Before opening a solution, write your answer and reasoning. In puzzles 1–6 every person is always truthful or always a liar: truthful people make true statements and liars make false ones. A false conjunction need not have both parts false. In puzzles 9–14 each person has exactly one job and each job belongs to one person: use only the clues, not stereotypes.
Truth-tellers, liars and deduction
Both liars?
A says: “We are both liars.” B says nothing. Who tells the truth?
Show solution and reasoning
A cannot be truthful, since that would make A a liar. A therefore lies: they are not both liars. B is truthful.
The same type
A: “B is a liar.” B: “We are the same type.” Identify both.
Show solution and reasoning
If A were a liar, B would be truthful but would falsely claim they were the same type. Thus A is truthful and B a liar; B’s statement is indeed false.
Three voices in a circle
A: “B is truthful.” B: “C is truthful.” C: “A and B are both liars.”
Show solution and reasoning
The first two statements force A, B and C to have the same type. If all were truthful, C would be false; if all were liars, C would be true. No consistent assignment exists: detecting impossibility is the solution.
Exactly one truth-teller
Exactly one of A, B and C is truthful. A: “B is truthful.” B: “C is a liar.” C: “A is a liar.”
Show solution and reasoning
A being truthful would make B truthful too. B being truthful would make C a liar, yet C would tell the truth about A. Thus C is truthful; A and B lie, satisfying every condition.
The “if” statement
A says: “If I am truthful, then B is truthful.” Use classical implication: it is false only when its premise is true and its conclusion false.
Show solution and reasoning
A cannot lie: making the statement false would require A to be truthful and B a liar. Therefore A is truthful; the premise holds and B must also be truthful. This relies on the stated logical meaning of “if”.
Two doors, one question
One door leads out, the other to a dead end. Both guards know the destinations and each other’s type; one is truthful, one lies. You may ask one guard which door the other would indicate. How do you escape?
Show solution and reasoning
Ask: “Which door would the other guard indicate if I asked for the exit?” Both indicate the wrong door: the truthful guard reports the other’s lie; the liar reverses the other’s true answer. Choose the other door.
Three wrong labels
Three crates contain only apples, only oranges, or both. The labels “apples”, “oranges”, “mixed” are all wrong. You may draw one fruit without looking from one crate. Which identifies all three?
Show solution and reasoning
Choose “mixed”: its wrong label means it contains one type only. If you draw an apple, it is the apple crate; “oranges” can contain neither oranges nor apples, so it is mixed; “apples” contains oranges. Reverse the fruit names if you draw an orange.
Exactly one true inscription
A prize is in exactly one of boxes A, B, C. A says “It is in B”; B says “It is not in B”; C says “It is not in A”. Exactly one inscription is true. Where is the prize?
Show solution and reasoning
A and B are opposites, so exactly one of them is always true. C must therefore be false: the prize is in A. Check: false, true, false.
Four people, four jobs
Jobs by elimination
Anna, Bruno, Carla and Diego each have one different job: doctor, cook, teacher, engineer. Anna is neither doctor nor engineer; Bruno neither doctor nor teacher; Carla neither cook nor teacher; Diego is the engineer.
Show solution and reasoning
Diego: engineer. Bruno must be the cook, so Anna is the teacher. Carla is the doctor. Every clue holds and no other assignment works.
Two possibilities each
The four jobs are lawyer, baker, vet and pilot. Anna is lawyer or pilot, but not pilot. Bruno is baker or pilot. Carla is lawyer or vet. Diego is neither pilot nor vet.
Show solution and reasoning
Anna: lawyer. Carla: vet. Diego: baker. Bruno: pilot. Each assignment removes an option for the next person.
Age as a clue
Anna is 20, Bruno 30, Carla 40 and Diego 50. Jobs: architect, journalist, chemist, electrician. The journalist is older than the architect but younger than the chemist. The electrician is 30.
Show solution and reasoning
Bruno is the electrician. The other three must be architect, journalist, chemist in increasing age order: Anna, Carla, Diego. No assumptions about suitable ages for jobs are needed.
Four people in a row
From left to right sit Anna, Bruno, Carla, Diego. Jobs: teacher, engineer, doctor, cook. The teacher is left of the engineer; the engineer is left of the doctor; the doctor sits immediately left of the cook.
Show solution and reasoning
The order is teacher, engineer, doctor, cook: Anna, Bruno, Carla, Diego. “Left of” need not mean “immediately left of”, but four ordered roles fill all four seats here.
Jobs around the table
Around a square table sit Anna, Bruno, Carla, Diego clockwise. Jobs: programmer, cook, nurse, photographer. Carla is the photographer. The programmer sits opposite the nurse; the next clockwise seat after the programmer belongs to the cook.
Show solution and reasoning
Diego: programmer; Anna: cook; Bruno: nurse; Carla: photographer. Anna as programmer would make Carla the nurse; Bruno would make Carla the cook; Carla cannot be programmer. Only Diego remains.
One correct card out of four
Jobs: artist, baker, chemist, dentist. Carla is the dentist. Anna is artist or dentist; Bruno is baker or chemist. Four cards say “Anna artist”, “Bruno baker”, “Carla chemist”, “Diego dentist”. Exactly one is correct.
Show solution and reasoning
Carla is dentist and Anna artist, so the first card is true. “Bruno baker” must be false, making Bruno chemist. Diego is baker. The last two cards are false as required.
Paradoxes: contradiction, infinity and meaning
“This sentence is false”
Consider just the sentence “This sentence is false”. Can you consistently assign it true or false while allowing it to state its own truth status in the usual way?
Show solution and reasoning
If true, its content makes it false; if false, its assertion of falsity is correct, making it true. This is the liar paradox: those assumptions allow no consistent classical assignment. Calling the speaker a liar does not resolve it; several theories address the problem.
The impossible barber
A barber, himself a man in the village, shaves all and only the village men who do not shave themselves. Who shaves the barber?
Show solution and reasoning
If he shaves himself, he belongs to those he must not shave. If he does not, he belongs to those he must shave. The answer is not to find a second barber: no man can satisfy that rule when it also applies to himself.
When does a heap begin?
One grain is not a heap. Adding one grain seems unable to turn a non-heap into a heap. Repeating this, even a million grains would not be a heap. Which step needs examination?
Show solution and reasoning
The second premise, applied without exception to every quantity, yields the paradox. “Heap” is vague: ordinary language supplies no precise numerical boundary. A stipulated threshold makes the rule exact but does not uncover a hidden universal boundary. This is the sorites paradox, not an arithmetic error.
The full hotel with room for more
A mathematical hotel has rooms 1, 2, 3, …, all occupied. Can it house one new guest? What about a bus with guests numbered 1, 2, 3, …, without evicting anyone?
Show solution and reasoning
For one guest, move the occupant of room n to n+1, freeing room 1. For countably many guests, move each occupant to 2n, freeing all odd rooms; new guest k takes 2k−1. This is a mapping between infinite sets, not a physical procedure for a finite hotel.
Does Achilles catch the tortoise?
The tortoise starts 100 m ahead and moves at 1 m/s; Achilles runs at 10 m/s. Whenever he reaches its previous position, it has moved on. Must infinitely many stages prevent him catching it?
Show solution and reasoning
He catches it after 100/(10−1) = 100/9 seconds. Stage times are 10 + 1 + 0.1 + 0.01 + … = 100/9. Infinitely many positive terms can have a finite sum: infinitely many stages do not imply infinite time.
The ship of Theseus
A ship’s planks are replaced one by one until none remain. The old planks are then reassembled into a second ship. Which is the original ship?
Show solution and reasoning
There is no unique answer without defining “the same ship”. Continuity of history and use favours the first; identity of material favours the second. The puzzle pits criteria of identity against each other: it is not a matching puzzle with one compulsory hidden answer.
Not every puzzle asks for a name or number: sometimes the correct conclusion is that the assumptions conflict or a definition is missing. Each of the six job matchings does have a unique solution, checked against all 24 permutations.