Subterranea: Day-Knights and Night-Knights
Some logic puzzles inspired by those found in Raymond Smullyan's To Mock a Mockingbird.
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The city of Subterranea is completely underground; the inhabitants have never seen the light of day. Clocks, watches, and all other timepieces are strictly forbidden. Yet, the inhabitants have an uncanny sense of time, they always know when it is day or night. Each inhabitant of the city is either a day-knight or a night-knight. Day-knights tell the truth during the day and lie at night, while night-knights tell the truth at night and lie during the day.
Suppose you are visiting Subterranea - naturally, you loose your sense of time and would like to know whether it is day or night. You encounter two inhabitants, and each makes a statement...
There are two types of statements - simple statements and compound statements. Compound statements join two simple statements with an 'and' conjunction.
Some simple statements provide a lot of information.
If an inhabitant simply says "it is day," then they must be a day-knight: day-knights will say this during the day (being truthful), and will also say it at night (when lying). Night-knights can never simply say "it is day" (either truthfully, or when lying).
If an inhabitant says "I am a day-knight" and nothing else, then it must be day. A day-knight will say this truthfully during the day, and a night-knight will also say this during the day, but as a lie. At night, no one makes a simple statement claiming to be a day-knight.
Beware, however: these simple rules do not apply when simple statements are combined into compound statements... but compound statements are also very helpful.
For example:
If an inhabitant says "I am a day-knight and it is night" they cannot be telling the truth. Moreover, you know they cannot be a day-knight during the day, or a night-knight during the night (as then they would have to be truthful): you can conclude that it is day and that they are a night-knight.
If an inhabitant says "I am a day-knight and it is day" they may or may not be telling the truth, but you can at least rule out the possibility that they are a night-knight at night.
You may be able to work out some other rules that will help you solve these puzzles...