Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 32A / b / iii / Solution

An itinerary fixed by the eighth shift has least period dividing $8$. Those with period dividing $4$ are exactly the itineraries fixed by the fourth shift. Thus the primitive length-eight itineraries formpointed words. Each cycle has eight cyclic shifts, so they givedistinct $8$-cycles. Therefore at least five such cycles are forced.