It's come up before on meta, a lot, it seems.
However, I don't know whether there's a set of specific rules that define what is a maths problem and what isn't one.
For example:
Question x is a objectively a maths problem, if:
a
b
c
etc.
It's come up before on meta, a lot, it seems.
However, I don't know whether there's a set of specific rules that define what is a maths problem and what isn't one.
For example:
Question x is a objectively a maths problem, if:
a
b
c
etc.
You are asking, "When is something a math problem, and therefore not a puzzle?" I would argue the following:
Every math question is a puzzle, even textbook ones. They're just really bad puzzles.
The reason I say this is because math problems fit the pattern that a lot of valid puzzles have. You start with some initial state, there are a certain number of allowed "steps" you can take, and the goal is to reach an end state. For example, for solving the equation $\frac{8x+3}7=5x+2$, the steps you would use are adding or multiplying both sides by a number, and the goal is an equation of the form $x=\_$. This paradigm, of starting state, allowed steps, goal, is shared by rubiks cubes, chess puzzles, lights out, sokoban, mazes, peg solitaire, and so on and so on.
Why is the puzzle bad? Because for us, it's easy to find the path from start to goal; we're used to the steps, as we've been doing them since middle school. It's a rote problem for us, but for those just learning algebra, it's challenging, as they aren't comfortable with the steps yet. For some people, $\frac{8x+3}7=5x+2$ is a genuine challenge, and requires a bit of cleverness to navigate the unfamiliar maze of allowed algebraic operations. For that reason, you can't objectively say that it is not a puzzle; you can only subjectively say that you don't find it interesting.
By saying that a math problem must have a "clever or elegant solution" or "unexpected result," you tread on thin ice. These terms are subjective, not only from person to person, but also form era to era. Consider the question
What is the area under the parabola $y=x^2$ and between the lines $x=0$, $x=1$, $y=0$?
Nowadays, this problem is rote and uninteresting, easily solved by anyone taking Calc 1. But, in 1635, this problem was much more challenging for Cavalieri, requiring the ingenuity to approximate the area by smaller and smaller rectangles, and further to exactly compute these approximating sums using the non-obvious identity $1^2+2^2+\dots+n^2=n(n+1)(2n+1)/6$. Certainly, this problem is interesting, the solution is clever, and the simplicity of the area being just $1/3$ is surprising. But those descriptions aren't true anymore, because we've thought about similar problems over and over in the unpleasant setting of calculus homework.
In my mind, something that was a puzzle yesterday should still be one today, even if our opinion of it has changed. Therefore, the concepts of "surprising" and "elegant," which are destroyed when the same puzzle is considered over and over, can't be included the definition of a math puzzle.
From the canonical meta post on maths problems vs maths puzzles (which is linked to from every single question closed for this reason):
So, what makes something a math[s] puzzle rather than math[s] problem? I think there's a few features.
- Clever or elegant solution, often an "aha" moment
- Unexpected problem statement.
- Unexpected or counterintuitive result.
[...]
In contrast, math[s] 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[s] 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[s] olympiads like the Putnam exam, which are designed to have clever solution[s].
My answer in the second link you gave is objective and seemed to be acceptable to a majority of voters.
A math puzzle must:
be more than plain calculation
not require heavy calculation (unless tagged computer-puzzle)
and most importantly,
- require some insight that makes the problem easier to solve