Green eyes
On an Island there is a brutal regime; All 100 inhabitants wish to escape. The leader of the regime has told them that if a green eyed inhabitant comes to him, they will be set free. Anyone else coming to him will be put to death. The inhabitants can come to him only at night, when they are separated form the other inhabitants (assuming they aren't coming to see the leader).
All 100 inhabitants are extremely cautious, perfect logicians and most importantly green eyed. There are no reflective surfaces on the island and none of the inhabitants can ask another what colour their own eyes are but they can see everyone elses.
A diplomat comes to the Island and the leader tells her that she may issue one statement, under the condition that it doesn't give any residents new information. Which of these options would tell the residents enough to let them leave without breaking this rule.
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1 solution
The simplest way is to eliminate the impossible options: You all have green eyes tells them something new so that's out.
The others see what you see also tells them something (That everyone sees 99 green eyes) You may all leave now also tells them something (that they can leave)
Telling them that all of the green eyed people can leave isn't telling them anything but it's also completely useless. Now lets examine the correct option (Starting with a different population, say 1 green eyed person and 99 blue eyed people). When he hears this he knows that he must have green eyes; he can see everyone elses so he leaves on the first night.
If there were two green eyed people then on the first night neither could leave: They don't know if the other is the only green eyed person. On the second night he would know that the other isn't the only one and by extension that he was the other. This means that both of them leave on the 2nd night.
This pattern can be continued until we see that on the 100th night all 100 people would leave after being told that at least one of them had green eyes.