Broadly speaking, yes, the point of an exam is to test the examinee's knowledge. However, that is a very vague and imprecise statement. In principle each exam item is meant to assess a specific piece of knowledge. For example, the example of finding leap years is likely intended to get students to demonstrate an ability to iterate using a for loop with a nested conditional. One could simply tell the examinee that this is the intention, but then you are essentially telling them how to answer the question, which reduces the validity of the assessment (to what extent is, of course, a matter of debate). At the end of the day, the exam writer needs to know if an examinee can actually recognize a situation which calls for iteration, and whether or not they actually code up a valid for loop.

On the other hand, if a student comes up with a valid though unexpected solution, the fault is with the writers of the exam item, and not the examinee. The correct course of action in such a case is probably to give the examinee credit for their correct answer, or to remove the exam item from the computation of scores for all examinees. Hopefully, the knowledge which the invalid exam item was meant to assess is tested elsewhere—one or two problematic exam items should not seriously threaten the overall validity of an exam which is made up of many questions.

No... if I wanted to solve that DE, I would integrate both sides with respect to $x$. On the right, the integral is easy enough to compute. On the left, it comes down to an application of the fundamental theorem of calculus. The Leibniz notation is convenient in this case, since it lets you treat the differential as a number, but the usual exposition relies on actual theorems which justify this kind of manipulation. It should also be noted that Abraham Robinson went over this in the 60s...

Shinichi Mochizuki has a solid history of producing good mathematics. While it is possible that he is trying to pull a fast one, that seems quite unlikely, given his reputation. The most charitable explanation is that he has invented a new branch of mathematics (the "inter-universal Teichmüller theory") in order to resolve the ABC Conjecture, and that the "newness" of this approach is causing difficulty for outsiders.

This is not at all true. First off, mathematics itself has many different highly specialized sub-fields, many of which don't communicate effectively between each other. An complex analyst and a homotopy theorist speak very different mathematical languages, and may have difficulty communicating their ideas to each other. It is reasonable to suggest that this represents a different "cultural" background (as per Tylor's definition, these differences are differences in knowledge and belief, as well as differences in language).

Additionally, mathematicians from different parts of the world conduct mathematics differently. The internet and the wide-spread adoption of English as the de facto language of discourse has ameliorated this problem some in recent history, but there are still very significant cultural differences between American, European, Russian, and Japanese mathematics (for example). The approach that one takes in tackling a mathematical problem does depend quite a bit on where one learns it. As a historic example, Ramanujan was interesting to Hardy not just because he was producing interesting results, but because he was producing these results in an idiosyncratic way which differed immensely from the British approach.

Hence the original question

is it obtuse because he's trying to pull a fast one, or does it appear obtuse because he's form a different cultural background than?

is entirely reasonable.

There is where you are wrong. Wonder Woman is clearly historical fiction, grounded in reality. It is neither science fiction nor fantasy, because everything in that movie is based well documented historical fact. Or are you calling Herodotus a liar?

My bachelor's degree, from the University of Nevada Reno is a Mathematics BA (Statistics emphasis). The difference between that degree and a Mathematics BS (Statistics emphasis) is that the BA requires a foreign language, does not require any CS, and is not required to take numerical modeling (which is essentially a CS class). Otherwise, the requirements for the two degrees are identical. The same structure exists for the other possible emphases in mathematics: the BA requires a foreign language and does not require any CS.

I'm not disputing anything that Mandelbrot said. He himself walked back from a formal definition of a fractal, and ultimately chose not to give a precise mathematical definition. From the second edition of Mandelbrot's book (on page 459):

...to leave the term "fractal" without a pedantic definition, to use "fractal dimension" as a generic term applicable to all the variants in Chapter 39, and to use in each specific case whichever definition is the most appropriate.

In the first edition of the book, Mandelbrot suggested that a fractal should be any set with Hausdorff dimension strictly larger than its topological dimension (note that this does not mean that the Hausdorff dimension is non-integer; only that it is larger than its topological dimension). However, it was pointed out that there are lots of sets that we intuit as fractals, but which fail to satisfy this property (the Devil's staircase, for example, has both topological and Hausdorff dimension equal to 1 (it is a rectifiable curve)).

Falconer, who (in my opinion) is a much better source for a rigorous mathematical discussion of fractals, also chooses not to give a formal definition. From the introduction Fractal geometry. Mathematical foundations and applications (pp. xx--xxi):

In his original essay, Mandelbrot defined a fractal to be a set with Hausdorff dimension strictly greater than its topological dimension... This definition proved to be unsatisfactory in that it excluded a number of sets that clearly ought to be regarded as fractals. Various other definitions have been proposed, by they all seem to have this same drawback.

My personal feeling is that the definition of 'fractal' should be regarded in the same way as the biologist regards the definition of 'life'. There is no hard and fast definition, but just a list of properties characteristic of a living thing...

When we refer to a set F as a fractal, therefore, we will typically have the following in mind.

1. F has a fine structure, i.e. detail on arbitrarily small scales.
2. F is too irregular to be described in traditional geometric language, both locally and globally.
3. Often F has some form of self-similarity, perhaps approximate or statistical.
4. Usually, the 'fractal dimension' of F (defined in some way) is greater than its topological dimension.
5. In most cases of interest F is defined in a very simple way, perhaps recursively.

It all depends on what you mean by "fractal." The term has no precise mathematical meaning.

On the other hand, car theft is almost a non-issue anymore, thanks in large part to the technology that you decry.

Where in the constitution of WSFS (the organization that gives the awards) is it written that Hugos must be given to works that are strictly science fiction? Indeed, the constitution states

3.2.1: Unless otherwise specified, Hugo Awards are given for work in the field of science fiction or fantasy appearing for the first time during the previous calendar year.

This specifically includes fantasy.

To the contrary, in a recent interview (maybe on the Jist or Marketplace? I can't recall exactly where I heard this...) the author mentioned that the distinction between "nigga" (a common lyric) and "ni**er" (not a common lyric) made it easier to distinguish potentially racist searches from others. On the flip side, the author ran into trouble when trying to study sexist/misogynistic searches, as many of those are people looking for porn.

It should also be noted that the punchline is not "people who search offensive phrases are racist." The punchline is that seemingly racist searches correlate (i.e. the effect is statistical, rather than individual) with other variables (such as regions where Obama underperformed when compared to other Democrat candidates and/or polling) that seem to indicate some underlying racism.

The actual book appears to be pretty nuanced. The Vox interview linked above is also appears to be relatively nuanced. The Slashdot summary and the paragraphs preceding the interview on Vox are sensationalist, click-baity claptrap.

I lived in Reno for over a decade, and referred to the mountains as ``the Sierras'' for years. I am so embarrassed that no one ever corrected me. :(

To anyone who knows what they are doing, both pi and tau are simply constants. Absorb either into some constant term, or carry it around with you wherever you go---it doesn't matter. Real engineers, mathematicians, scientists, etc can handle either constant without difficulty. The real advantage that tau has over pi is pedagogical. It is much easier to communicate the relation between angle measure and arc length with tau. Since trig functions deal with lengths more readily than area, it makes sense to teach people trigonometry using a constant that is more closely related to length.

This is not the paper described in the summary, but rather an older paper with some of the same authors. The paper referenced in the summary was published online yesterday in Nature Climate Change. I'm sorry that I can't give a direct link to a .pdf (yay for paywalls keeping all of the non-ivory tower plebs out! huzzah!), but for those with access, the paper can be found at Influence of high-latitude atmospheric circulation changes on summertime Arctic sea ice. For those without access to an academic library, the first author provides an email contact. One presumes that a polite request would yield the full text of the paper.

What is there to compromise if I don't share you veneration of dead bone?