Second-guessing Puzzles

Good Puzzles are forever. Here are some on a single theme – knowledge of others’ knowledge. The first one is at least 45 years old, read it in बहुरंगी कर्मणूक (Bahurangi Karmnuk). The second one is given in “Elements of Discrete Mathematics” by C. L. Liu, McGraw Hill (1987). The Third one is adapted from “Impossible – surprising solutions to counterintuitive conundrums” by Julian Havil, Princeton University Press (2008).

1. The king summoned the three best mathematians in the kingdom to the palace. The king told them that he had placed either a red or white hat on each one’s head; they may look at, but not talk to, one another (they cannot see their own hats). Each one must raise his hand if he sees at least one red hat on the others, and to lower his hand once he deduces the color of his own hat. First one to deduce the color of his own hat will be rewarded.
Now all three have been given red hats, and each one raises his hand after seeing the others. But none lower their hands for several minutes. Then the smartest one of them lowers his hand, and announces that he has a red hat. How did he deduce this?

2. The king summoned the best mathematicians in the kingdom to the palace. The king told them that he had placed white hats on some and red hats on the others; they may look at, but not talk to, one another (they cannot see their own hats).  The king would leave the room and return every ten minutes. Every time he returns, he wants the mathematicians who have deduced that they are wearing white hats to come up and inform him.
If there are n white hats and m red hats, predict what is going to happen.

3. The king summoned the two best mathematicians in the kingdom to the palace. The king whispered a natural number in each mathematician’s ear. He told them that he had given them two consecutive small natural numbers. They were not to communicate with each other. The king would leave the room and return every ten minutes. Every time he returns, he wants the mathematicians who have deduced the other’s number to come up and inform him.
If the two numbers are n and n+1, predict what is going to happen.

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