Comment by jakelazaroff

I think their issue is with the end of the sentence: "and in some cases, no method will work", which I interpret to say there are some programs for which no method can determine whether they will halt.

That's obviously incorrect; the program either halts or it does not. So if you had function that always returned true and another that always returned false, one of those functions would correctly tell you whether a given program will halt — for every program that could possibly exist.

The hard part is figuring out which function to use :)