"The princess is the king's daughter."
Translation:Die Prinzessin ist die Tochter des Königs.
According to Duden only Königs, wiktionary lists both forms as genitive sing. More common is the one without e.
As a rule of thumb, it is more common to add the e for short, monosyllabic words; longer ones often go without. Most of the times, however, one or the other form is more often used for no particular reason other than custom, but neither one, with or without e, is strictly wrong.
Why "Die Prinzessin ist eine Tochter des Königs" not accepted? The king may have more than one daughter and the English original does not differentiate between these two possibilities: "the" in the English sentence belongs to "the king", not "the daughter".
I think the definiteness of ‘daughter’ in this case is debatable, since a genitive or possessive generally makes a noun at least somewhat definite (take, for example, how you can't use the indefinite article with genitive or possessive pronouns and you have to reword the sentence, like ‘a daughter of the king’, ‘one of the king's daughters’, ‘a friend of mine’, ‘one of my friends’). Aside from this, even if the king had more than one daughter, I believe you could still say ‘sie ist die Tochter des Königs’, ‘sie ist eine Tochter des Königs’ puts much more stress on the indefiniteness, closer to ‘one of the daughters of the king’.
"Die Prinzessin ist des Königs Tochter."
Should be an accepted answer as it works perfectly fine in German and, with all respect, is an even more direct translation than Duolingo's suggestion.
- the king's daugher = des Königs Tochter / die Königstochter
- the daugher of the king = die Tochter des Königs