Problem of the Day #18

POTD December 7, 2023
Author: munch

Munch is standing in front of an evil ATM. He has 100 credit cards on him, and he knows that for every positive integer k \le 100 one of the cards contains exactly k dollars. However, the cards are visually indistinguishable and Munch does not know which card contains how many dollars. Whenever Munch inserts a card into the ATM and punches in a positive integer n, one of two things happens. If the card contains fewer than n dollars, the ATM eats it without giving Munch any of the money in it. Otherwise, the ATM gives Munch exactly n dollars and then eats the card anyway. What is the largest number of dollars that Munch can be sure to gain?