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I don't understand. How do the 12 Federal Reserve Banks determine the flow of dollar bills?

Banks return their bills to their regional Fed bank for counting/cleaning/exchange/destruction/etc. So if you're from Dallas and you spend a dollar in Northern Louisiana that dollar is probably going to end up back in Dallas because Northern Louisiana is in the Dallas Fed's district.

http://www.frbservices.org/help/coin_and_currency.html#a1

Businesses that collect large numbers of dollar bills bundle them up and give them to their banks, these wind up in the hands of the local federal reserve that checks them and delivers them to businesses that give out large numbers of dollar bills inside of the same district.

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