 problem" step by step with sliders.

Zeno's Paradox Lab

See why an infinite geometric series can add up to a finite number, and meet the diverging harmonic series too — Achilles and the tortoise included.

Polar Graphing

Explore how polar coordinates create beautiful patterns such as rose curves, spirals, and cardioids by plotting points using only angle and distance.

Logistic Map and Chaos Lab

Observe how the simple equation xₙ₊₁ = r × x × (1 − x) can produce stability, cycles, or complete chaos depending on the conditions, and see the butterfly effect for yourself.

Pi Series Race

Compare how quickly the Leibniz and Nilakantha infinite series approach π as the number of terms increases, and see why some series converge faster than others.

Wheat on a Chessboard

Put one grain of wheat on the first square and double it on every square after that. Click your way across the board and feel just how unimaginable the 64th square becomes.

Fractal Zoom Lab

Explore the Koch snowflake, Sierpiński triangle, dragon curve, and more — zoom in and increase the depth to see self-similarity.

Spirograph Lab

Change two circles' radii and a pen position to draw spinning patterns, and see how the number ratio decides the shape.

Tessellation

Gather regular polygons around a single vertex and see which shapes tile a plane with no gaps or overlaps.

Straightedge & Compass Construction Lab

Step through six classic constructions, then take on 7 challenges with only a straightedge and compass.

Conway's Game of Life Lab

Just 4 rules, yet self-moving patterns like the glider emerge — draw your own and watch it play out.

Triangle's Five Centers Lab

Drag a triangle's three vertices freely and watch how the centroid, circumcenter, incenter, orthocenter, and excenter form and move.

Dodecahedron Net Folding Lab

Manipulate a flat net of 12 regular pentagons as it folds into a solid dodecahedron with a single slider.

Fractal Dimension Calculator

Divide a shape into smaller and smaller grid cells and measure how quickly the number of cells it passes through increases. Calculate for yourself why fractals such…

Koch Snowflake: Infinite Perimeter, Finite Area

Build a Koch snowflake by repeatedly adding pointed bumps to the sides of a triangle, and see the paradox: its perimeter grows without bound while its area remains…

Vanishing Point Perspective Lab

Explore the perspective principle artists use to create depth on flat paper by drawing one-point and two-point perspective with lines converging at a vanishing point.

Voronoi Diagram Lab

Click points to automatically divide the plane into regions based on the nearest point. Create a Voronoi diagram and explore how it can find the nearest location, such as a convenience store.

The Monty Hall Problem Lab

Play the 3-door and 10-door versions yourself and watch why switching doors doubles your odds.

Monte Carlo Pi Lab

Scatter random points inside a square and estimate pi from the fraction that land inside a circle.

Birthday Paradox Lab

See why the odds of a shared birthday pass 50% with just 23 people, with sliders and a live simulation.

Simpson's Paradox Lab

Treatment A wins in every group, yet B wins overall — pull the real data apart and see how the group mix flips the result.

Central Limit Theorem Lab

See for yourself one of the key theorems in statistics: no matter how skewed the original distribution is, repeatedly taking samples and averaging them makes the…

Conditional Probability Tree Diagram Lab

Draw situations where probabilities change step by step, such as disease testing, as a tree diagram and see why even a positive test result can correspond to a…

Pólya Urn Model Lab

Use a rule that adds one more ball of the same color after each draw to simulate how a completely random process can settle into an unpredictable bias toward one side.

Random Walk Lab

Simulate a drunk person taking random steps based on coin flips in one and two dimensions, and see how the probability of eventually returning to the origin differs.

Secretary Problem Lab (37% Rule)

When you meet candidates one by one and must decide on the spot whether to hire them, simulate how many initial candidates you should simply observe to maximize the probability of choosing the best one.

St. Petersburg Paradox

Play this probability paradox and see the contrast between a mathematically infinite expected value and the small amount people are actually willing to pay.

Platonic Solids Lab

Adjust the angle gathered at a vertex to discover why there are exactly 5 Platonic solids, then rotate each in 3D.

Möbius Strip & Klein Bottle Lab

Draw a line on a Möbius strip and fill a Klein bottle with water to see a surface with no inside or outside.

Sphere Volume Lab — Archimedes' Proof

Slice a hemisphere and a cylinder-minus-cone shape at the same height to see why a sphere's volume is (4/3)πr³.

Magic Square Lab

Fill in the blanks with 1–9 so every row, column, and diagonal adds up to the same total.

Tower of Hanoi Lab

Move a stack of disks to another peg and watch the minimum move count explode as you add disks.

The Konigsberg Bridges Lab

Try to cross all seven bridges exactly once, then find the odd-vertex rule that says when it can be done.

Benford's Law Lab

Explore Benford's Law by counting the first digits in real-world number sets such as the Fibonacci sequence and powers of 2, and see for yourself why 1 appears most…

Bézier Curve Lab

Drag the control points and see how the smooth curves used in fonts and animations are calculated.

Binary Search Number Guessing Game

Try to find a hidden number from 1 to one million by repeatedly cutting the range in half, and see for yourself why binary search is as fast as a logarithm.

Caesar Cipher Lab

Turn the alphabet wheel to encrypt and decrypt messages with the Caesar cipher used since Roman times, and see why trying just 25 possibilities makes it easy to crack.

Fair Cake-Cutting Lab

Explore the divide-and-choose algorithm for sharing a cake fairly even when two people have different preferences.

Celsius–Fahrenheit Conversion Lab

Use the slider to explore how Celsius and Fahrenheit are connected by a single linear function.

Locus of the Centroid of an Inscribed Triangle

Move one of the three vertices on a circle and see for yourself that the path traced by the triangle’s centroid is actually a smaller circle.

Packing Squares Inside a Circle

Solve a packing puzzle by placing as many non-overlapping squares as possible inside a circle, then compare your arrangement with a regular grid.

The Birth of e: Compound Interest Lab

Explore how the final amount approaches a limiting value as interest is compounded more frequently, revealing the natural constant e.

Cycloid Lab: A Circle Rolling Inside Another Circle

Create the star-shaped and flower-like curves traced by a point on a small circle as it rolls without slipping inside a larger circle.

Dijkstra Shortest Path Lab

Watch an animation showing, step by step, how Dijkstra’s algorithm finds the shortest path between cities on a map.

Distance–Speed–Time Formula Triangle Lab

Use a simple triangle mnemonic to find distance, speed, or time when the other two are known.

Divisibility Rules Lab

Learn how to tell whether a number is divisible by another number just by looking at its digits, without doing the division yourself.

Egyptian Fractions Lab

Ancient Egyptians used only fractions with a numerator of 1. Enter any fraction and see how it can be broken down into a sum of distinct unit fractions.

Eight Queens Lab

Try solving the classic puzzle of placing 8 queens on a chessboard so that none can attack another, and discover that there are 92 solutions.

Epidemic Spread Simulation (Exponential Function Lab)

Simulate how an infection spreads across a grid, and see the explosive exponential growth at first followed by a slowdown in the graph.

Eulerian Path Checker

Draw your own graph and instantly check whether it can be drawn in one continuous stroke without lifting the pen, using Euler's rule based on the number of odd-degree…

Exam Mistake Checklist Lab

Explore common types of mistakes where the calculation seems right but the answer still goes wrong by spotting the errors in incorrect solutions.

Fake Coin Balance Puzzle

Solve a classic logic puzzle by finding the one heavier fake coin with just a few weighings on a balance scale.

Fermat’s Last Theorem: A Mini Story

Search for yourself to see that there are no natural numbers satisfying aⁿ+bⁿ=cⁿ when n is 3 or greater, and find out why Pythagorean triples exist only when n=2.

15 Puzzle (Sliding Puzzle) Lab

Try solving the 15 puzzle by sliding the numbered tiles into order, and discover the parity of permutations: depending on how the puzzle is shuffled, exactly half of…

Fraction Comparison Shortcuts Lab

Compare two fractions instantly without finding a common denominator, using cross multiplication or the 1/2 benchmark.

Fraction–Decimal Memory Tricks Lab

Learn an efficient way to memorize frequently used fraction–decimal pairs by grouping fractions with the same denominator into families.

Galton Board Lab

See for yourself how repeatedly bouncing beads randomly left or right off pegs hundreds of times eventually produces a bell-shaped normal distribution.

Gambler’s Fallacy Lab

Use a simulation to see why the common belief that the other side becomes more likely after the same side appears repeatedly is wrong.

Gambler's Ruin Problem

Simulate the gambler's ruin problem and see for yourself why, even in a fair coin game, the player with less starting capital is much more likely to go broke first.

Golden Ratio Rectangle Checker

Enter the width and height of objects around you to see how close they are to the golden ratio (about 1.618), and watch the golden spiral being drawn.

Half-Life Lab

Simulate hundreds of atoms randomly decreasing by half each half-life and see why the exponential decay curve always takes this shape.

Heron’s Formula Lab

Explore Heron’s formula and see how to calculate a triangle’s area using only the lengths of its three sides, without knowing any angles.

The Secret of the Honeycomb: Hexagon Optimization Lab

Isoperimetric Inequality Lab

Kaleidoscope Symmetry Drawing Lab

Draw in just one section and watch rotational and mirror symmetry automatically repeat your lines into snowflake-like kaleidoscope patterns.

Knight's Tour Lab

Find a path for a chess knight that visits every square on the board exactly once using only L-shaped moves, and watch the computer solve it automatically.

Knot or Not? Lab

Look at tangled curves and decide whether they are true knots or shapes that can actually be untangled, then use three-coloring to prove that a true knot really is a…

Stochastic Fractal Tree Simulation

Create a natural-looking fractal tree that grows like a real tree using only a rule where the angle and length vary slightly at each branch, adding a touch of randomness.

Laser Reflection Puzzle Lab

Place mirrors using the rule that the angle of incidence equals the angle of reflection, and solve the puzzle by sending the laser precisely to the target.

Linear Programming Profit Maximization Lab

Explore the point that maximizes profit within the feasible region that satisfies multiple constraints. The optimal solution always occurs at a vertex of the region.

Mandelbrot Set Zoom Exploration Lab

Zoom in on the Mandelbrot set again and again to see for yourself how the self-similarity of fractals keeps producing similar patterns no matter how far you zoom in.

Markov Chain Weather Lab

Explore a Markov chain as a weather model in which tomorrow's weather depends only on today's weather. Simulate many days and watch the long-run proportions of each…

Math Tips for Faster Calculations

A collection of practical math tips that you do not have to know, but that can make calculations much faster. Explore divisibility rules, mental math techniques, and…

Maze Generator Lab

Watch an algorithm randomly break down walls to create a maze, then automatically find the shortest path from the start to the finish.

Mental Math Division Tricks Lab

See how dividing by 5 or 25 can be turned into multiplication for much easier mental math.

Mental Math Multiplication Tricks Lab

Learn quick mental-math tricks for multiplying by 11, multiplying by 5, and squaring two-digit numbers without a calculator.

Peg Solitaire Puzzle

Solve the classic logic board game yourself: jump over pegs on a cross-shaped board, removing them one by one until only one remains.

Polygon Diagonal Count Formula Lab

Change the number of sides of a polygon and instantly calculate how many diagonals it has using the formula.

Powers of Two Memory Lab

2, 4, 8, 16, 32… Memorizing powers of two is useful in everything from counting subsets to understanding computer storage sizes.

Measure Height with Shadows Lab

Recreate Thales’ method of measuring a towering building like a pyramid using just a stick, its shadow, and similar-triangle ratios.

Simple vs. Compound Interest Lab

See how the gap between simple and compound interest grows over time, even with the same principal and interest rate.

Slide Rule Lab

Before calculators, slide rules used two logarithmic scales and a cursor to multiply numbers. Explore the principle of logarithms by operating one yourself.

4D Cube (Tesseract) Lab

Project a 4D cube, or tesseract, onto a 2D screen and rotate it to get an intuitive feel for a shape one dimension beyond the cube.

Huffman Coding Compression Lab

Experience Huffman coding compression by giving shorter codes to frequent characters and longer codes to rare characters, reducing file size.

Knapsack Problem Lab

Choose items for a weight-limited knapsack to maximize total value, then compare your choice with the computer's optimal solution in this hands-on knapsack problem experiment.

Map Coloring Lab (Four Color Theorem)

Explore the Four Color Theorem by coloring a map yourself and see how four colors are enough to color adjacent regions differently.

Minimum Spanning Tree Lab

See how to connect several villages with the shortest possible total road length through a Kruskal algorithm animation.

Sorting Algorithm Race Lab

Watch Bubble Sort, Selection Sort, and Quick Sort race to sort the same bars, and see why their speeds differ by comparing the number of comparisons.

Mental Percentage Math Lab

Learn how 10%, 5%, and 1% can be combined to calculate almost any percentage mentally.

Square-Cube Law Lab

Explore the square-cube law: when an animal or object is scaled up by 2, its surface area becomes 4 times larger while its volume becomes 8 times larger, and see why elephant legs are so thick.

Trilateration Lab

Explore the principle behind GPS by using only distances from three stations to infer a hidden position.

Vector Addition Lab

Drag two vectors and add them using the parallelogram method, while seeing their components and magnitude calculated in real time.

Traveling Salesman Problem Lab

Try solving the traveling salesman problem yourself — the shortest route that visits every city once and returns home — and feel how the number of possible routes explodes as cities are added.

24 Game Lab

Drag the four operations between four numbers and try to make exactly 24.

Four Fours

Use the digit 4 four times, with square roots, factorials, and grouping, to build any number from 0 to 32.

Mini Sudoku Lab

Solve 4×4 and 6×6 sudoku puzzles by process of elimination — no number repeats in a row, column, or box.

Tangram Lab

Drag, rotate, and flip seven classic pieces to build a square, a triangle, or a shape of your own.

Number Baseball Lab

Gather strike and ball clues to deduce a secret number made of different digits.

Number Detective

Check clues one at a time and cross off suspects by process of elimination to find the secret number.

Venn Diagram Set Operations Lab

Drag two circles over each other and watch in real time which region intersection, union, and difference each refer to, and how many elements land in each one.

The Game of Nim Lab

Take turns removing stones from piles and find the unbeatable winning strategy using the XOR operation.

Tic-Tac-Toe Winning Strategy Lab

Play tic-tac-toe against a perfect computer and discover for yourself the game-theory conclusion that when both players play their best, the game always ends in a draw.

Prisoner's Dilemma Lab

Choose whether to cooperate or defect and explore this classic game-theory example: why do players end up defecting even when mutual cooperation benefits both?

Optical Illusion Lab

Match five classic illusions with a slider, then verify the real values — a hands-on lesson in why intuition can mislead.

The Finger Multiplication Trick

Fold down one finger and instantly get the answer to a times-9 multiplication — see why the trick works.

Number Base Lab

Group the same count of stars by 2s, 3s, 5s, or 10s and see how the number written changes with the base.

Curious Digit Numbers Lab

Palindromes, happy numbers, narcissistic numbers — numbers that follow surprising rules just from playing with their own digits.

Divisor Chain Numbers Lab

Perfect numbers, amicable numbers, sociable numbers — repeatedly summing proper divisors sometimes loops back to the starting number.

Elliptic Curve Cryptography Lab

From the curious rule for adding points on an elliptic curve to the hard problem that turns it into a real-world cipher.

RSA Encryption Lab

Build a key from two tiny primes, encrypt a number and decrypt it back, and see why factoring is the whole defence.

Huge Numbers & Infinity Lab

From googol and googolplex to Hilbert's Hotel and Cantor's diagonal argument — infinity comes in different sizes.

Prime Number Mysteries Lab

Twin primes, the Goldbach conjecture, Fermat primes — explore famous unsolved problems about prime numbers.

Curious Number Properties Lab

Kaprekar's constant 6174, figurate numbers, Mersenne primes and perfect numbers — patterns that appear the more you calculate.

Special Numbers Lab

π, e, the golden ratio φ, and transcendental numbers — see exactly what makes each of the world's most famous numbers special.

Catalan Numbers Lab

Matching parentheses, triangulating polygons — see why totally different-looking problems all land on the same Catalan numbers.

Why You Cannot Divide by Zero Lab

Gradually bring the divisor closer to 0 and see, in two different ways, why you can never divide by 0.

GCD & LCM Venn Diagram

Prime-factorize two numbers and place the factors in a Venn diagram to see how the overlap gives the GCD and the whole diagram gives the LCM.

There Are Infinitely Many Primes: Euclid's Proof Lab

See for yourself that no matter how long a list of primes you make, you can always find a new prime that is not on the list.

Hilbert's Infinite Hotel

Explore the paradox of an infinite hotel that can keep accepting new guests even when infinitely many rooms are already full, and discover the surprising properties of infinity for yourself.

Kaprekar Constant 6174 Lab

Try any four-digit number with digits that are not all the same and repeat the process of sorting the digits high-to-low and low-to-high, then subtracting. You will always reach 6174.

Modular Arithmetic Lab (Day Calculation)

See how to calculate the day of the week 100 days from now using only a remainder operation, with a circular day-of-the-week clock.

Palindrome Numbers Lab

Experience the fascinating process of repeatedly adding a number to its reversed form to create a number that reads the same forward and backward.

Perfect, Deficient & Abundant Number Classifier

Discover the mysterious perfect numbers whose proper divisors add up to the number itself, and see how they differ from deficient and abundant numbers.

Square Root Estimation Lab

Learn a quick way to estimate a square root without a calculator by locating it between nearby perfect squares.

Speed Math Challenge

A 60-second time-attack game — solve as many problems as you can and build combo streaks for bonus points.

Mathematician Story Cards

Flip through 100 stories about Gauss, Archimedes, Ramanujan and more, and uncover the math concept hiding in each one.