Synonyms for wondering hard

Can someone help me to identify some English synonyms/idioms meaning to "wonder so hard"? The word or phrase I'm looking for could be used in a situation where somebody tries to solve a very confusing riddle or is unable to manage what is going on.

I thought one example might be "breaking [somebody's] head open" (like in other languages: "den Kopf zerbrechen" or "rompersi la testa"), but I realized that the English meaning is literal in that case.

Here are some ideas for you.

I had to really scratch my head about that one.

I was completely lost in that lecture. For the life of me, I could not figure out what the heck the professor was talking about.

I had to wrack my brain to solve that riddle.

A word like ponder or ruminate would be synonyms, but neither has the intensity I think you're looking for. There's an idiom, "crack one's brains," or "rack one's brains," that might work. I personally use a phrase that involves smoke coming out of my ears.

P.S. "Wrack one's brains" is often used; according to Grammarist rack is correct.

I've ruminated over the problem for months.

The problem is driving me crazy.

The problem is puzzling me.

I'm cogitating about the problem.

I'm still mulling over it.