# Are math-textbook-style problems on topic?

There's been some recent questions on the site that are problems from math textbooks, or problems in that style.

Are these on topic? What do we about them?

Related discussion: Should mathematics questions really be on-topic here?

• Looking at how the site has changed in few years I just would like to live this small unconstructive comment here: God, people, please don't forget that puzzles are what makes your mind work, and thereby the most important part of them is LOGIC, not Story. And this is exactly what math is about. Please don't make this site to be about riddles and only riddles! – klm123 Apr 14 '16 at 17:22
• I agree. Math should be a part of puzzling here. – Rigidity Feb 17 '19 at 22:54

## 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.

• I might posit that a well-known procedural solution makes something a math problem, while a little-known or self-discovered solution makes something a puzzle. (Edit: I may turn this into an answer.) – user20 Jan 14 '15 at 2:12
• @Emrakul Please do write an answer. I think I have the same feeling, but don't really know how to explain what makes a solution procedural. – xnor Jan 14 '15 at 2:32
• I think you've failed to draw a distinction. At least two of the four examples you claim as puzzles are standard textbook questions to test comprehension of the material, and all of them could be used for that purpose. – Peter Taylor Jan 14 '15 at 11:36
• A major problem with math questions is that they sometimes get spectacularly bad yet high-scoring answers. If we aren't capable of making the correct answers stand out and getting the bad answers to a negative score, we have no business answering math questions. – Gilles 'SO- stop being evil' Jan 14 '15 at 19:36
• @Gilles I think that's from the dark ages of the site and hasn't been true lately. People have been aggressively downvoting bad answers. – xnor Jan 14 '15 at 23:29
• I've been thinking: if we were to add a 'math problem' close reason, would "I know it when I see it" be sufficient? – user20 Feb 17 '15 at 21:19
• @Emrakul So, I think it would suffice for most cases, but would also lead to some unhappy arguments when people disagree, or when people don't want to see their question closed. – xnor Feb 17 '15 at 23:56
• @xnor That's fair, though the people who don't want their questions closed will be unhappy either way. The reason I suggest this is because it may be a place to start; the community thinks math problems shouldn't be on topic, and this might help develop better consensus. – user20 Feb 18 '15 at 0:06
• @Emrakul If you think it would help make consensus, I think it would be worth trying out. An option to migrate to math.SE would also be nice. – xnor Feb 18 '15 at 0:14
• @xnor In that case, there's actually a specific reason why SE doesn't usually open migration paths. It prevents the syndrome of "well, we don't really want it, and it's sort of a math problem, so here, you guys deal with it." (This happens all too frequently with current migration paths.) See also: meta.stackexchange.com/a/203326/206222 – user20 Feb 18 '15 at 0:20
• note that math textbook things and theorems are what was the most beautiful and practical puzzles for our predecessors (and children). Many math theorems still are very good puzzles (with clever and elegant solution), for those who doesn't know them. – klm123 Apr 14 '16 at 17:26

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:

• is it obvious that one requires maths to solve it (and potentially non-trivial maths)? If so, if it is clearly a mathematical problem and nothing surprising, creative or fun - please post it at maths!
• Or is it in any aspect puzzling? (i.e. would your old, dusty and joyless math's teacher just point you the door when you'd ask him the question...) Then you might be on he right site to post it! But make it clear that you really seek an answer. Otherwise it could be misinterpreted as badly written challenge...
• 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.

• I think "exists a common-knowledge routine" is the right idea, but too restrictive a criterion. A math problem can be dry and non-puzzley without belonging to a class of problems general enough for a common-knowledge routine to exist for solving it (as there is for integrals or algebraic equations). – xnor Jan 15 '15 at 6:16
• @xnor I have yet to see a strictly-math puzzle (something mathematicians might call a puzzle) on the site, so I'd be reluctant to preclude them from topicality quite yet... I do see what you mean, though. – user20 Jan 15 '15 at 6:17
• I'm not sure what you mean by strictly-math puzzle then -- I would have thought my newest three questions would qualify, as would some of the examples I gave. Could you give an example? – xnor Jan 15 '15 at 6:21
• Maybe we should move to chat? (I see this becoming a more extended discussion pretty quickly.) – user20 Jan 15 '15 at 6:23
• Now define "common-knowledge routine". Most people haven't taken college calculus, but a criterion which only excludes simple arithmetic isn't very useful. – Peter Taylor Jan 15 '15 at 12:50
• In retrospect, this answer is a lot less useful than I'd hoped it would be. – user20 Jan 16 '15 at 1:28
• Please tell whether this one is on topic or not? The previous one also has some votes to close. – ABcDexter Apr 16 '16 at 7:22

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!!!

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.

• I don't think this criterion works: we accept questions of the form “I read this puzzle in a book and the solution was given as … with no explanation. Why is it that?” — but if the puzzle can be modeled mathematically then your criterion would exclude it. – Gilles 'SO- stop being evil' Jan 15 '15 at 18:03
• I think that "Prove this is true..." is just a mathy wording, and there's puzzley questions that, taken formally, are asking for a proof. This includes things like "how can first player can win the following game" or "can you connect the dots in fewer than 12 lines" or "figure out which sister is lying". – xnor Jan 15 '15 at 20:12
• @xnor, the mathy wording is more what I was getting at than the mathematical nature of the question. E.g. your planets question is phrased as a maths exercise, not as a puzzle, and as a result it feels to me to be clearly way past the blurry line. – Peter Taylor Jan 15 '15 at 21:59
• @PeterTaylor If I instead asked "Is it true that ...?", would you be OK with the question? What about "What is the maximum possible hidden area achievable?"? – xnor Jan 15 '15 at 22:02
• @xnor, I think either of those changes would move it into the grey area. – Peter Taylor Jan 15 '15 at 22:04