2024 SSMO Relay Round 1 Problems/Problem 2

Problem

Let A circular necklace is called if it has black beads and white beads. A move consists of cutting out a segment of consecutive beads and reattaching it in reverse. It is possible to change any necklace into any other necklace using at most moves. Find . (Note: Rotations and reflections of a necklace are considered the same necklace).