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This question is inspired by a comment on a recent popular question:

Which area is larger, the blue area, or the white area?

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Warning: Spoiler Below! - Don't keep reading if you want to solve this yourself!

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There are several ways to arrive at the answer that the regions have the same size, as seen with straightforward derivations in the answers to the question linked above. Some users valiantly slog through the tedium of geometric arguments and others have a slicker way of arriving at the answer.

However, this highly upvoted comment attached to the question itself gave me pause:

It's a general rule of thumb, that in all puzzles which ask "which area is larger" the answer always is that both areas have the same size, even when one is visibly enormous and the other visibly tiny.

As time has gone on and I've seen more questions that fit the spirit of the comment above, I think I've become slightly disillusioned with such questions/puzzles. Don't get me wrong: I love the satisfaction of a long derivation/proof to demonstrate something, say, the relative sizes of the areas of differently colored shaded regions. But my intuition and inclination to try to find a "slick" answer somewhat removes the satisfaction of working out a proof more explicitly.

What I'm saying is that for these types of questions, a slick intuition can "spoil" the answer because you don't really need to go through a full derivation to get the correct answer. If I choose to ignore the slick way and work out a non-slick derivation for my own satisfaction, it feels like I'm merely verifying an answer as opposed to discovering the solution.

Here's some simple optical illusions that ask, respectively, "Which middle line is longer?", "Which white rectangle is longer?", "Which sector is larger?", and "Which vertical strip is longer?"

enter image description here enter image description here enter image description here enter image description here

I'm not saying these such puzzles/questions could be answered without a measuring device or rigorous phrasings of what's given and what's unknown. But the more I see these illusions/problems, the more they feel...cliche.

On the other hand, there are problems that might require some derivations to discover the correct answer. E.g., resolving the impossibility of the Magic Chocolate Bar. Such a problem is about halfway to the type I'm seeking, because physical intuition already rules out that such a bar can exist; the challenge (derivation) is to understand and reconcile an nearly-imperceptible visual trick can fool your brain.

My Question: What I'm looking for is problems (not just limited to geometric or related to optical illusions) that require one to actually work out different derivations and then finally compare things at the end.

A simple example that at least has the right spirit is asking "Which is larger?: $\pi$ or $22/7$?" If you know that $\pi \approx 3.14159$ and that $22/7 = 3 \frac{1}{7}$, this becomes a matter of "showing" that $1/7$ has a (repeating) decimal expansion of $1/7 = 0.142857...$.

A more complicated question in the same spirit is asking: "Without a calculator, determine which is larger: $e^\pi$ or $\pi^e$."

I understand this question still be perceived as vague or too subjective for MSE. I'll further clarify what I mean by "non-trick" questions if prompted with a particular suggestion.

[Note: I've tagged this question with the "intuition" tag, but what I'm really going for is finding problems that are more in the spirit of "anti-intuition".]

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    $\begingroup$ by intuition seems like the area is the same for both (blue and white) :) $\endgroup$ – user441848 Jul 24 '17 at 20:50
  • $\begingroup$ Draw the other diagonal. $\endgroup$ – BruceET Jul 24 '17 at 20:56
  • $\begingroup$ @AnneliseToft Thank you for your intuition, but the point of my question is not about the blue-white-region problem. $\endgroup$ – Xoque55 Jul 24 '17 at 21:03
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    $\begingroup$ Interesting question. But another 'rule of thumb' is that in response to vague questions you must expect Comments that you do not find relevant. $\endgroup$ – BruceET Jul 24 '17 at 21:08
  • $\begingroup$ The question "which is larger, $\pi^e$ or $e^\pi$" is not actually difficult. In general, if $a,b \gt 1$, then $$\operatorname{max}(a^b,b^a)=\operatorname{min}(a,b)^{\operatorname{max}(a,b)}$$ $\endgroup$ – Franklin Pezzuti Dyer Jul 25 '17 at 13:24
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Below is a plot of the density function of the probability distribution $\mathsf{Exp}(\text{rate}=1).$ Areas beneath the curve on either side of the vertical blue line are equal. The density function is $f(x) = e^{-x},$ for $x > 0$ and the line is at $-\ln(1/2),$ the median of the distribution.

The solution is by elementary calculus. (If there is a 'slick' answer, I have not been bright enough to discover it.) Similar, slightly messier, versions involve other heavily skewed distribution for which it is difficult to visualize the area under the long tail extending to $+\infty.$ Depending on the speed of 'decay' either side of the median may appear to have larger area.

enter image description here

Addendum: OK then. Two more.

$\mathsf{Gamma}(shape=.3,rate = 1),$ where there is the additional complication of judging area next to a vertical asymptote. Software may be required for solution for some members of the gamma family of distributions.

enter image description here

qgamma(.5, .3, 1)
## 0.07313114        # median, vertical blue line
pgamma(.07313114, .3, 1)
## 0.5               # area to left of median

$e^X \sim \mathsf{LogNorm}(0,1),$ if $X \sim \mathsf{Norm}(0,1).$ Usual parameters of lognormal distribution are mean and standard deviation of the corresponding normal. Obviously, median of lognormal is $e^0 = 1.$ [By calculus, mean is $\sqrt{e},$ mode is $1/e.$]

enter image description here

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