So here's the latest problem I've been playing with. You have a set of numbered cards and a set of blank cards. Here's a process - choose any two of your numbered cards, write their (positive) difference on one of your blank cards, and replace the two original cards with your new card. (So if you started with n cards, you now have n-1 cards.)
Keep doing this until you only have one card left.
The original problem that was posed to me was as follows: if you start with cards numbered 1-100, prove that when you have one card left it has an even number on it.
Once I'd proved that, I played around a bit. Here's some questions I enjoyed exploring: What sets of cards could I start with that guarantee finishing with an even number? What even finishing numbers are possible with the 1-100 set in the original problem? Given a set of cards with the numbers 1-n, what could the last card have on it?
Would love to hear any other interesting questions that occur. And of course, always interested to hear other people's solutions too!