There is a room with 100 logicians, and each of them have an infinite number of hats in a tower upon their head. Each hat can be either white or black, and the sequence of hats upon a given logicians head is random. Standard hat rules apply of course.
At a specified time, each logician must simultaneously write down a number. For each logician, the hat of the number they wrote down is checked (so if Bob wrote down 13, we look at the 13th hat on his head), and if that hat is white, that logician is "correct". If all the logicians are correct, they win, if any of them are incorrect they will play the game again tomorrow with new random hats.
Before the game, they may strategize. The naive strategy gives you a 1/2100 chance to win, so it will take about 2100 days to be released, but they do not want to wait that long. Find a strategy that will probably get them out within the year.
Because the hats are random you may use a strategy that only will specify a number with probability one. For example, Bob could look at Alice's hats and look for "the first occurrence of 578 consecutive black hats", which will appear somewhere on her head with probability one.
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