2966: Exam Numbers

2024-08-01T15:25:31Z

Umnikos: Full solution for the integral

{{comic
| number = 2966
| date = July 31, 2024
| title = Exam Numbers
| image = exam_numbers_2x.png
| imagesize = 553x400px
| noexpand = true
| titletext = Calligraphy exam: Write down the number 37, spelled out, nicely.
}}

==Explanation==
{{incomplete|Created by 10 MORE THAN AVERAGE MATH TEACHERS - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

The comic depicts various similarly formatted examination questions that might appear on test papers at various points in a student's potential academic career. While they all share a similar feel, they are asking for different things, some of which might be considered more serious and examinable proofs of study than others. The joke is that the first and last questions are the same, but have very different interpretations based on the context.
{| class="wikitable"
|+ Exam numbers
! Exam level !! Question !! Answer
|-
| {{w|Kindergarten}} math || Write down the biggest number you can think of || At a kindergarten-level education, it is assumed that an individual might write down a relatively small number like 300, depending upon whether they have encountered the concept of hundreds. It might also be interpreted as "what's the highest number that you (think you) can count up to". Given the nature of a child's exuberant glee at learning about ''really'' big (but otherwise normal) numbers, they may even try an answer something like "a million billion squillion gazillion". It is not certain what criteria would be used to mark this question correct or otherwise, it may actually be a stealth question in child psychology or a question that everybody "gets right" so long as they answer it.
|-
| Pre-{{w|algebra}} || Write down the value of x if x=3x-8 ||

''3x'' refers to the multiplication of 3 and the originally unknown number ''x'', as a convenient shorthand.

By subtracting ''3x'' from both sides, ''-2x = -8''. Divide both sides by -2 to find ''x''=4. Alternately, subtract ''x'' from both sides to give ''0 = 2x - 8'', and as taking 8 from two ''x''s makes it zero, one ''x'' is half that

(Verify by plugging ''x''=4 into the original equation. '''4''' = (3*'''4''') - 8 -> '''4''' = 12 - 8 -> '''4''' = 4.)
|-
| {{w|Calculus}} || Write down the value of ∫0π x sin2 x dx ||
The integral can be solved using a trigonometric identity:

∫0π x sin2 x dx

= ∫0π x (1-cos 2x)/2 dx

= ½ ∫0π x dx - ½ ∫0π cos 2x dx

= ¼ x2 |0π - ¼ ∫0π cos 2x d2x

= ¼ x2 |0π - ¼ sin 2x |0π

= ¼ π2 = 2.4674...

|-
| PhD {{w|Cosmology}} || Write down the Hubble constant to within 1% || The Hubble constant is a component of {{w|Hubble's law}}, which describes the relationship between the distance between galaxies and their speed of separation. Its exact value is not known to this level of accuracy; it is about 2.3×10-18 Hz. Different methods of measuring it have come up with significantly different values, and resolving this difference (the {{w|Hubble's law#Hubble tension|Hubble tension}}) is one of the great challenges of modern cosmology. PhDs involve advancing their field, so it seems this particular PhD student has completed a thesis solving this problem. This question might have a different acceptable answer in the future, depending upon further developments in cosmology (although the 'constant' itself changes over time).
|-
| {{w|Game Theory}} || Write down 10 more than the average of the class's answers || Game Theory studies "games" (and 'game-like' situations) in which two or more participants take actions that will succeed or fail based on other participants' decisions. In this case, all students' answers will be averaged (what kind of average is not stated, but the common assumption would be an {{w|arithmetical mean}}), and the highest- (or perhaps only) scoring answer will be one that is 10 more than that average.

If a student knew what everyone else was thinking, this would incentivise them to answer 10 more than the consensus (taking into account their own forthcoming answer), which would not necessarily be the largest number written down. For instance, if the answers end up being 30, 40, 50, 60, and 70, the (mean) average would be 50, making 60 the most correct answer. Since they presumably do not know each other's answers, they will have to guess what those answers are likely to be, factoring in that each of the other students will also be guessing everyone's answers and responding accordingly.

In reality, most game theory exams test your understanding of game theory as an academic subject, not your ability to win games. (A type of class where actual results may result in better grades is a business negotiation class where the results of practice negotiations can determine one's grade on the assignment.)

Something similar to this question is found in the title text of [[2385: Final Exam]].
|-
| Postgraduate Math || Write down the biggest number you can think of || This question echoes the very first example, but would be expected to be answered very differently (unlike a revisiting of most of the others).
Postgraduate math students can probably think of ''very'' large numbers. While a tempting answer could be "{{w|infinity}}", most mathematicians do not consider infinity to be a number, but rather a class of numbers. (Writing down "Infinity" in this context would be as wrong as writing down "Primes" or "Positive integers"). Students familiar with the field of [https://googology.fandom.com/wiki/Googology#History Googology] may give an answer such as Rayo's Number, which was the winning entry in the [https://googology.fandom.com/wiki/Big_Number_Duel Big Number Duel]. Even if infinity is an acceptable answer, some infinities are bigger than others.

This might heavily depend upon the branch of mathematics you are studying. Named (finite) numbers, or ones with specific and useful notations, might satisfy some questioning contexts, whilst the existence of a whole further set of trans-finite numbers (i.e. increasingly large types of "infinity") would be important considerations in others. For those associated with more computational mathematics, any infinity would be {{w|NaN|Not a Number}}, and their answer might instead be the ceiling of some binary representation (typically ''28n-1'' for some value of ''n''), the largest unsigned value reliably storable in a given byte form for an integer (e.g. a {{w|Integer (computer science)|double quadword}}). On the other end of the spectrum, many abstract algebraists might answer with some variation of "What ring are we working in, and is it even well ordered?" It also might be a trick question: if you can envision a real number greater than one, are you even doing real math (in a given field)?

As with the kindergarten question, there may be no previously anticipated "correct" answer. It could be another "correct just so long as you answer it" (or perhaps "sensibly" so) or the mark goes only to those giving the greatest valid number across all submissions.
|-
| {{w|Calligraphy}} (title text) || Write down the number 37, spelled out, nicely || Calligraphy is the art of artistic writing. The title text expands the joke outside the realm of math and points out that since calligraphy does not require any math skills, the only way a calligraphy exam would even mention numbers is if one had to write them out in such a way as to showcase their calligraphic skill and aesthetic judgement (choosing a form and adornement of script that is "nice", which may be a highly subjective choice). In this case, it could be rendered as "thirty-seven" or "thirty seven" (or possibly, "one score and seventeen" in old-fashioned writing). {{w|37}} is a number that some people believe [http://thirty-seven.org mysteriously appears more often than it should]; this was a subject of a [https://www.youtube.com/watch?v=d6iQrh2TK98 recent Veritasium video].
|}

This style of final exam question, un-numbered and therefore possibly the ''only'' question upon the whole of each final paper, in some ways (for some instances) echoes the question "What is your name?" that Randall will be aware was the sole question given to Discworld's {{w|List of Discworld characters#Victor Tugelbend|Victor Tugelbend}} in an attempt to ensure he comprehensively passed (or utterly failed) his final student-wizard's exam, after many prior times of deliberately not-quite-passing.

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[6 different math test questions.]

:[The first panel:]
:Kindergarten math final exam
:Q. Write down the biggest number you can think of
:A. [empty box]

:[The second panel:]
:Pre-algebra final exam
:Q. Write down the value of x if x=3x-8
:A. [empty box]

:[The third panel:]
:Calculus final exam
:Q. Write down the value of [integral sign, from 0 to pi] x sin^2 x dx
:A. [empty box]

:[The fourth panel:]
:PhD cosmology final exam
:Q. Write down the Hubble constant to within 1%
:A. [empty box]

:[The fifth panel:]
:Game theory final exam
:Q. Write down 10 more than the average of the class's answers
:A. [empty box]

:[The sixth panel:]
:Postgraduate math final exam
:Q. Write down the biggest number you can think of
:A. [empty box]

{{comic discussion}}

[[Category:Math]]
[[Category:Cosmology]]

2841: Sign Combo

2023-10-15T19:27:20Z

Umnikos: AoC is not needed to prove 45∈65

{{comic
| number = 2841
| date = October 13, 2023
| title = Sign Combo
| image = sign_combo_2x.png
| imagesize = 322x334px
| noexpand = true
| titletext = Speed Limit: 45 MPH / Minimum: 65 MPH
}}

==Explanation==
{{incomplete|Created by a TRAPPED DRIVER - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

This comic depicts a trio of road signs that provide legal instructions that are, in combination, impossible to follow: "Do not enter" precludes passing the point of the sign, the No U-Turn sign precludes turning around, and the "No Stopping Any Time" sign precludes both halting before the sign as well as reversing back (even ignoring that this is usually already illegal with or without signs explicitly forbidding it). A driver approaching this sign combo would seemingly be forced to violate at least one of the three.

Of the three, "No U-Turn" is the one with the largest wiggle room, as it can be defined more narrowly/specifically as driving in a U-shape; thus, a driver might be able to get around it by [[207|drawing a more elaborate path.]] However, since the lane dividers on the road are solid until the signs, this potential loophole is preemptively closed.

Depending on the jurisdiction, signs may only apply to the road ''after'' them, so you could validly stop just before it.

This no-escape scenario could be done more easily with just "Stop" and "No Stopping Any Time" signs. But by combining 3 signs, the joke is presumably that it was done accidentally, without noticing the contradictions.

The title text adds even more to the dilemma, posting a 45 miles-per-hour maximum speed limit, but also a ''minimum'' required speed of 65 MPH. Since 45 is lower than 65 (as can be proven without using the Axiom of Choice, with the exact proof left as an exercise to the reader), this is quite the perplexing contradiction.

For another of Randall's adventures in road signage (he does live in the Greater Boston area, after all), see [[1116: Traffic Lights]].

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}
:[A two-lane highway stretching out to the horizon, with the lanes separated by a double solid white line in the foreground which becomes a partially solid, partially dotted double white line as it approaches the horizon. To the right of the highway is a pole with three traffic signs.]
:[Top sign, featuring a white rectangle on a red circle on a white background:]
:DO NOT
:ENTER
:[Middle sign is the No U-Turn symbol, a U-turning arrow with the red prohibition circle symbol.]
:[Bottom sign with red text:]
:NO
:STOPPING
:ANY
:TIME

:[Caption below the panel:]
:Oh no.

{{comic discussion}}

[[Category:Comics with color]]

explain xkcd - User contributions [en]

2966: Exam Numbers

2841: Sign Combo