A Turing Machine is an idealized computer. But no computer/TM can find out which of the possible 61.519 4-state TMs can write the longest string of 1's on a blank tape before halting.
If you're talking about the halting problem now, humans also don't know whether the TM they are manually simulating will halt eventually.
And you still haven't said specifically what the thing is that the humans do and computers can't.