How He Won the Office of Grandvizier

A king wanted to appoint a grandvizier. Three candidates competed for the king’s nomination to this high office, but the king was unable to decide which one of them was the most intelligent, and the most suitable for the post. At last, he decided to subject them to a competition, the winner of which would be chosen as grandvizier.

The king summoned the three candidates and told them that he had prepared five balls, three of which were white and two were black. He said he would place one ball on the turban of each of the three men. Each of them would be able to see the balls on the heads of the other two, but would have to guess the colour of the ball placed on his own head. The first one of them to make the correct guess would be appointed grandvizier.

The king then placed one ball on the turban of each of the candidates, whom he asked in turn to guess the colour of the ball placed on his head. Two of the men were unable to make any guess at all, and remained silent. The third man then answered the question of the king and correctly guessed the colour of the ball on his own head. The question of course is: What colour did the third man mention? How was he able to make a correct answer?

The third man said: ‘I have a white ball on my head’. He had guessed correctly and the king asked him to explain how he was able to make a correct answer. The successful candidate explained that he first discarded the hypothesis that the king would place two black balls on the heads of the candidates, because the third man would, on seeing them, immediately guess that he had a white ball on his head. Next he eliminated a second possibility, namely that the king would place a black ball on the head of one of the candidates. To arrive at this conclusion he reasoned as follows: if the black ball were placed on the head of one of the three men, this would simplify the guesswork of the other two candidates who, seeing a black ball on the head of the first man, and discounting the hypothesis that the king would place two black balls on the head of two of the candidates, would immediately conclude that they had white balls on their heads. With these two hypotheses being eliminated, it was evident, therefore, that the king could only have placed white balls on the heads of all three candidates.

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