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(Terry Tao version on wordpress ) A tribe of islanders: 1000 people of various eye colors, 100 of which are blue-eyed. If a tribesperson discovers his.

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Presentation on theme: "(Terry Tao version on wordpress ) A tribe of islanders: 1000 people of various eye colors, 100 of which are blue-eyed. If a tribesperson discovers his."— Presentation transcript:

1 (Terry Tao version on wordpress ) A tribe of islanders: 1000 people of various eye colors, 100 of which are blue-eyed. If a tribesperson discovers his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village for all to witness. A blue-eyed foreign visitor, not knowing the customs, addresses the entire tribe one evening: “how unusual it is to see another blue-eyed person like myself in this region of the world”. What impact does the address have on the tribe?

2 1. Nothing happens 2. All blue-eyed commit suicide 100 days after the address Assumption: All the tribes people are highly logical, highly devout, and they all know that each other is also highly logical and highly devout (and they all know that they all know that each other is highly logical and devout, and so forth).

3 The visitor has brought no new information, and they would end up killing themselves regardless. Any tribes person can imagine that such a visitor making the address without being questioned of the veracity of the speech Can the foreigner reduce or stop the casualties if he realizes his mistake on the next day or a few days after his speech?

4 A number of married couples live in a small village. Every time some husband cheats on his wife, every woman except the wife finds out about it. If one can deduce that her husband is a cheater, she must kill him. An old lady that never lies tells everyone that there is a least one cheater in the village. What happens next?

5 A related puzzle Alice and Bob each have a number glued to their forehead for the other one to see. The numbers are N and N+1 for some positive integer N Alice and Bob both know that, but they don’t know what N is and they don’t know which of them has the N and which has the N+1. They are repeatedly interrogated about their state of knowledge – first Alice, then Bob, then Alice again, and so on. For various values of N, what happens?

6 Simulated conversations Alice 1, Bob 2 A: I don’t know B: I know A: I know now Alice 2, Bob 1 A: I know B: I know too Alice 2, Bob 3 A: I don’t know B: I know A: I know now Alice 3, Bob 2 A: I don’t know B: I don’t know A: I know now B: I know now Alice 3, Bob 4 A: I don’t know B: I don’t know A: I don’t know B: I know now A: I know now The person with the big number knows first Alice 4, Bob 3 A: I don’t know B: I don’t know A: I don’t know B: I don’t know A: I know now

7 Martin Gardner “the impossible problem” Mathematical Games Two unknown whole numbers, m and n, both greater than 1, and less than 50. One mathematician, Mr. Product is given the product of these two numbers, while another mathematician, Mr. Sum is given the sum of these two numbers. The following conversation takes place: Mr. Product: (I do not know the numbers.) Mr. Sum: I knew you didn't know the numbers. (I don’t know either.) Mr. Product: Now I know the numbers Mr. Sum: Now I know the numbers, too.

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9 (Peter Winkler, Nick Reingold) Each of resident of Dot-town carries a red or blue dot on his/her forehead. If he/she figures out what color it is  commits suicide Each day the residents gather A stranger tells them something nontrivial about the number of blue dots. Prove that every resident commits suicide. (nontrivial means: something true about the number of the blue eyed that disallows at least one possible number)

10 It is assumed that the people in the puzzle (Simona, Peter and Danielle) can solve any solvable problem instantaneously. Peter, Simona and Danielle have to find out two numbers. They have the following information: both numbers are integers between 1 and 1000 and it is also possible that both numbers are identical. Peter knows only the product of the numbers, Simona the sum, and Danielle the difference of the two numbers. The following sequence of statements passes between them. Peter: I don't know the numbers. Simona: You didn't need to tell me that, I knew you didn't know the numbers. Peter: Now I know the numbers then. Simona: Now I also know them too. Danielle: I don't know the numbers. I can only guess one number that is probably there but I don't know for certain. Peter: I know what number you are thinking but it is wrong. Danielle: OK, then now I also know both numbers. The puzzle is: What are the two numbers? Answer: 9, 64, 73

11 Public pronouncement: written or oral ---common knowledge knowledge


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