Prisoner Solution

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 situation.

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