But that doesn't really answer his actual question, which requires more than just sheer numbers of passes.
Puzzle boxes are given to you upon meeting other Miis within rooms you create. The logic behind these encounters is something like:
1) If the Mii you encounter also has Monster Manor, you will be asked if you would like to team up. If you decline the teamup, you will be given a random item from that person's inventory. If the person does NOT have Monster Manor, you are usually given gems.
2) If you have tagged that person two or more times, the person will offer you a puzzle box. If the person has Monster Manor, a puzzle box that person has completed will be offered to you, with their completion time as your target time. If the person does NOT have Monster Manor, a puzzle box you have not yet completed will be offered, with the default target time.
So the most efficient means to collect puzzle boxes for someone with a second 3DS without the game is to set up multiple encounters with that second 3DS's Mii. This usually involves building up a large queue (I just used play coins to hire randoms if I was not in an area where I was receiving passes regularly), dropping the second Mii's piece first so it is free for encounters, then using the rest of the queue to create rooms to try to start an encounter (larger rooms are better).