The famous and ruthless explorer Wyoming Wilbert reports in the third part of his memoires that he once visited an island inhabited by jokers and truth tellers. Truth tellers always tell the truth, whereas jokers sometimes lie and sometimes tell the truth. Furthermore when a joker is killed than his body turns green, while the corpse of a dead truth teller decomposes in the ordinary fashion.
Wilbert spent several weeks in a small village with 155 inhabitants. Every village inhabitant knew exactly who the jokers were and who the truth tellers were, but Wilbert did not know the identity of a single inhabitant. However, Wilbert happened to know the exact number $j\ge2$ of jokers in the village.
On the first day, he asked every inhabitant a single yes-no question. He analyzed the answers and then killed one of the inhabitants; the corpse showed Wilbert that this guy had been a joker. On the second day, Wilbert repeated this procedure with the remaining 154 inhabitants: he asked every survivor a single yes-no question, and afterwards killed one of them. And so on, day by day, until he decided to stop.
Wyoming Wilbert reports in his book that he had designed all his questions meticulously. They guaranteed him that after several days all the jokers would be dead, while none of the truth teller had been killed.
Questions: For what values of $j\ge2$ can this story possibly be true?
What was the questioning strategy of Wyoming Wilbert?