- If at some point in solving this, you come across something that would be implausible or even impossible in reality, that's fine. This problem is not meant to be realistic.
- This problem is meant to be solved entirely without a calculator (or other computational aids). There's a certain theorem involved in solving this problem that you'll have to look up if you've never seen it before (you'll likely recognize when you'll need to do this), but aside from that, electronic aids are unnecessary.
- Yes, this problem is very contrived. That's because I wrote it with a certain fact in mind that can be used to simplify part of it.
You are playing a game with 32 rounds, numbered 0 through 31. In each round you are asked a question. If you get the question in round n correct, you receive coins (assume all coins are identical). You continue until you get a question wrong (you don't lose coins for getting a question wrong).
At the end of the game, you are told to put your coins into piles, with each pile containing the same number of coins. Let's call that number n. Having done this, you are then presented with (n-1)! coins and told to put them into n piles, each with the same number of coins. If you have exactly one coin left over upon doing this, you win n-1 coins. If not, you win nothing.
What is the maximum number of coins you can win
1) if you get every question right?
2) by getting any number of questions right? (you may have to look something up for this)