[Nocturnal]

Tougher riddle similar to the bracelet riddle.

ACME Technologies has just shipped your school a new weights and measures system. Comprised of a super-accurate beam balance and a 40 lb. rock, this system will allow you to accurately weigh items from one to forty pounds, and every integer value in between. Some dis-assembly required, of course.

The forty pound rock will have to be cut up with a high-powered diamond-tipped saw (not included), because as it is, it will only let you accurately weigh other forty pound rocks.

What is the fewest number of pieces the rock will need to be cut into to measure any item weighing an integer amount from one to forty pounds, inclusive (assuming you can manage to get the rock into the precise fractions you want)?