The three purple necklace pieces are collected at certain times or following certain events in the game. The first piece is a "Day 2 reward" from the daily reward calendar, while the second is found in a crack on day 5+ and the third in a crack on day 10+.

The items found in the crack are 100% random, including the chance a villager might not find anything. The only way to guarantee that you get something from the cracks is with the Lemur.