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I mostly use the following for "What do you think?"
как ты думаешь?
I understand there can be many variants of the aforementioned question, and I have seen a few of them. I was recently following a thread on Reddit and somebody was called out for using
что ты думаешь?
Maybe its the artless state of my Russian, but I cant understand why that would be incorrect? Does it not mean the same or at least something similar?
Что ты думаешь... ? is an unfinished or incomplete question. Its complete form is something like Что ты об этом думаешь... ? or Что ты думаешь по этому поводу (на этот счёт, об этом)? Some words must be included in the question itself, while the question Как ты думаешь? does not need that. However, it still needs a previous context, typically a more detailed question, so Как ты думаешь? is about someone's personal opinion on the subject and it much better corresponds to the mentioned English version.
Both these questions are normal in Russian, but there is a difference in usage.
The question with что is used when you ask to express an opinion about something. You introduce the subject with о preposition or with words like насчёт, по поводу:
Что ты думаешь о глобальном потеплении? - What do you think about global warming?
The question with как is used when you ask to make a choice of something (including questions you can answer with yes/no). There is no preposition:
Как ты думаешь, завтра будет дождь или нет? - What do you think, will it rain tomorrow or not?
Notable that use of что was normal in this case a hundred years ago or so.
When you just said something and want your interlocutor to reply in some way, both questions will work:
[А] как ты думаешь? (more usual, less formal)
[А] что ты думаешь?
If in doubt, opt for что. In the worst case you will sound old-fashioned.
Simple "what do you think?" should be translated as "что думаешь?" or "как думаешь?", but the context should be clear.
If it's "what do you think [about something]?", than it should be translated like other guys already said.
