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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.

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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.

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  • $\begingroup$ Could you name examples of maths puzzles that aren't maths problems? $\endgroup$ – Buffer Over Read Sep 18 '16 at 18:18
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    $\begingroup$ @TheBitByte There is an island with 13 red, 15 blue, and 17 green chameleons. When two differently colored chameleons meet, they change to the third color. Is it possible for all chameleons to become the same color? $\endgroup$ – Mike Earnest Sep 18 '16 at 18:21
  • $\begingroup$ What makes this a puzzle rather than a maths problem? What is the exact definition of "maths puzzle" that excludes maths problems? $\endgroup$ – Buffer Over Read Sep 18 '16 at 18:22
  • $\begingroup$ Sorry, I can't really think of a good answer. I think all math problems are puzzles, but I'm not sure what to think about the other way around. $\endgroup$ – Mike Earnest Sep 18 '16 at 20:26
  • $\begingroup$ If all maths problems are puzzles, they don't all have to be good puzzles, though. However, what is "good" and what isn't, that is not an objective classification. $\endgroup$ – Buffer Over Read Sep 19 '16 at 20:35
  • $\begingroup$ @TheBitByte I agree, it is not objective at all. You asked for an objective set of rules, and I don't think I can give any. Maybe none exist? $\endgroup$ – Mike Earnest Sep 20 '16 at 0:07
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    $\begingroup$ Well, I think this close reason needs to be removed until we get a clear set of rules about it. @Deusovi, what do you think? $\endgroup$ – Buffer Over Read Sep 20 '16 at 0:36
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    $\begingroup$ @TheBitByte No, the point of having humans apply a close reason is that we can collectively decide—when presented with a new post—what is and isn't a puzzle. There's nothing wrong with having subjectivity here. Mike (and rand in his answer) have done a good job listing some general criteria. If enough Puzzling users of > 3000 rep think it looks like a math problem and not a puzzle, then that's what matters. An OP can always argue/explain/edit to make his/her case, but having 100% clarity in close reasons isn't realistic. How would one ever 100% define what is "too broad"? $\endgroup$ – Dan Russell Sep 22 '16 at 2:19
  • $\begingroup$ @DanRussell Yes, good general criteria, just too broad. It needs to be refined a bit. Subjective thinking is fine, but you need a clear set of rules to supplement that. $\endgroup$ – Buffer Over Read Sep 22 '16 at 15:54
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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].

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    $\begingroup$ @TheBitByte It's nearly impossible for any close reason to be 100% objective. That's why the closing of questions is handled by human beings rather than an automatic algorithm. Is there an objective criterion for "too broad" or "primarily opinion-based"? There will always be some level of subjectivity, and borderline cases. $\endgroup$ – Rand al'Thor Sep 17 '16 at 23:45
  • $\begingroup$ @TheBitByte: PPCG still has edge cases that they handle with close votes. $\endgroup$ – Deusovi Sep 17 '16 at 23:48
  • $\begingroup$ My vote of disapproval is part of my ongoing disagreement with the singling out of mathematics as a close reason. $\endgroup$ – humn Sep 18 '16 at 0:02
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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 )

and most importantly,

  • require some insight that makes the problem easier to solve
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    $\begingroup$ Gotta say I disagree with the first of these bullet points - and judging from the comments on that answer, so do many others. I once posted a very nice and interesting puzzle (well, IMO, anyway - that's why I posted it, after all!) which required the Green-Tao theorem to solve. $\endgroup$ – Rand al'Thor Sep 17 '16 at 23:48
  • $\begingroup$ What is "plain calculation"? Addition and subtraction, etc, only? Or is square root allowed? What about factorials? $\endgroup$ – Buffer Over Read Sep 17 '16 at 23:50
  • $\begingroup$ @TheBitByte: Anything that can be solved with a simple algorithm. $\endgroup$ – Deusovi Sep 17 '16 at 23:54
  • $\begingroup$ What is a "simple algorithm"? Are factorials simple? Is a square root simple and a cube root or above not simple? $\endgroup$ – Buffer Over Read Sep 17 '16 at 23:56
  • $\begingroup$ @TheBitByte: Okay, let me rephrase that. Anything that can be solved purely with an algorithm is not a puzzle. So "calculate √10" is not a puzzle, even if you replace the square root with a cube root. $\endgroup$ – Deusovi Sep 17 '16 at 23:57
  • $\begingroup$ As an example, "add 2+2" is technically an algorithm. You haven't defined the scope of the algorithms you're talking about $\endgroup$ – Buffer Over Read Sep 17 '16 at 23:59
  • $\begingroup$ @TheBitByte: Yes! It is! Which is why that's not a puzzle. Look, the close reasons are slightly subjective by design. It's so we don't have to have long, tedious conversations clarifying every minutia of the rules. $\endgroup$ – Deusovi Sep 18 '16 at 0:01
  • $\begingroup$ My vote of disapproval is part of my ongoing disagreement with the singling out of mathematics as a close reason. $\endgroup$ – humn Sep 18 '16 at 0:02
  • $\begingroup$ @Deusovi, "which is why is not a puzzle" <--- that doesn't mean all algorithms are not allowed, though. $\endgroup$ – Buffer Over Read Sep 18 '16 at 0:02
  • $\begingroup$ @humn: We talked about this already. $\endgroup$ – Deusovi Sep 18 '16 at 0:03
  • $\begingroup$ And chatted, which I appreciate (very much!), but still disagree that mathematics should have a special status. $\endgroup$ – humn Sep 18 '16 at 0:05
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    $\begingroup$ @humn: If we had people regularly asking us to analyze poetry or battles of WWII, then we would also have close reasons for literature and history. $\endgroup$ – Deusovi Sep 18 '16 at 0:06
  • $\begingroup$ @Deusovi meta.puzzling.stackexchange.com/questions/4938/… $\endgroup$ – Buffer Over Read Sep 18 '16 at 0:07
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    $\begingroup$ @TheBitByte: I've removed that part of the close reason. $\endgroup$ – Deusovi Sep 18 '16 at 0:34
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    $\begingroup$ @randal'thor: Fair enough! I took that part out. $\endgroup$ – Deusovi Sep 18 '16 at 0:35

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