Brain Teasers
One Hundred People at a Party
Exactly one hundred people were at a party. Each either always tells the truth or always lies. After the party, one of the one hundred people said, "I shook hands with exactly one truthteller at the party." Another person says, "I shook hands with two truthtellers," and a third says, "I shook hands with three." And so on until someone says, "I shook hands with 99 truthtellers." The hundredth and final person says, "I shook hands with no truthtellers at all!" How many truthtellers were at the party?
Answer
There was only one truthteller.Let the person who said he shook hands with one truthteller be Mr. One, the one who said he shook hands with two truthtellers be Mr. Two, and so on.
Suppose Mr. Ninety-Nine is a truthteller. This means the others are all truthtellers and he shook hands with them all. But if Mr. Zero and Mr. Ninety-Nine are both truthtellers, Mr. Zero would not have shaken hands with Mr. Ninety-Nine. So Mr. Ninety-Nine must be a liar.
Suppose Mr. Ninety-Eight is a truthteller. This means he shook hands with everyone else except Mr. Ninety-Nine. But if Mr. Zero and Mr. Ninety-Eight are both truthtellers, then Mr. Zero would not have shaken hands with Mr. Ninety-Eight. So Mr. Ninety-Eight must also be a liar.
Using similar logic, we can determine that everyone between Mr. Ninety-Seven and Mr. One is a liar.
What about Mr. Zero? Since everyone else was a liar, he really did not shake hands with any truthtellers. Therefore, Mr. Zero is the only truthteller.
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