Learn Polish: W nocy nic nie widzę na korytarzu. - At night I see nothing in the corridor.

Polish requires what we call "double negation." Whenever you use a negative pronoun like nic (nothing), nikt (nobody), or nigdzie (nowhere), you must also put nie (not) in front of the verb. Literally, this sentence says "At night I do not see nothing."

Could I change the word order and say nie widzę nic instead?

Yes, nie widzę nic is grammatically correct. However, putting the negative pronoun (nic) before the negated verb (nie widzę) is extremely common in Polish and often sounds more natural to native speakers.

I learned that negated verbs make the object take the Genitive case. Why is it nic here instead of the Genitive form niczego?

You remembered the rule correctly! The strict Genitive form is indeed niczego, and saying niczego nie widzę is perfectly correct and formal. However, in everyday spoken Polish, nic acts as a common exception and is frequently used as the object of a negated verb without changing its form.

Why does the sentence use na korytarzu (on the corridor) instead of w korytarzu (in the corridor)?