I think just "nights" should be acceptable for Arabian Nights. While the latter is the more common name in English, the former is the original, and while I understand this is an English-language quiz, I think that should be accepted as an answer so long as it's not entered in the answer box in Arabic. :)
I agree. As Koopa pointed out below, it's referred to as either 1001 Nights (even in English, cf. Wikipedia) or Arabian Nights, but I've never heard 1001 Arabian Nights.
At least, not yet.