This challenge is about cutting cards. And therefore about probability and combinatorics.
You have a deck of cards. For convenience, we’ll just use A-9 to have a nice round 40 cards. You cut some number of cards. At least one quarter and less than 3/4. That’s a neat 20 possible number of cards. To keep things simple, make them all have a probability of 5%. Now, from this cutting you cut a number of single cards, shuffling in between cuts of course. Formally, that’s random selection with replacement. At some point, you will cut a card that you have seen previously. At this point, you stop. This cuts short the data you receive about the deck of cards; you haven’t seen half of them. Of course, one can always estimate the remaining number of cards. So say you cut 10 cards before a repetition (11 cards total), and that for some reason you cannot visually approximate how many cards are in the deck. The challenge is to determine the expected number of cards in the deck.
The expected number of cards, to put it simply, is the “best” guess for how many cards are in the deck given your information. “Best” means minimizing the average difference between the actual number and your guess.