No spoilers needed yet, I haven't solved it. But I am prepared to
kick around some ideas.
>
> The warden meets with 23 new prisoners when they arrive. He tells them,
> "You may meet today and plan a strategy. But after today, you will be in
> isolated cells and will have no communication with one another.
>
> "In the prison is a switch room, which contains two light switches labeled
> A and B, each of which can be in either the on or the off position. I am
> not telling you their present positions. The switches are not connected to
> anything.
>
> "After today, from time to time whenever I feel so inclined, I will select
> one prisoner at random and escort him to the switch room. This prisoner
> will select one of the two switches and reverse its position. He must move
> one, but only one of the switches. He can't move both but he can't move
> none either. Then he'll be led back to his cell.
>
> "No one else will enter the switch room until I lead the next prisoner
> there, and he'll be instructed to do the same thing. I'm going to choose
> prisoners at random. I may choose the same guy three times in a row, or I
> may jump around and come back.
>
> "But, given enough time, everyone will eventually visit the switch room as
> many times as everyone else. At any time anyone of you may declare to me,
> 'We have all visited the switch room.'
>
> "If it is true, then you will all be set free. If it is false, and somebody
> has not yet visited the switch room, you will be fed to the alligators."
>
> Here's the question:
>
> What is the strategy the prisoners devise?
For the moment, I'm taking this seriously. One strategy is
for everyone to keep track of how many times they have been to the
switch room, and just declare that everyone has been to it on their
100th (or whatever) visit. Depending on what "everyone will
eventually visit the switch room as many times as everyone else"
is supposed to mean, a strategy like this might give an arbitrarily
low probability of being fed to the alligators. But it feels too
cheap.
One thing I'm thinking is that most of the time the
prisoners will be just flipping switch A, saving switch B for
"special occasions". Since they are not allowed to do nothing
when they go to the switch room, flipping A would take its
place.
---David
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