Hint #1
It might be tempting to think the answer is 4, but it’s not.
Answer
3

Bob cuts all three links from a single 3-link chain.  That leaves three 3-link chains still solid.  If we refer to the individual broken links as X and the solid 3-link chains as OOO, then Bob arranges the pieces to form the desired 12-link chain like so:
X OOO X OOO X OOO