Math puzzles are on topic, math problems are not
Let me first give some examples to illustrate the distinction I mean.
Math problems:
Solve for $x$: $2x+3=7$.
My friend gave me a riddle: She went to the store and bought some
apples. Then, she went to the store and bought an equal number more apples.
Then, she picked three more apples off her apples tree. Now, she has
7 apples. How many apples did she buy on her first trip?
At a party, every attendee has someone at the party that they know. Is it necessarily the case that there's someone at the party who knows every attendee?
Let $S$ be a metric space. Prove that $S$ is connected if and only if
any locally-constant function from $S$ to $\mathbb{R}$ is a constant
function.
I also think all the problems linked in the question are examples of math problems, though less archetypal than these examples I made up (Can the car or the bike travel further? is borderline.)
Math puzzles:
So, what makes something a math puzzle rather than math problem? I think there's a few features.
- Clever or elegant solution, often an "aha" moment
- Unexpected problem statement.
- Unexpected or counterintuitive result.
For the example math puzzles (spoilers ahead):
In contrast, math problems tend to be "textbook". And by that I don't mean that they have to come from textbooks (or that textbooks can't contain math puzzles), but that they use standard, staightforward methods than anyone familiar with the subject is expected to know. They can be difficult, but their goal is to test comprehension of the material, not ingenuity. This doesn't apply to problems from math olympiads like the Putnam exam, which are designed to have clever solution.
Math problems should be closed and directed to math.SE (by the way, can we get support for migration?). I think answering these questions is well-intentioned but counterproductive, as they are liable to be homework questions. The poster doesn't learn from being given a solution, and we undermine math.SE's policies of avoiding giving full solutions and requiring the poster to show what they've tried.
Now, I intentionally chose examples that I think illustrate the two sides of the spectrum, and there's lots of grey area in between. So I'd like to see where this discussion goes.