add "prisoners and a light bulb" riddle

Author: whame

riddles/bulb.md

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+Prisoners and a light bulb
+==========================
+
+A prison guard is giving 100 prisoners a second chance for their death
+sentences. The guard tells them that everyday, starting from tomorrow, he will
+pick one prisoner randomly from his cell and bring him to his office. In the
+office there is a light bulb wired to a power switch. The light bulb is
+initially off and only has two states, on or off, triggered by the power
+switch. While the prisoner is in the office, he has to do one of the following
+things here: either _flip the power switch_ of the light bulb, and thus altering
+its state (if it is off, this turns it on, and vice versa), or do
+_nothing_. After the prisoner has chosen his action, he is brought back to his
+cell and locked up. The process is then repeated the next day with another
+random prisoner (it could be the same one, since the pick is random).
+
+Now, if any prisoner, that is currently in the office, knows for certain that
+every other prisoner has been in the office at least once, he can tell that to
+the guard. If his claim is true, they are all spared, otherwise the game is over
+and they all get killed!
+
+The guard is a very patient man and has thus no time limit for his little
+game. He will continue bringing a random prisoner to his office every day, until
+some one dares to make the claim. His random pick is [uniformly
+distributed](https://en.wikipedia.org/wiki/Uniform_distribution_(continuous)),
+meaning that every prisoner has the _exact_ same probability to be picked
+(1/100). Moreover, each one of the prisoners is contained in a different
+isolated cell. There are no ways for them to communicate with each other at
+all. They can also not leave any kind of "message" in the office for the others;
+the guard only lets them flip the power switch for the bulb or do nothing.
+
+The guard lets the prisoner discuss a strategy for a short period of time before
+locking them in their respective cells for tomorrow's game. What is a winning
+strategy for the prisoners, that spares them from their death sentences?
+
+**[Solution](../solutions/bulb_solution.md)**

solutions/bulb_solution.md

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+Solution to prisoners and a light bulb
+======================================
+
+One prisoner is assigned as a _counter_. The role of this counter is to keep
+track of how many prisoners that have entered the office. The counter thus has
+to count 99 other prisoners before he can tell the guard.
+
+Now, when one of the other 99 prisoners enters the office and the light bulb is
+on, he does nothing. If it is off, he turns it on but and only if he has not
+done this before. When the counter enters the room and sees the bulb on, he
+switches it off and increments his count with one. Since prisoners only switches
+the light bulb on if they have not done this before, the switched on light bulb
+encountered by the counter is a different unique prisoner. Thus, after the
+counter has switched the light bulb off 99 times he can now tell the guard that
+every prisoner has entered the office.
+
+**Remark**: This solution, for 100 prisoners, has an expected run time of
+10417.74 days (28.54 years). There are more complex and efficient
+solutions. However, the question of optimality is still open. For more details
+please see this
+[paper](https://www.ocf.berkeley.edu/~wwu/papers/100prisonersLightBulb.pdf).