Think about it for a minute, it's a 'toy' that requires all the pieces. So you sell them to kids and they lose the pieces and buy more. It's better than legos!
You know who probably invented puzzles? Moms!
I can just see it now. At home a child is crying because his little brother ripped up his favorite picture, and mom can't find the tape because her daughter wanted to do a science fair project that required three full rolls of tape. She is trying everything she can think of to help the kid feel better when suddenly she has a brilliant idea! She rips the picture into small pieces and makes it a game. "Let's see if we can put it all together just like new."
And thus the first puzzle was born.