Final answer:
option Tower of Hanoi , The Tower of Hanoi backup rotation scheme uses a mathematical puzzle to optimize unique days of backups stored, reducing the amount of necessary storage media.
Step-by-step explanation:
The backup rotation scheme that uses a complex mathematical puzzle to extend the number of unique days of backups stored with the least amount of tapes is the Tower of Hanoi. This scheme is based on the mathematical puzzle of the same name and its approach to data management minimizes the quantity of storage media required to store backups over an extended period.
Unlike the other mentioned schemes such as FIFO (First-In, First-Out) Backup, Grandfather-father-son, and the 3-2-1 backup strategy, Tower of Hanoi incorporates a mathematical algorithm to determine the backup rotation. This results in a backup schedule where backups are cycled through at different intervals, creating a balance between the number of backups and the diversity of data saved without requiring an excessive number of tapes.