This is a word problem for game theory. This a non-cooperative two-player game with three discrete ex interim stages that allow perfect recall. The game is also finite, and ends after the third stage.
Alice is purchasing alcohol at a local station, when Bob comments that she looks too young to legally purchase the beverages. Alice proposes a bet with Bob, saying if Bob can guess her age she'll give him ten dollars (the cost of the purchase). Bob is also a customer, so he can't check her identification but is allowed to verify Alice's age with the clerk ex post. If Bob loses he has to give Alice $10 (the cost of the purchase).
The rules Alice sets are that Bob has three guesses, but can only modify the first guess using one of two numbers, and either subtracting or adding the numbers to the original guess. Alice also picks the numbers that Bob can use to add or subtract from his first guess. Bob gives Alice the rule that she can only select integers in the range between 1 and 9.
My working assumptions are that Bob knows the legal age is 21, since this is common knowledge. Bob does not know if Alice is 21 yet, since the purchase hasn't been complete. If Bob is rational, he will start his guess at 21.
If Alice is 37 years old, and the legal age to purchase alcohol is 21 years old what numbers should Alice give Bob to use in his last two guesses? If there is more than one solution, would adding an additional number to Bob's moves increase the likelihood for Alice to win?
This question is drawn from experience, and is not a homework assignment.