BB(n) isn't computable, even given unlimited computational resources. Running all n-state machines won't work, because you may have some machines that continue

BB(n) isn't computable, even given unlimited computational resources. Running all n-state machines won't work, because you may have some machines that continue indefinitely, but without repeating. (Remember that while the number of states is finite, the tape is infinite.) No matter how long you run them for, you can't be sure whether they are going to terminate at some point in the future, or if they'll continue forever without halting. This is why computing BB(n) for arbitrary n is equivalent to solving the halting problem.