Logic, Truth-tellers and Liars Problems
These are classic logic puzzles involving a group of individuals, some of whom always tell the truth and others who always lie. The goal is to deduce who is who, or ascertain a specific fact, based on their statements. These require careful, systematic deduction.
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Question
My father always tells the truth, and my son always lies. Is there a question you can ask them both, so that they will give identical answers to that question?
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Question
A tourist arrived at an island inhabited by truth-tellers and liars. Truth-tellers always tell the truth, and liars always lie. The tourist is traveling with a local guide, and they see a farmer working in the field. The tourist asked the guide to find out if this man is a truth-teller or a liar. The guide spoke with the farmer and then said to the tourist: "He said that he is a truth-teller."
A. What can be deduced about the farmer?
B. What can be deduced about the guide?
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Question
One day, Harry Potter found a strange notebook in which the following one hundred sentences were written:
"In this notebook, there is exactly one sentence that is false."
"In this notebook, there are exactly two sentences that are false."
"In this notebook, there are exactly three sentences that are false."
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"In this notebook, there are exactly one hundred sentences that are false."
Are there any true sentences in this notebook, and if so, how many? Justify your answer!
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Hans the Brave and the Cruel Law
In a magical kingdom, there once lived a cruel king. Around his palace was a bridge, and he placed a guard on the bridge who received the following orders: he must ask everyone who crosses the bridge why they have come here. If the person answers falsely, he must be hanged, and if he answers truthfully – he must be beheaded.
One day, Hans the Brave needed to pass through this place. When the guard asked him: "Why have you come here?", Hans gave such an answer that the guard was forced to release him. What exactly did Hans say to the guard?
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Three Friends
There are three friends, Weiss, Schwartzman, and Rotenberg. This trio of friends is special because one of them is blonde, one is ginger (redhead), and one has black hair. One day, the blonde guy said to Schwartzman: "We are a very special trio of friends! Notice that it's not only that the last names of the three of us mean colors, but also that each of our last names does not match his hair color."
What hair color does each of these three people have?
Note: "Schwartzman" means "black man", "Rotenberg" means "red mountain" and "Weiss" means "white" (from Yiddish).
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Question
On an island live liars and truth-tellers (truth-tellers always tell the truth, and liars always lie). What question should you ask a random person from the island's inhabitants to find out if they keep a crocodile as a pet?
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Who Broke the Glass?
One of the following four students: Avi, Benny, Gili, and Danny – broke a glass. The principal asks them who did it. Here are the answers she received:
Avi: I know for certain that whoever broke the glass, it was not me or Benny.
Benny: I know for certain that whoever broke the glass, it was not me or Danny.
Gili: I know for certain that whoever broke the glass, it was not me or Benny.
Danny: I know for certain that whoever broke the glass, it was not me or Avi.
It is known that only one of them broke the glass, and it is also known that three of the students told the truth, and one lied. So who broke the glass?
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Hans the Brave and the Cruel Law
In a magical land, there once lived a cruel king. Near his palace was a bridge, and he placed a guard on the bridge who received the following orders: he must ask everyone who crosses the bridge why they have come there. If the person answers a lie, he must hang him, and if he answers the truth – he must behead him.
One day, Hans the Brave needed to pass through this place. When the guard asked him: "Why have you come here?", Hans gave such an answer that the guard was forced to release him. What exactly did Hans say to the guard?
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The Magic Octopuses
In the magic sea live octopuses who can talk. Each octopus either always tells the truth or always lies. One day
the following conversation took place between four octopuses, Avi, Benny, Gidi, and Danny:
Avi: I am a green octopus
Benny: I am not green
Gidi: All green octopuses are liars
Danny: Only a green octopus can be a liar
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It is known that only one of these four is a liar, and the rest are truthful.
a. Who is the liar among the four friends? Explain!
b. Is it possible to know what his color is? -
Circle of Liars - The Truth Claim
In a circle, `n` people are seated, each of whom is either a liar or a truth-teller.
The people are looking towards the center of the circle. A liar always lies, and a truth-teller always tells the truth.
Each of the people knows exactly who is a liar and who is a truth-teller.
Each of the people says that the person sitting two places to their left (that is, next to the person sitting next to them), is a truth-teller.
It is known that in the circle there is at least one liar, and at least one truth-teller.
a. Is it possible that `n = 2017`?
b. Is it possible that `n = 5778`?
(Solution format: "word, word" for example "cat, puppy")
Sources:Topics:Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction Logic -> Truth-tellers and Liars Problems- Gillis Mathematical Olympiad, 2017-2018 Question 1