User blog comment:DWither/Doubt about busy beaver/@comment-5529393-20180904212819

As you say, it is indeed possible to check for program "loops", i.e. whether the "full state" (tape, state, and head location) of the Turing machine is a repeat of a previous full state. The problem is, there are many Turing machines that never halt, but do not ever repeat a previous full state. A simple example is a Turing machine that keeps moving the head to the right without ever halting.