10
$\begingroup$

I posted a puzzle, The "e" in apple. If I remember correctly, initially it received 11 upvotes and 0 downvotes. Then another user posted an answer that I accepted, and soon after I posted my own version of the answer. Then the question received 7 downvotes (mingled with a few more upvotes).

I am curious about why so many people didn't like the puzzle after seeing the answer.

Based on comments under the accepted answer and comments under my answer, it seems that the cause of the dissatisfaction was that the answer involved taking a ratio of areas, and raising it the power of another ratio of areas:

$$\lim\limits_{r/R\to 0}\left(\frac{r}{y}\right)^\frac{R}{Y}$$

I guess a lot of people didn't like this maneuver, maybe because it does not have any obvious geometric meaning, or maybe because it's an unorthodox hybrid of geometry and algebra. But to me, the limit expression just seems like a natural object of investigation (note the symmetry in the expression).

I have asked many well-received questions on MSE/MO/PSE that involve unorthodox objects of investigation. For example:

What was it about my question about e in apple, that caused so many people to downvote?

I welcome downvotes, if they can teach me something and make me a better puzzle-maker.

$\endgroup$
2
  • 7
    $\begingroup$ Thank you for bringing this to meta in a calm, curious manner. I won't be able to help you much (my expertise is decidedly not in the math puzzle realm), but hopefully others will see this question. $\endgroup$ Commented Oct 15 at 4:29
  • 1
    $\begingroup$ This is a great reflection post. 👍🏼 $\endgroup$ Commented Oct 19 at 18:47

3 Answers 3

Reset to default
8
$\begingroup$

's are hard to write! I didn't vote on this question, but I have some thoughts on why the answer feels like a let-down to me.

The second hint spoils the mystery

The hint spells out that we should form the mathematical constant $e$ from the four quantities listed. This leaves the puzzle not very enigmatic, with you having answered what the "e" means rather than the solver. Algebraically, there aren't many options to consider, with taking respective ratios being a natural start.

Now, I presume you were heavy-handed because the puzzle remained unsolved after a couple days even with you dropping a small hint. This suggests the initial puzzle missed the mark and was too obscure. Even at that point, I would have preferred an edit expanding the question rather than an ugly spoiler block. Perhaps, explicitly point out the respective areas in the diagram shows in a 3D cross section, and add some wordplay about powers.

The setup is underused

My first idea to solve the puzzle was to find a formula akin to the "pizza equation" connecting the letters a, p, p, l, e, where, say, $a$ is an area and $l$ is a length. Wouldn't this be cool? But, only the $e$ is used, and as you say, the apple could be some similar object.

Taking powers is unmotivated geometrically

@Pranay and @justhalf pointed out that there doesn't seem to be a good geometric meaning for the power of the respective ratios. And in my view, the power is the source of the $e$, with the areas on the sphere more like "conduits" with the right asymptotics to allow it. But just $1+1/x$ and $x$ produce $e$ in the limit using powers, which would let you just as well find "e" in a rectangle of area 1 whose width is then increased by 1.

$\endgroup$
3
  • $\begingroup$ Thanks! I think this is a fair assessment. The second hint was pretty blunt. The apple motif was meant to make the puzzle more "puzzly", but it felt a bit artificial or something. I am still struggling to grasp this issue of "no geometric motivation". For example, I guess an infinite product of areas is also geometrically meaningless, but that didn't seem to be an issue in that question. But you make a good point that we can also get e from a rectangle of area 1. $\endgroup$ Commented Oct 15 at 9:31
  • 1
    $\begingroup$ @Dan In that question, I wouldn't say that that product is """in""" the diagram - it's a very 'artificial' thing to consider. (As opposed to, say, the total of the chord lengths, which has a much better claim to being """in""" the diagram.) But that wasn't an issue for that question, because the point of that question wasn't to find something in the diagram. That quantity was part of the premise of the question, not something that you were asking people to search for! $\endgroup$ Commented Oct 15 at 16:58
  • $\begingroup$ Both answers so far (from @xnor and @Bass) are great. I wish I could accept both. I accept this one because of the last sentence, which showed me how e can be found in other shapes with some algebraic manipulation. $\endgroup$ Commented Oct 17 at 13:31
8
$\begingroup$

Surely it was mostly a case of unsuccessful expectation management.

First, the puzzle language promises a clever way to find e in an apple, using planes, spheres, and all the regular building blocks of great geometry puzzles, and there are no limits to what the solver is allowed to do. Sounds fantastic! Surely this is going to be great!

Then, the solution is (and I'm strongly exaggerating here to make the point clear) "there's actually no e in apple, but if you arbitrarily rearrange some letters like this, you get a formula for e."

One very handy way to manage the expectations of the solver is to better define the solution space and/or the scope of the possible operations; here an additional phrase along the lines of "I have found a powerful way to extract the e from an apple" would probably have made the exponentiation trick seem quite clever instead of completely arbitrary: now we're extracting e instead of trying to find it, so the solution is probably going to be some clever manipulation instead of a beautiful universal geometrical truth. Also, with the inclusion of the "powerful" hint, now raising random stuff to random powers is very much an acceptable thing to do.

$\endgroup$
1
  • 2
    $\begingroup$ Very helpful, thank you! I now realize that the presence of $e$ is more about algebraic manipulation to get the definitional limit expression (whereas $\pi$ is more about inherent geometrical structure). I think that was my main blind spot. Now I can better understand the feeling of disappointment that likely caused the downvotes... I like your example of wordplay on "power". That shows how a puzzle can be both mathematics and enigmatic. $\endgroup$ Commented Oct 15 at 12:49
4
$\begingroup$

This "$e$ in apple" feels extremely contrived. Consider the following puzzle for an analogy:

Question: I saw a mouse running by. I realized that there's an $e$ in the mouse. How?

Answer: If we let $t$ be the length of the mouse's tail, then as the mouse gets arbitrarily small,

$$\lim \left (1 + t \right)^{1/t} = e.$$

It's pretty clear that this way of obtaining $e$ from the mouse actually had nothing to do with the mouse at all; we merely wrote out a common formula for $e$ but involved the mouse in some arbitrary way. That's not a satisfying answer.

Your puzzle does better than this, but not much better. You identified some quantities proportional to $t$, $2 - t$, and $t (2 - t)$, and that's somewhat interesting. From there, however, there's nothing motivating us to build fractions and take a limit, except for the fact that we're trying to obtain $e$ and we can get $e$ that way.

We were expecting the puzzle to show us the number $e$ coming from an apple in a surprising and interesting way. "You can build ratios and take the limit and get $e$" isn't surprising or interesting at all.

$\endgroup$
3
  • $\begingroup$ Point taken with the mouse analogy. If a puzzle asked, "How is e hidden in Pascal's Triangle?" and the intended answer were this, how contrived would you consider that puzzle to be? $\endgroup$ Commented Oct 23 at 4:01
  • 2
    $\begingroup$ @Dan Much less contrived. I think the only problem is that there's no obvious motivation for taking the product of all the entries in each row. But still, "let's take the product of these entries and see if we can find e" is an interesting premise for an answer even if it doesn't feel well-motivated. On the other hand, "let's take the limit of the power of one ratio to another ratio and see if we can find e" just feels like cheating. $\endgroup$ Commented Oct 23 at 10:20
  • $\begingroup$ For Pascal's triangle, once we have taken the product of all entries, it's pretty natural to investigate it somewhat like this: "Okay, how fast does this sequence grow? Maybe exponentially? What's the limit of the ratios? Oh, the ratios actually approach infinity, so it grows superexponentially. But how fast do the ratios grow? Oh, those are growing exponentially. What's their growth rate? Oh cool, it's e." $\endgroup$ Commented Oct 23 at 10:26

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.