Top pickBeginner
paradox · doors · counting
Monty Hall
Should you switch doors when Monty offers?
Play
math-lab
Each card is a short interactive experiment.
- Pick a paradox or idea
- Play with the small interactive
- See why your first intuition is often wrong
- Read the short explanation
Everything is designed to be simple: small numbers, clear visuals, and one clear “aha” moment.
No accounts needed. Just explore.
Hi, I’m Mina Nessim.
I built math-lab to explore mathematical ideas that feel wrong at first — paradoxes, fallacies, and counterintuitive results — through small interactive experiments.
Guesses that feel right
The chance shows up sooner, lands differently than you expected, or feels obvious once the ending is known.
Top pickBeginner
paradox · counting · traps
Birthday paradox
A shared day shows up sooner than you think.
PlayTop pickBeginner
rates · traps · information
Base rate fallacy
A positive test, or a rare crime, is weak if the thing itself is rare.
Play
Tue
Intermediate
paradox · traps
Boy or girl paradox
The sentence you used already changed the sample.
Play
Beginner
paradox · counting · traps
Gambler's fallacy
A run of heads does not owe you the next result.
Play
GG
Intermediate
paradox · traps · doors
Bertrand's box paradox
A gold coin is showing. How often is the other side gold too?
Play
Beginner
traps · paradox
Conjunction fallacy
Extra details can make a story feel more likely than it is.
Play
2Advanced
paradox · traps · decisions
Two envelopes paradox
St. Petersburg too. An average win can be a bad reason to switch, or to play.
Play
1/21/3Advanced
paradox · traps · decisions
Sleeping Beauty paradox
The two answers are still both in the room.
Play
Advanced
paradox · traps
Bertrand paradox
Three fair recipes give three different answers.
Play
−Intermediate
fairness · paradox · traps
Parrondo's paradox
Each game loses. Switching between them can win.
PlayIntermediate
fairness · counting · statistics
Benford's law
In real lists, 1 is a first digit more often than 9.
PlayBeginner
information · rates
Bayes' theorem
A test says someone is ill. Count everyone it says that about, not only the people who are ill.
Play
Beginner
information · traps
Hindsight bias
Once the ending is known, that path feels like the only one.
PlayBeginner
paradox · traps · counting
Missing square puzzle
The same four pieces. One picture has an empty square.
PlayIntermediate
traps · logic
1 = 2
The steps start at 1 = 1. They seem to end at 2 = 1.
PlayThe cases you happened to see
A special case is easy to take for the whole story.
Beginner
rates · traps
Inspection paradox
You did not sample the gaps the way you think you did.
PlayBeginner
paradox · traps · social
Friendship paradox
Your friends have more friends than you do.
PlayIntermediate
statistics · traps · cause
Berkson's paradox
Will Rogers too. The group you put people into can fake an improvement.
Play
Top pickBeginner
statistics · cause
Survivorship bias
You are looking at the ones that lasted.
PlayIntermediate
statistics · cause
Missing not at random
The gaps are not a random sprinkle.
PlayBeginner
fairness · social · traps
Why your line is slower
The other lines look faster because you are stuck in this one.
PlayBeginner
social · statistics
Matthew effect
Being cited makes the next reader more likely to cite it.
PlayWhen the groups and the total disagree
Every group can fall while the total still rises.
Top pickBeginner
statistics · rates · cause
Simpson's paradox
Better in each group. Together it flips.
PlayIntermediate
statistics · cause
Lord's paradox
One class, and two ways of reading the gain.
PlayIntermediate
statistics · cause
Ecological fallacy
A pattern in the average can be false for each person.
PlayBeginner
social · fairness · statistics
A fair rule, an unfair total
Everyone is treated by the same rule, and the pile at the end is still lopsided.
PlayIntermediate
paradox · counting
Gabriel's horn
The inside of each slice gets small. The skin still adds.
PlayTwo people each make a choice
The result is the pair of picks, so your best pick depends on theirs.
Top pickBeginner
games · paradox
Prisoner's dilemma
Each of you does better by defecting. Together you both lose.
PlayBeginner
games
Game of chicken
Swerving is safe. If you both refuse, you both crash.
Play
Intermediate
games
Stag hunt
The big prize needs both of you. The hare you can take alone.
PlayTop pickIntermediate
games
Nash equilibrium
A pair of choices neither of you wants to leave.
PlayIntermediate
games
Mixed strategy
Sometimes the best plan is a coin flip you meant.
PlayBeginner
games
Coordination game
You both win if you pick the same thing, and lose if you miss.
PlayBeginner
games
Battle of the sexes
You want the same evening, and different plans.
PlayBeginner
games
Matching pennies
One of you wins when the coins match. The other wins when they do not.
PlayIntermediate
games
Iterated prisoner's dilemma
The same pair plays again, and reputation gets a turn.
PlayIntermediate
games · information
Cheap talk
A message costs nothing. That does not make it believed.
PlayWhen everyone uses the same thing
Taking a little more, all using one road, or agreeing because you think the others want to, can leave everyone worse off.
Top pickBeginner
games · social
Tragedy of the commons
Each extra bite is good for you and bad for the field.
PlayBeginner
games · social
Volunteer's dilemma
Someone should do it. Each person hopes it is not them.
PlayTop pickBeginner
fairness · paradox
Braess's paradox
An extra road can make the traffic worse.
PlayBeginner
voting · social
Abilene paradox
You go because each person thinks the others want to.
PlayDividing money, or a cake
Someone can force the shares, or the pieces can be cut so nobody wants to swap.
Beginner
games · decisions
Ultimatum game
You offer part of $10. A low offer can be rejected, and then nobody is paid.
PlayBeginner
games
Trust game
You send part of $10. It is tripled, and the other player can send some back.
Play
Beginner
fairness
Fair cake cutting
You cut the cake. The other person chooses a piece first.
PlayVotes that disagree with each other
A count can look fair and still lose to another way of counting.
Intermediate
voting · a circle
Condorcet paradox
Three restaurants. Every pair has a winner, and the winners go in a circle.
Play
Advanced
voting · fair rules
Arrow's impossibility
With three or more choices, no voting rule can keep every fairness promise.
Play
Intermediate
voting · two counts
Borda vs plurality
Count only the top name, or score 2, then 1, then 0. The winner can change.
Play
Beginner
voting · second choice
Strategic voting
You want A to win. A vote for A can leave C, the one you fear, still ahead.
Play
Intermediate
voting · two parties
Ostrogorski's paradox
Each person votes for the party they agreed with more often. More people agree with Party X on the school, the taxes, and the roads. Party Y still wins.
Play
Intermediate
voting · who loses
Anscombe's paradox
Every plan, such as a park or a road, can pass, while most people lose most of the time.
Play
Intermediate
voting · a bigger group
Condorcet jury theorem
Three people answer the same three cases. If each is right a little more often than wrong, the group gets more sure.
PlayIs this choice worth taking
A good average, a larger count, or one more look can still be the wrong reason to say yes.
Intermediate
decisions · paradox
Allais paradox
A sure thing changes your pick even when the extra chance is the same.
Play
Intermediate
decisions · paradox
Ellsberg paradox
A known risk feels safer than a risk with no stated odds.
Play
Beginner
decisions
Sunk cost fallacy
The money already spent is not a reason to spend more.
Play
Intermediate
decisions · games
Moral hazard
Once the loss is covered, the careful choice gets less careful.
Play
Advanced
decisions · logic
Pascal's wager
A decision tree with an infinite branch, and a reason to distrust it.
Play
Advanced
decisions · paradox
Newcomb's paradox
One box or both, and the predictor has already packed them.
Play
Top pickBeginner
decisions
Trolley problem
One track, five people, and a switch.
Play
Beginner
decisions
Explore vs exploit
Keep the restaurant you like, or try the new one.
Play
Intermediate
decisions · counting
Secretary problem
See them one by one. You cannot go back.
PlayWhat the others have already figured out
The puzzle starts from what was said: an announcement, a message, or an answer that might be a lie.
Top pickIntermediate
logic · paradox
Unexpected hanging
The surprise disappears the moment the last day is impossible.
Play
Intermediate
logic · information
Two generals
A message is not enough. The acknowledgement is not enough either.
Play
Advanced
logic
Blue-eyed islanders
They leave on the nth night, and only because they all heard the guru.
Play
Beginner
logic
Knights and knaves
One says: We are both knaves.
PlayKeep going one small step
Replace a plank, remove a grain, or aim a rule at itself.
Beginner
logic
Ship of Theseus
Replace every plank. Is it the same ship?
Play
Beginner
logic · paradox
Sorites paradox
Remove one grain. When did it stop being a heap?
Play
Beginner
logic · paradox
Liar paradox
This sentence is false.
Play
Intermediate
logic · paradox
Russell's paradox
A bag of every bag that is not inside itself. Does that bag go in?
Play
Beginner
paradox · counting
Coastline paradox
The ends stay put. The shore still has no single length.
Play
Advanced
paradox · logic
Banach-Tarski paradox
Turn one set of points and it fills a ball. The other set fills a second ball.
PlayWhen two things rise together
Rising together still does not say which one pushes.
Beginner
cause
Reverse causation
The arrow may run the other way.
Play
Intermediate
cause
Intervention vs observation
Watching the coffee drinkers is not the same as assigning the coffee.
Play
Intermediate
cause · statistics
Ice cream and drowning
Ice cream sales and drowning both go up in summer.
PlayNo problems match.