Seven differently coloured beads are strung on a thread to make a necklace. Since the necklace can be turned over, an arrangement and its mirror image count as one and the same. How many distinct necklaces are possible?

Beads round a closed loop can be rotated freely, which already cuts the count to (7 − 1)! = 6! = 720. A necklace can also be flipped over, and flipping maps each string onto its mirror image, so the 720 fall into pairs that look alike: 720/2 = 360. Trap 720 stops at the rotation step and forgets the flip. Trap 5040 = 7! is the count for beads in a straight row, where neither rotation nor flipping is allowed. Trap 180 divides by 2 a second time.