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At best, there's a 50% chance any one prisoner can find his number. It would seem like there's a probability of (½)¹⁰⁰ or 0.00000000000000000000000000008% that every prisoner could find his own number, if each were to check 50 envelopes at random.
They could certainly do even worse than this. If they all agreed in advance to open envelopes 1-50, then fifty of them would be guaranteed never to find their number and they'd all die. But could they come up with a strategy in advance which would improve their survival chances?
What makes this my favourite combinatorics problem is that there *is* a strategy which increases their likelihood of survival from 0.00000000000000000000000000008% to about 30%. So, what is the strategy, and why does it work?
...is the correct answer I didn't think of!
The answer is already elsewhere in this thread, so I'll try to give a hint. The prisoners can't communicate, but all prisoners gain access to new information that is common to all of them while they're opening the envelopes. They could use this to decide which envelopes to open.
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