There's been some recent questions on the site that are problems from math textbooks, or problems in that style.
Doubt : Pokemon Hunter and the Rogue Brook
Groups of Farmers
Cows eat too much
Who is corrupt in the Senate?
Can the car or the bike travel further?
how many times 12 men may shake hands?
Are these on topic? What do we do about them?
Related discussion: Should mathematics questions really be on-topic here?
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.
For the example math puzzles (spoilers ahead):
Digging a tunnel between random locations features a counterintuitive result : The likeliest distance is not the one corresponding to the biggest ring, but one corresponding to a single point!
In Infinite dwarfs wearing infinite hats of 2 colours, there's a surprising problem statement, that only finite many dwarves need to wrong. Also, the Axiom of Choice makes an unexpected appearance for an elegant reprsentative-element method.
In Ten numbers on a blackboard, it's surprising that the order of operations is irrelevant, and there's an elegant way to express the operations and the result.
Does the drunk man fall off the cliff? (a random walk problem) has a surprisingly compact expression for the answer, and there's multiple clever arguments for it that avoid the combinatorial summations of the direct approach.
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.
I think there is quite some similarity between this question and the question posted on Magic Trick Solving. At least, my answer is very similar.
I think we will have a hard time (oh, a puzzle! Or just a challenge?...) to find hard-core rules to differentiate and will therefore often have to decided on a very subjective case-by-case basis. However, isn't it lovely that StackExchange and its voting system was build exactly to facilitated that?
I think Xnor's question gives already some very good starting points for guidelines on which everybody voting on (or posting) questions can measure his or her decision.
Guidelines for authors in self-evaluation could/should be:
What is the purpose of the question?
Have you found it somewhere and just want a solution? Then ask yourself:
Have you created it with the idea of creating a puzzle and now want to test it (and others)? If so, make sure you present it as a fun thing to solve. There are various properties which can make it qualify, just make sure at least one is fitting! (Again, fun and or the aspect of surprise or ingenuity are nice benchmarks.)
What is the presentation of the question?
If your (mathematical) puzzle is just maths then make sure you present it as a puzzle. There should be a reason why anybody wants to solve it. Usually fun in solving, or curiosity for a solution are the driving forces. If your posted puzzle raises neither, it is wasted effort.
A good puzzle can be pure maths, i.e. it is perfectly possible, that e good puzzle requires pure maths to be solved. The puzzle can be that one has to find the mathematic which is needed. Or it might be, that the maths involved is unexpected. It may also be, that the language of the puzzle simply is maths like in "combine these mathematical operators to get XY" type of puzzles.
I'd offer that puzzles involving any manner of mathematics should be fine here - whereas questions that are simply mathematics aren't. (This is the same conclusion as xnor, but I made it for different reasons.)
In textbook mathematics, there's a defined process for reaching a solution that's almost algorithmic for every problem. For instance: $$\int x\ln{x}\;dx$$
This isn't a puzzle. Anyone who's taken college calculus has seen the routine to solve this, and it doesn't require reasoning to determine a solution.
The criterion I propose is: a mathematics-based question is not a puzzle if there exists a common-knowledge routine for producing a solution.
In other words, a math puzzle is one which requires you to think about the process you're using. A math problem is one for which you already know the process, and simply need to figure out how to apply it.
Having just reviewed my $?^{th}$ close vote review giving this as the reason for voting to close, I too began to wonder what the difference between a question on maths and a maths puzzle is, so that I can justify my decision at least to myself.
Then I remembered Martin Gardner and friends, and so this question has been debated before.
The difference is generally cited as mathematics versus recreational mathematics, where the difference according to Wikipedia is:
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity.
They give a list of number theory topics that can be considered as recreational mathematics here, and a list of people considered to be recreational mathematicians here.
A puzzle is the story that comes with it and the interest it creates. The idea of a puzzle is to stimulate your urge to get the result. There is a kind of eureka moment - ahhhh. It is enough to get interest of 10 people to make the puzzle a puzzle - no need of the "experts" to make judgment in such case!!!
I believe the grey overlapping area (math puzzles math problems) is significant and, moreover, on MSE (math stack exchange) these questions are often closed as well because of different etiquette there. In fact I was once kindly asked on MSE to move such question to PSE (puzzle stack exchange).
I do not want to generalize but on PSE typically one challenges others to find answers, and one hides parts of the answer to not spoil the challenge, whereas on MSE typically one asks for help solving a problem or verifying a solution and one does not need to hide anything.
If some math questions are not welcome on ME and SE, I feel that's an unfortunate pity. In fact, then it's no longer a grey area but rather a black hole.
After all, one can use several math tags on PSE .... and one can even add puzzle tag on MSE.
Agreed, question should not be presented pure mathematical on PSE. A bit of (puzzling) story is minimum extra required ...
By the way: one could try to ask these questions on mathoverflow SE, but that's rather a research exchange and the questions we are considering here are typically not research questions but are recreational questions. And, in fact, very often on PSE the answer to the question is known by the OP.
There's bound to be some level of "I know it when I see it", so I think that the best one can hope for is a set of white-listing criteria, a set of black-listing criteria, and an understanding that some questions will fall into a grey area.
Within that context, I think that one black-listing criterion should be:
If a question says words to the effect of "This is true: prove it", it belongs on maths.SE rather than here.
