The "math problem" in the All in the Family episode "Pay the Twenty Dollars" is a comedic sequence of circular logic used to settle multiple debts triggered by a counterfeit bill.

The Core Problem
Archie accidentally pays George Jefferson with a counterfeit $20 bill for dry cleaning. George gives Archie $18 in real change plus $2 in services, meaning George is out $20 in real value. To fix this, Louise hides a real $20 bill in Archie's pocket so he’ll "find" it and pay George.

Mike’s "Mathematical" Solution
Mike (the "Meathead") intervenes with a confusing series of transactions to "square" everyone. According to plot summaries and video transcripts, the logic follows these steps:

Initial Standing: Archie is considered $40 "ahead" because he has the $18 change, $2 in cleaning, and the $20 George found and returned from his pocket.

The First $10: Archie gives George $10. Mike calculates Archie is now only $30 ahead ($40 - $10).

The Owed $8: Since George originally gave Archie $18 in real cash and only got $10 back, Archie still "owes" George $8.

The Outcome: Mike calculates that if Archie pays that $8, he is still $22 "ahead" ($30 - $8).

Circular Debt Clearing: Mike then incorporates other debts:
George owes Archie $2.
Archie and Mike owe Lionel $7 for ballgame tickets.
Mike owes Archie $5 for a pizza.
Gloria owes Mike $10 for a plumber.

Mike directs the characters to pass various bills around—including a $5 bill and two $10 bills—until he declares everyone is "even".

The Irony: After the rapid-fire exchange, everyone leaves satisfied except Archie, who realizes he is left holding only the original phony $20 bill.