Choose the correct one. ___ they ___ (buy) a house next year?

The sentence is in the future tense, indicated by "next year". The correct structure for a future question with 'will' is 'Will + subject + base verb'. Therefore, 'Will they buy' is the correct option. The other options are grammatically incorrect for this context: 'Will they bought' uses the past participle, and 'Did they buy' and 'Do they buy' are for past and present tenses, respectively.

Answer: Will they buy