What's Depeche Mode's famous song?!
"It's a question of time" ... and it can only be solved keeping this in mind.
The following situation could immediately and easily be solved:
Prisoner no. 3 could easily conclude that he is assigned to an "X" since only
two "O"s have been distributed and he sees both of them. Note that this
situation is the only instance which can easily be solved.
All other possible distributions cause "headaches" and the solution cannot
directly be found.
The unsolvable situation looks like this:
All prisoners do not have enough information to solve the problem.
These situations described above are not the problems you had to solve, right? However, you have
to take the thoughts above into account when you want to solve your problem:
Prisoners no. 2, 3 and 5 can find out the letter they are assigned to.
It is done like this:
All prisoners try to solve the problem. Minutes will pass and nobody will make
a guess since there is no direct/obvious solution. Having waited for an
extended period of time, all prisoners will realize that it is not an obvious
solution, i.e. it is not the situation described first on this page.
Therefore, prisoner no. 3 sees an "X" and an "O" and he knows that it is not an obvious
situation. If he had been assigned to a "O", prisoner no. 5 would have solved
the problem and would have brought it to everybody's attention. But because
prisoner no. 5 stayed quiet in the first place, prisoner no. 3 knows that he cannot
be assigned to a "O", i.e. he concludes to have an "X".
Prisoners no. 2 and 5 can use the same logic to find out their letters.
Note that prisoner no. 1 and 4 cannot solve the problem, because they do not have enough
information to distinguish between the solvable and unsolvable
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