If you use couldn’t does it mean you can/ do now?
1. I tried to fix the machine but I couldn’t figure out the problem. (Does it mean I do now?)
2. I was admitted to hospital but they couldn’t work out what the problem was.
Is it also possible to say?
1. I tried to fix the machine but I can’t figure out the problem.
2. I was admitted to hospital but they can’t work out what the problem is.
Your first examples do not logically preclude the possibility that you can now, but neither do they imply it. At the time you were fixing the machine, you could not figure out what the problem was. Most people would infer that nothing else has happened since which might change your ability to fix the problem. Similarly, the inference would be that if you returned to the same hospital for the same investigations, they would still not be able to work out the problem.
In your second situation, the tenses create a different situation. “I was admitted to hospital but they can’t work out what the problem is.” suggests that you are still at the hospital and they haven’t given up on you yet. (It reads a little like a Facebook update, for example.) With the machine, it’s harder to reconcile the mixed tenses. Perhaps “I am trying to fix the machine but I can’t figure out the problem.”
Another useful phrasing is “I have yet to figure out the problem” = I am still trying to and haven’t given up.