Always use Haus, even in the dative case, unless using the idioms: 'nach Hause' = 'home' (when talking about going there) or 'zu Hause' = 'at home'. They are phrases that have stuck around from when Hause was correct in the dative case, now it is not except for these two phrases.
I accidentally translated this in the negative ("They are not allowed to go home"), my mind still on a previous sentence (whoops!) and now I am curious as to where the "nicht" would go, exactly? (I would guess either before or after "nach Hause") ...or could it be a matter of emphasis?
THank you for your answer =D But what I meant was something like this, "In this patch they(game producers) allowed going home after your energy bar is empty, but sleeping now costs you more coins." I mean "they" made the allowing decision.
Other example would be "They(the authorities) allowed hunting last summer but I'm not sure for this one" Again Not that they are allowed but they allowed it in the past =D thank you for your time tho =D
I think--and, mind, my research consists of looking it up in the dictionary--that those sentences would be more likely to use the verb "erlauben" or something similar.
Remember your English teacher correcting you "It's not 'can we go outside?'; you're perfectly able. 'May we go outside?'; do you have permission?"? (or was that just me? ;) Nah, I'm kidding; it's a pretty common experience I think in English speaking nations but it might not happen elsewhere. And "can" is casually correct, as Duolingo's acceptance of it in some sentences suggests. )
...anyway, I'm pretty sure duerfen is a lot more like the English auxiliary "may" than "allow".