Cashiers always place money in the cash register "face up". Stamp the back side.
There'a guy in Phoenix who actually puts "Please return me to XXXXX, Phoenix, AZ" on $1 and $5 bills, then he goes out grocery shopping or whatever. He has an average of a 50% return of those bills every month. He even had one that was sent back from Holland. The legality of "defacing" the bills was addressed by a local attorney....nothing illegal about it. (BTW: he is disabled, he's just making lemon aid out of lemons)
Because so many people use debit and credit cards, cash in a register will be more likely to be circulated back into the buying public because the stores aren't going to get much cash to begin with, and they always need a certain amount of cash in their drawers every morning when they open. If businesses are making bank deposits, they probably only consist of checks, for the most part. Debit and credit cards are electronically "placed" in their account at the POP.
Honestly, it sounds like a good idea!
