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Headbands I
You are part of a group of 500 people who have been captured by an evil tribe. The tribe leader, who appreciates a person with good logic, decides that he will propose a challenge of many headband problems.
The first challenge is that each person will get a random headband of one of two colors, black or white. There are 500 of each (500 black and 500 white). If you look at yours or say a word after you have been given your headband and before you have guessed, the evil people will kill everybody. Also, if you make any movement with your body other than your eyes, you will ALL die.
At the count of 10, each person will shout out what color they think their headband is. If you or anyone else is right, you will survive for the next challenge. If not, you will be killed immediately. You are given some time to discuss your strategy; then you are given your headbands.
How many survivors can you guarantee, assuming the tribe leader is not lying and that each person will follow your instructions? For example, if you tell everybody to guess, you can guarantee zero survivors because you have a slight chance of getting zero survivors.
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