Logic puzzles
Constraint-based problems, deductive reasoning, and clear step-by-step solution strategies.
New section
Recreational Math
A space for logic, puzzles, numerical curiosities, and elegant challenges explored with a creative mindset.
Paradoxes and curiosities
Surprising ideas, intuitive geometry, and thought experiments that keep mathematics playful.
Number challenges
Sequences, combinatorial games, and classic problems revisited in a modern, accessible format.
Coming soon
This section is ready to grow with new posts, interactive exercises, and thematic mini-paths.
First puzzle
The Three Ages Problem
Two mathematicians meet again after many years. The first says: “I got married and I have three children.”
The other asks: “How old are they?”
The first replies: “The product of their ages is 36 and the sum of their ages is equal to the house number across the street.”
The second looks at the number, thinks for a moment and says: “With this information I still cannot determine their ages.”
Then the first adds: “The oldest has blue eyes.”
At that point the second exclaims: “Now I know how old your children are.”
Question: how old are the three children?
Show answer
Let us list all positive integer triples whose product is 36:
1, 1, 36→ sum 381, 2, 18→ sum 211, 3, 12→ sum 161, 4, 9→ sum 141, 6, 6→ sum 132, 2, 9→ sum 132, 3, 6→ sum 113, 3, 4→ sum 10
Since the second mathematician still cannot decide after seeing the house number,
the sum must be ambiguous. The only repeated sum is 13, coming from
1, 6, 6 and 2, 2, 9.
The clue “the oldest has blue eyes” implies there is one unique oldest child,
so it cannot be 1, 6, 6 (which has two oldest children aged 6).
Final answer: 2, 2, and 9 years old.
Second puzzle
The Crossroads of Truth-Tellers and Liars
Statement
An explorer lands on an island divided into two villages.
In the first live truth-tellers, who always tell the truth; in the second live liars, who always lie.
The two groups speak the same language, have the same appearance, and cannot be distinguished externally in any way.
At a crossroads, the explorer knows that one road leads to the truth-tellers’ village and the other to the liars’ village. He meets one inhabitant, but does not know from which village that person comes.
If he can ask only one question, how can he identify with certainty which road leads to the truth-tellers?
Show solution, variants, and logic note
Classic solution
The most elegant question is:
“If I asked you which road leads to the village of truth-tellers, which one would you point to?”
The explorer should then follow the indicated road.
Why it works
If the person is a truth-teller, they answer honestly and indicate the correct road.
If the person is a liar, they lie about what they would say and, precisely for that reason, still end up indicating the correct road.
So in both cases the final indication is the same.
Equivalent variant
Another question that works in exactly the same way is:
“If I asked someone from your village which road leads to the village of truth-tellers, which one would they indicate?”
In this case as well, the explorer must follow the indicated road.
Explanation
If the person is a truth-teller, someone from their own village would tell the truth and indicate the correct road.
If the person is a liar, someone from their own village would lie and indicate the wrong road; but the liar, when reporting that answer, lies again and ends up indicating the correct road.
Famous textbook variant
The most famous formulation is:
“If I asked someone from the other village which road leads to the village of truth-tellers, which road would they point to?”
In this case, however, the answer you get is the wrong road. The explorer must therefore take the other one.
Why
If the person is a truth-teller, they will honestly report what someone from the other village (a liar) would say, and therefore indicate the wrong road.
If the person is a liar, they will lie about what someone from the other village (a truth-teller) would say, and again indicate the wrong road.
The two cases still coincide, but now they produce the indication opposite to the correct one.
Yes/No variant
The explorer may point to one road, for example the left one, and ask:
“The left road leads to the village of truth-tellers if and only if you are a truth-teller?”
- If the answer is yes, the left road is the correct one.
- If the answer is no, the correct road is the right one.
This variant is more refined, but less immediate.
Key idea
The logical core of the puzzle is to construct a question that neutralizes lying.
This can be done in two ways:
- ask what the person themselves (or someone from their own village) would say: in this case, you can follow the answer;
- ask what someone from the other village would say: in this case, you must invert the answer.
Common mistake
The most frequent mistake is believing that all similar formulations directly give the correct road.
That is not the case:
- question about the person themself → follow the answer;
- question about someone from the same village → follow the answer;
- question about someone from the other village → take the opposite road.
Final formula
Recommended question
“If I asked you which road leads to the village of truth-tellers, which one would you point to?”
Action to take
Follow the indicated road.