Brain Teasers
Cleverest in the Land
In the village of Einstown, every villager thought they were the cleverest. They used to argue about it all the time, so eventually the leader devised a clever plan. He pretended to hold a vote for who was the cleverest person, where people had to nominate other villagers to go up to the vote. Everyone in the village would know of everyone nominated except themselves, and they all knew that at least one person got nominated.
Once the nominations were over the leader announced his plan. He said that the first person to come forward and say that they were nominated with proof would be the cleverest person in the village. He told them to go to the village hall at exactly 7:00 pm once they had worked it out. None of the people told each other if they got nominated, as they all wanted to win.
After the first night nobody had come forward. Nobody came forward on the second night either. Nor the third, fourth fifth etc. Eventually on the 30th night a certain number of people came forward claiming to have been nominated. They were all right.
How many people came forward on the 30th night and why? (Assume that they all were clever enough to work the method out).
Once the nominations were over the leader announced his plan. He said that the first person to come forward and say that they were nominated with proof would be the cleverest person in the village. He told them to go to the village hall at exactly 7:00 pm once they had worked it out. None of the people told each other if they got nominated, as they all wanted to win.
After the first night nobody had come forward. Nobody came forward on the second night either. Nor the third, fourth fifth etc. Eventually on the 30th night a certain number of people came forward claiming to have been nominated. They were all right.
How many people came forward on the 30th night and why? (Assume that they all were clever enough to work the method out).
Answer
Call the number of people nominated N.Any person nominated would know of N-1 people nominated, as they would not know that they themselves were nominated. If there was 1 person nominated, the nominee would know of nobody who was nominated, and therefore would know that they were nominated. Because nobody came forward on the first night there was more than one person nominated.
Using the same logic, anyone on the second night who knew of one person nominated would then be able to work out that they were nominated as there was at least two people. Because nobody came forward on the second night there was at least three.
This carries on all the way up. Nobody came forward on the 29th night, meaning there was at least thirty. People now came forward the next night, so everyone who came forward knew of 29 nominees, meaning 30 people came forward on the 30th night.
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