Math problems
Math paradoxes · probability

Puzzles where intuition gets it wrong

The answers to these puzzles feel wrong even after they're proven. So instead of just explaining them, each one lets you play it, test it with thousands of simulations, and take it apart several ways, from a table to a formula.

Pick a puzzle Start with Monty Hall
  1. 1Try itPlay and note what your gut says.
  2. 2Test itRun thousands of experiments in the browser.
  3. 3Get itPick the explanation that clicks for you.

The puzzles

Each puzzle has its own page: the setup, a game, a simulation, five explanations and harder variants.

Probability · 3 doors

The Monty Hall problem

You're on a game show and pick one of three doors. The host, who knows where the car is, opens another door to reveal a goat and offers a switch. Should you take it?

Intuition50 / 50
Actually2/3 to switch

The most famous puzzle about conditional probability. In 1990 thousands of readers, including mathematicians with PhDs, insisted the correct answer was wrong.

  • the scene from 21
  • a three-door game
  • 10,000-game simulation
  • five explanations
  • 20 and 100 doors
Work through the Monty Hall problem →
Try it here

Exponential growth · 30 days

The duckweed lake puzzle

Duckweed doubles every day, and one plant covers a lake in 30 days. How long do two plants take?

Intuition15 days
Actually29 days

The classic lily pad problem about exponential growth and doubling: why half the lake fills only in the last day.

  • guess the answer
  • a lake filling up
  • curves for 1–1,000 plants
  • five explanations
  • bacteria in a bottle
Work through the duckweed puzzle →
Try it here

Combinatorics · 365 days

The birthday paradox

How many people must be in a room before two of them probably share a birthday?

Intuition≈ 183 people
Actually23 people

A puzzle about how fast the number of pairs grows. The same idea powers the "birthday attack" on cryptographic hashes.

  • guess the answer
  • a room on a calendar
  • 10,000-room simulation
  • five explanations
  • PINs, hashes, triples
Work through the birthday paradox →
Try it here

Bayes' theorem · medical tests

The false positive paradox

1% of people have a disease. The test is 99% accurate, and you test positive. What is the chance you're really ill?

Intuition99%
Actually50%

A puzzle about the base rate fallacy: when a disease is rare, even an accurate test raises as many false alarms as true finds.

  • guess the answer
  • a crowd of 1,000
  • 100,000-test simulation
  • five explanations
  • retests, spam, fraud
Work through the false positive paradox →
Try it here

Optimal stopping · 1/e

The 37% rule: the secretary problem

100 candidates arrive one at a time, and you must answer on the spot. What chance of hiring the best?

Intuition≈ 1%
Actually37%

The classic optimal stopping problem: let the first 37% of candidates go, then take the first one who beats everyone before.

  • guess the answer
  • a 20-candidate game
  • simulation for any cutoff
  • five explanations
  • top three and a second chance
Work through the 37% rule →
Try it here

Statistics · a 2×2 table

Simpson's paradox

Treatment A beats treatment B for small kidney stones and for large ones. Can B be better for all the patients together?

IntuitionNo, A wins
ActuallyB: 83% vs 78%

A puzzle about how combining groups can reverse a conclusion, and why decisions need a causal picture, not just overall percentages.

  • pick a treatment
  • a 2×2 table editor
  • thousands of random tables
  • five explanations
  • baseball and Berkeley
Work through Simpson's paradox →
Try it here

Conditional probability · 2 children

The two-child paradox

A family has two children, and at least one is a boy. What is the chance both are boys? And what if it is a boy born on a Tuesday?

Intuition1/2
Actually1/3 and 13/27

A puzzle showing that the answer depends on how you got the information. Even the weekday of birth changes the odds.

  • three sentences, three answers
  • a deck of random families
  • 100,000-family simulation
  • the 14 × 14 grid
  • rare details, three children
Work through the two-child paradox →
Try it here

Geometric probability · π

Buffon's needle

A match as long as a floorboard is wide drops onto a wooden floor at random. How often does it cross a seam between boards?

Intuition≈ 50%
Actually2/π ≈ 63.7%

A random experiment with no circle in sight that produces the number π. One of the earliest examples of the Monte Carlo method.

  • guess the share
  • a floor and matches
  • a million throws
  • where π comes from
  • grids, rings, points
Work through Buffon's needle →
Try it here

Networks · averages

The friendship paradox

Who has more friends: you, or your friends on average?

Intuitionabout the same
Actuallyyour friends

Why in almost any network your friends are more popular than you: a biased sample, the formula E[d²]/E[d], and the same effect in classes, buses and epidemics.

  • guess the share
  • a 36-person network
  • networks of thousands
  • five explanations
  • classes, buses, flu
Work through the friendship paradox →
Try it here

Geometry · 40,075 km

The rope around the Earth

A rope around the equator is made 1 metre longer and lifted evenly. How big is the gap between it and the ground?

Intuitionmicroscopic
Actually15.9 cm

A puzzle about circumference: one extra metre of rope lifts it by 1/(2π) m around any sphere, from an orange to the Sun.

  • guess the gap
  • five spheres to scale
  • a chart from orange to Sun
  • five explanations
  • a 121 m tall tent
Work through the rope puzzle →
Try it here

Permutations · 100 boxes

The 100 prisoners problem

Each of 100 prisoners may open 50 of 100 boxes to find their number. All go free only if every one succeeds. What are the odds with the best strategy?

Intuition≈ 0%
Actually31.2%

A puzzle about how a random permutation splits into cycles, and how to link the prisoners' fates so they win together.

  • guess the answer
  • play a prisoner
  • 10,000-prison simulation
  • four explanations
  • a malicious warden
Work through the 100 prisoners problem →
Try it here

Psychology of thinking · 3 questions

The bat and ball: the cognitive reflection test

A bat and a ball cost $1.10 together. The bat costs a dollar more than the ball. How much is the ball?

Intuition10 cents
Actually5 cents

The three questions of Shane Frederick's test, where the fast answer feels obvious and turns out wrong. About System 1 and System 2 and the habit of checking your first thought.

  • a timed test
  • bar model and factory
  • session statistics
  • five explanations
  • new trap puzzles
Work through the cognitive reflection test →
Try it here

Probability · 50 stickers

How many stickers to fill the album

An album has 50 slots, and the stickers in the packs are random. On average, how many stickers do you have to buy to fill it?

Intuition≈ 70 stickers
Actually≈ 225 stickers

The coupon collector's problem: why the last stickers in an album cost the most, and how swapping duplicates helps.

  • guess the answer
  • an album and packs
  • 10,000-album simulation
  • five explanations
  • swaps and rare stickers
Work through the coupon collector's problem →
Try it here

Statistics · skill + luck

Regression to the mean

After praise players do worse, and after criticism they do better. Does that mean criticism works?

IntuitionYes, it works
ActuallyIt would happen anyway

A puzzle about why an unusually good result is usually followed by a worse one and a bad one by a better one, and why medical trials need a control group.

  • guess the answer
  • coach 30 players
  • 3,000 players and a luck slider
  • five explanations
  • pills, covers, heights
Work through regression to the mean →
Try it here

Probability · reasoning

The two envelopes paradox

One envelope holds twice as much money as the other. You took one. Should you switch?

Intuitionswitch: +25%
Actuallyno difference

The sum "the other holds 2A or A/2, so 1.25A" looks airtight. We find where it breaks and when switching really does pay.

  • guess the answer
  • a game with a host
  • 100,000-round simulation
  • five explanations
  • Cover's threshold, neckties
Work through the two envelopes paradox →
Try it here

Probability · 100 flips

Coin streaks

How long is the longest run of one side when you flip a fair coin 100 times?

Intuition3–4 flips
Actually≈ 7 flips

Why randomness doesn't look random: long streaks, the gambler's fallacy and the hot hand in basketball.

  • guess the answer
  • fool the detector
  • 100,000 sequences
  • five explanations
  • roulette and the hot hand
Work through coin streaks →
Try it here

Geometric probability · chords

Bertrand's paradox

A chord of a circle is drawn at random. What is the chance it is longer than the side of the inscribed equilateral triangle?

Intuitionone answer
Actually1/3 · 1/2 · 1/4

Joseph Bertrand's 1889 puzzle about why "at random" means nothing without a procedure. Three ways to pick a chord give three correct answers.

  • guess the chance
  • three ways to draw a chord
  • 100,000 chords
  • arc, radius, inner circle
  • Jaynes's straws
Work through Bertrand's paradox →
Try it here

About

This is a collection of interactive explanations of famous math paradoxes and probability puzzles: the Monty Hall problem, the birthday paradox and more. Each page combines the puzzle, a game, a simulation, several independent explanations and harder variants.

Everything runs in the browser, with no sign-up, in English and Ukrainian. The site was built together with AI, and new puzzles are added over time.

The puzzles also work well in job interviews. They show how a person reasons: whether they spot a wrong assumption, whether they are ready to doubt the obvious answer, and whether they can explain their thinking. After the conversation, the interactive page helps you go through the puzzle together.

StudentsSee probability at work, not just memorise a formula.
TeachersReady-made classroom interactives: game, simulation, explanation.
InterviewersGive a candidate the puzzle, hear their reasoning, then check the answer together with the game and the simulation.
Anyone who arguedA link that settles the argument better than words.
About the author Dmitry Kudrenko

Dmitry Kudrenko

Co-founder of

In job interviews I like to discuss logic puzzles where intuition pulls you towards the wrong answer even though there is an exact mathematical solution. Whether the candidate gets it right has never mattered much to me. What matters is whether they can question a wrong assumption in conversation, and, when they are right, how well they can explain their idea or take apart someone else's.

One day I was explaining one of the best-known of these puzzles, the one from the film 21, to my son, and later to friends. I noticed that each person needed a different explanation, depending on how they think. That gave me the idea for this project: visualise these puzzles, offer several simple explanations, and push them further for the mathematically curious.

I built the first few puzzles together with AI, just to test the idea. But I liked the project so much that it was hard to stop, so there will be more puzzles here.

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New puzzles, explanations and discussions

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