Problem 27 · 2010 Math Kangaroo
Stretch
Logic & Word Problems
casework
In Tautostadt there are only nobles and liars. Every sentence spoken by a noble is true, and every sentence spoken by a liar is false. One day some of them meet in a room, and three of them speak as follows:
The first one says: “There are no more than three in this room. We are all liars.”
The second one says: “There are no more than four in this room. We are not all liars.”
The third one says: “In this room we are five. Three of us are liars.”
How many people are in the room, and how many of them are liars?
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Answer: C — four people, two of which are liars
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Hint 1 of 2
Nobles speak only truths and liars only falsehoods — a liar's whole sentence is false.
Still stuck? Show hint 2 →
Hint 2 of 2
Notice that anyone claiming everyone is a liar cannot be telling the truth; start there.
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Approach: test the speakers' truth values for consistency
- The first speaker's claim includes that everyone is a liar, which a noble could not say, so he is a liar and his whole sentence is false.
- With four people in the room, the second speaker's claim (at most four, not all liars) is true, so he is a noble, while the third speaker's claim of five people is false, so he is a liar.
- That gives a room of four people, two of whom are liars (option C).
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