What makes a "good" challenge problem?

I've been giving my calculus students challenge sets every couple of weeks or so (8-10 total for the semester). I'm requiring that they complete at least 5 of them, and their challenge set grade will count for 10% of their final grade. Many students work on all of them (for some combination of enjoyment and extra credit, I think) and I've learned that before each set is due there's a group of them who loiter at a local chain establishment and argue for hours about the problems, probably spending a grand total of $1 on a bottle of water. I'm not going to lie -- this brings back fond memories of math study groups as an undergraduate, and it makes my heart smile to know that they've started their own little geek-fest.

It turns out that some of my students are pretty demanding and tell me when I've given them a "good" problem and when I haven't. They didn't like their last set so much - it basically asked them to investigate the limits of a function similar to f(x)=sin(1/x). I have lots of thoughts as to why they didn't enjoy this one as much, and it's primarily because they didn't really understand what I was asking for and started it too late to ask me any questions. However, it makes me incredibly happy that they even have an opinion one way or another about the problems: passion in the classroom = a math teacher's dream.

I fear that I've ventured far, far away from the course content with these challenge sets. On one hand I feel like I should be able to come up with some honest-to-goodness calculus challenge problems; on the other hand, it's actually more important to me to generate some excitement about math, period. It's probably also not a bad idea to give the kids who aren't loving calculus another point of entry into math.

I really, really like this week's challenge problem:

Alice and Bob need your help! They have been captured by pirates and will only be released if they can accomplish the following task:

A pirate will deal Bob five random cards out of a standard deck of 52 playing cards (no jokers). He gets to choose one card to put aside as the “mystery card.” He must use the other four cards to communicate the identity of the mystery card to Alice. They may not talk or look at each other’s faces once the cards are dealt. However, they can communicate and agree on a strategy beforehand.

Alice gets one guess – if she’s right (about the number and the suit of the card), the couple will get released. If she’s wrong, it’s bad news for Alice and Bob.

What should their strategy be?

Note: Your strategy shouldn’t contain anything shady like “If Bob points his index finger a certain way, the card is a heart.” There is at least one purely logical/mathematical strategy.

I'm trying to figure out what makes a "good" challenge problem - one that's rich with mathematical ideas but also approachable and engaging. So far, I've got:

• Anything having to do with infinity (like this back-handed approach to the harmonic series);
• Something they can "play around with"/brute force until they figure out a more "elegant" solution

Finally, I also tapped into current grad students in my old math graduate program for help. Every grad student has their favorite puzzles and a desire to do anything other than their research, so not surprisingly I got a ton of great problems from them. Here they are, for your perusing pleasure.