Logic Puzzles

Each prisoner may open 50 of 100 boxes to find their own number. Random guessing succeeds with probability (1/2)^100; the optimal strategy does astronomically better.

One bottle in a thousand is poisoned, tasters can be sacrificed, and you may test in three successive rounds. How few tasters do you need?

Planes carry fuel for half the way round the world and can pass fuel in mid-air. How few planes are needed to get one all the way around and everyone home?

Three players guess their own hat colours with no option to pass, and the team needs a majority right. How high can a clever, coordinated strategy push the odds?

One reading from an exact scale must reveal which bags hold fakes, with any mix possible. Taking at most 30 coins per bag, how many bags can you handle?

Two ropes each burn for 60 minutes, but unevenly. Measure exactly 45 minutes: how few times must you put flame to a rope end?

Albert knows the month, Bernard knows the day, and three cryptic statements pin down the date. When is Cheryl's birthday?

Five failed guesses at a four-digit lock, each with a count of exact and misplaced digits. What is the code?

You cannot tell whether the native is honest, and you do not know which word means yes. One cleverly nested question still gets through.

Eight claims, each describing the total number of true claims on the list. Which total is self-consistent?

Alice and Bob each see the other's number but not their own. Seven times in a row nobody knows. Then Bob does. What was on his forehead?

A river has one small boat that carries one soldier or up to two boys. How many crossings does it take to ferry a whole squad and the boys?

Six islanders, one hidden spy who may say anything, and a web of claims about who is honest.

Five houses in a row, five colors, jobs, pets and drinks, eleven clues, and nobody mentions the dog. Who owns it?

Each claim has the form 'if this, then that'. A false conditional pins down both halves at once.

Five married couples must cross a river in one small boat, and no wife will tolerate another man's company without her husband. How few crossings suffice?

What is the fewest queens that can be placed on an 8×8 chessboard so that every square is occupied or attacked?

You may put weights on both pans of a balance. With only five weights to choose, what is the heaviest guaranteed-weighable range of whole-number masses?

In FORTY + TEN + TEN = SIXTY each letter is a different digit. What number is SIXTY?

Four conference speakers, four jobs, four cities, four days. Eight clues leave exactly one assignment; where does the pilot live?

The twelve-coin puzzle has an elegant three-weighing solution. Give yourself one more weighing, and ask how large the pile can grow.

Fill a 4 by 4 city block with towers of heights 1 to 4 using six sightline clues. What is the top row?

Everyone at a round table claims to have a knave beside them. How many honest people can the table seat?

A balance scale, three weighings, and one coin that is secretly heavier or lighter. How many coins can you guarantee to sort out?

Nine numbered claims that talk about one another and about the tally. Exactly one assignment of truth values works; how many claims are true?

100 prisoners in a line, each guessing their own hat color. How many can be guaranteed to survive with the right strategy?

Normal islanders may say anything at all, which makes their statements worthless and the deduction subtle.

Find a secret number from 1 to 1000 with yes/no questions, knowing the answerer may lie once. How many questions, adaptively chosen, are needed in the worst case?

At most one of 1024 bottles is poisoned, and it is a single round of tasting. The usual ten tasters turn out to be one short.

100 prisoners, one light switch, and a fatal declaration to time perfectly. How does the designated counter know when everyone has visited?

Eight runners of distinct speeds; each race is a head-to-head duel that shows who is faster. How many duels guarantee a complete ranking?

Seven players, red or blue hats, pass allowed, one wrong guess sinks the team. Remarkably, error-correcting codes let them win far more often than a single lucky guesser could.

Six people at a round table, eight clues that say only who is, or is not, beside or across from whom. Who faces Ann?

Each of six coins is independently real or fake, and the scale shows an exact total. How many weighings are needed to label every coin?

Ten islanders each make a claim about how many knaves there are. Can the claims police themselves?

On this island some inhabitants alternate between truth and lies. One of them says three things in a row.

In the muddy children puzzle every child can see the others. Here one cannot. Does the blind child's muddy forehead change when anyone first speaks up?

A visitor tells 100 blue-eyed perfect logicians something they all already "know." On which night does everything change?

Four people crossing at speeds 1, 2, 5, and 10 minutes share one flashlight, two at a time. What is the fastest total crossing?

The calendar of 2026 is identical to which later year, the first time every date falls on the same weekday again?

A camel must move 3000 bananas across 1000 km, carrying 1000 at a time and eating 1 per km. How many bananas can arrive?

Three children's ages multiply to 36 and sum to the house number next door, yet that isn't enough. One more clue settles it.

One coin may be flipped to tell your partner which square hides the key. For which board sizes can the message always get through?

An unmarked 3-gallon jug, an unmarked 5-gallon jug, unlimited water. How few steps to exactly 4 gallons?

Five hikers share one torch and cross two at a time, but the two quickest cannot stand to be alone together. How fast can everyone cross?

Five runners, no ties, six clues about who finished next to whom. Who came second?

A fly zigzags at 75 mph between two trains closing at 100 mph from 100 miles apart. How far does it fly?

One road leads to safety, one to doom, and the guard is either a truth-teller or a liar. How many yes/no questions do you need?

A wristwatch's hands coincide every 65 true minutes, but a perfect clock needs a little longer. How much does it gain per day?

Person k toggles every kth locker, for k = 1 to 100. After all the passes, how many lockers are open?

An assistant shows four of five chosen cards in a row and the magician names the fifth. How large can the deck be before no scheme can possibly work?

Five runners of distinct speeds, and only head-to-head duels. How many duels are enough to single out the one in the exact middle?

Three jars (apples, oranges, mixed) and every label is wrong. How few fruit draws guarantee correct relabeling?

Find the ten-digit number whose first digit counts its zeros, second digit its ones, and so on up to nines.

A meadow regrows grass steadily. 10 cows clear it in 24 days, 20 cows in 8. How many cows could graze it forever?

On an ordinary analogue clock, how many times a day do the hour and minute hands stand exactly at right angles?

A big round table, a simple claim about each person's two neighbours, and a number-theoretic twist hiding in the count.

Fill a 3×3 grid with 1 to 9 so every row and column sums to 15, diagonals ignored. How many fillings are there?

A knight, a knave and a spy walk into a room. Two of them confess to being the spy.

What is the smallest positive integer whose digits multiply to 10! = 3628800?

S knows the sum, P knows the product, and four cryptic statements identify two secret numbers. Freudenthal's "Impossible Puzzle."

The counter must know when every prisoner has visited the room, but nobody can tell whether the bulb started on. How long a tally is enough to be sure?

Twelve identical-looking balls, one counterfeit of unknown weight, and a balance scale. How few weighings always suffice?

Another sharing problem with an awkward set of jugs. This one forces a long shuttle of pourings before the wine splits evenly.

Twenty players of fixed, distinct strengths; each match reveals the winner. How many matches always suffice to identify both the best and the runner-up?

A man lies on a fixed set of weekdays and says two things about the days ahead.

Eight coins, three of them fakes that weigh the same as each other and a little less than the real ones. How many weighings on a balance are needed?

Three logicians draw hats from a bag of two red and three blue, then take turns saying whether they know their colour. On average, how long until someone does?

Three logicians each see two positive integers and know one of the three is the sum of the other two. Six rounds of silence later, you can work out Alice's number.

Three switches downstairs, one incandescent bulb upstairs. How many inspection trips do you need to match switch to bulb?

25 horses, a 5-horse track, no timer. How many races to guarantee finding the fastest three?

I am twice as old as you were when I was as old as you are now. Together we are 63. How old am I?

With only two identical eggs and 100 floors, how many drops guarantee you find the critical floor?

Two players each wear a hat in one of five colours, see only the other's, and must both guess their own. How likely can they make it that at least one is right?

Exactly two bottles in eight are poisoned and all testing happens in one round. The number of outcomes is easy to count, but is the count enough?

One sister lies on Monday, Wednesday and Friday, the other on the remaining weekdays. Each says something about the other's lying days.

Over one full 400-year Gregorian cycle, on how many occasions does the 13th of a month fall on a Friday?

Four teams, one round robin, only the final points table survives. Who beat the team that finished on top?

Five children, one broken vase, and a rule that exactly two of them are telling the truth.

A farmer, a one-item boat, and three mutually hungry passengers. What is the minimum number of river crossings?