Shady B.S.: How The Show robbed (and continue to rob) Dreamcatcher of their first win

On the 204th episode of SBS MTV The Show, aired last September 24, 2019, Everglow took home their first music show win for “Adios,” squeaking past Dreamcatcher and “Deja Vu” with a score of 7861 to 7850. However, this win was not without controversy, as InSomnia (Dreamcatcher fans) believed that Dreamcatcher should have won, and presented mathematical evidence to support this belief. The initial narrative was that there was a computation error that gave Everglow the win by a mere 11 points. However, digging deeper strongly suggests that while indeed, “Deja Vu” should have won, there was no mistake. Only a deliberate, concerted effort to rob Dreamcatcher of their first win.

How The Show calculates scores

On The Show, the total score is broken down into a 90% prefinal score and a 10% score from live voting. The 90% prefinal score consists of 50% sales, 20% YouTube MV views from all official distributors, 15% expert picks, and 5% pre-voting on StarPlay (formerly StarPass). The 50% sales are broken down into 40% digital sales from Korean web streaming services such as Melon, Genie, Naver, Bugs, etc., and 10% sales of physical albums as reported by the Hanteo charts. Sales and views are counted from Monday until Sunday of the previous week.

Due to the nature of the digital sales, which draws from multiple music charts which in themselves are not exactly transparent about the number of streams recorded, this is not the easiest category to deal with. Data on physical sales, however, is pretty easy to obtain and verify.

Scores on The Show are numerical, with an upper limit of 10000 points; thus, 1% corresponds to 100 points. For instance, the 5% pre-voting corresponds to 500 points, while the live voting corresponds to a maximum score of 1000 points. Category scores are relative, i.e., all actual figures are scaled down to the maximum score. For instance, if the top group in physical sales sold 2000 copies that week and got the full 1000 points, a runner-up that sold 1500 copies would get 1500/2000 × 1000 = 750 points in the physical sales category.

Frustratingly, in the final display of scores, the individual category scores (except for the live vote) are not seen; instead, several categories are lumped together, as can be seen in the three rows. While this is ostensibly to provide a much cleaner display, it also serves well to camouflage any wrongdoing.

The first row corresponds to the sales; it is the 40% digital sales + 10% physical sales. The maximum possible score attained here is 5000 points.

The middle row consists of the 20% YouTube views + 15% expert preferences + 5% pre-voting results, for a total of up to 4000 points.

The last row is the live voting, where, unlike with pre-voting, we have literally no way to verify that what is displayed reflects the actual voting result; we take it at face value. The highest possible score here is 1000.

Sample computation: Episode 187

Let’s look at the May 7, 2019 episode (Ep. 187) of The Show.

This was yet another example of a close victory, where The Boyz squeaked past N.Flying on a score of 7032 to 6930:

The pre-voting data (from the StarPass days), as shown here, had N.Flying winning the 5% category.

This means that, for pre-voting, The Boyz got a score of 500 × 399263/433019 ≈ 461.

Assuming full scores for MV and expert preferences, they got a total score of 461 + 2000 + 1500 = 3961. Apparently, this was rounded up to the next ten, 3970. This in itself is pretty bad practice, given that some contests have been decided by fewer than 20 points and that this is inconsistent with the last row, which allows for single-digit differences. Checking other episodes reveals that they appear to be consistent with this practice, though, so we will assume this moving forward. We will similarly assume that the first row is rounded up.

(Important note: With N.Flying vs. The Boyz, we cannot establish wrongdoing, simply because based on tweets from around that time period, N.Flying got utterly waxed at YouTube MV views; TBZ garnered 7 million views for “Bloom Bloom” at some point within that week, while N.Flying have not even broken a million with “Spring Memories” as of today. The judges did favor The Boyz, but, based on some estimates I did, within reason: N.Flying got something like 1000 points out of 1500. Not the worst thing in the world. If my fellow N.Fia want N.Flying to win with “Good Bam,” we have to do a lot better at YouTube streaming.)

Back to The Show 204: physical and digital sales

In the first row, which is the combined physical and digital sales, Dreamcatcher got a score of 5000 to Everglow’s 3980.

We look first at the physicals, worth a total of 1000 points. Due to Chuseok interrupting Hanteo reporting, I will count Everglow’s sales from September 12 to September 20. They sold 12509 – 11544 ≈ 965 copies. On the other hand, Dreamcatcher’s physical sales at the end of the 20th were 6198.

HANTEO: Daily chart (190911)#Everglow – Hush: 977 copies (11,544 total)#Bolbbagan4 – Two Five: 137 copies (1,266 total)#RedVelvet – ReVe Festival Day 2: 71 copies (68,446 total)#ITZY – IT'z ICY: 53 copies (66,206 total)#BLACKPINK – Square Up: 43 copies (158,031 total)

— girlgroups album sales (@girlgroupsales) September 11, 2019

HANTEO: Daily chart (190920)#Dreamcatcher – Raid of Dream: 1,184 copies (6,198 total)#RocketPunch – Pink Punch: 282 copies (12,567 total)#Laboum – Two of Us: 273 copies (318 total)#Saturday – IKYK: 263 copies (1st day sales)#Everglow – Hush: 183 copies (12,509 total)

— girlgroups album sales (@girlgroupsales) September 20, 2019

Dreamcatcher won this category (CLC’s Devil, being a digital single, had no album sales) so they indeed got the full 1000 points; Everglow should have gotten 965/6198 × 1000 = 156 points. This means that Everglow have a digital score of at least 3970 – 156 = 3814. That is, digitally, they sold more than 3814/4000 ≈ 95% of what Dreamcatcher did.

What do the charts say about this?

Deja Vu’s first week charting is perhaps the highest Dreamcatcher have ever attained. While it seems they did not get as close to charting in Melon as they did with Piri (peaking at 101), they peaked at 11 in Bugs, they also charted on Genie, Naver, and Soribada for the first two days:

[BUGS] #DejaVu has debuted at #12 on the Bugs Chart! #드림캐쳐 #Raid_of_Dream #Dreamcatcher #데자부 pic.twitter.com/w3zVpdJO8H

— 7 DREAMERS | 드림캐쳐 (@7_DREAMERS) September 18, 2019

For comparison, a cursory Twitter search suggests that “Adios” had fallen off all aforementioned charts after the first week of September. (Their international sales/Spotify streams are still going strong, to their credit, but that doesn’t count here.) Again, due to the opaque nature of and sheer number of variables involved in computing the digital score, it is pretty much impossible to establish wrongdoing in this part of the score. But this alone should have you on edge.

The contested second row: expert preferences + MV views + pre-voting

Let’s now look at the second row, which is most fans’ main point of contention, and where we can establish beyond reasonable doubt that wrongdoing took place. I will not be doing the calculations myself here, as others have beaten me to it.

As can be seen in the calculations linked, for 2/3 of the groups, the scores obtained agree with the scores on The Show… with one massive caveat.

The problem isn’t just that Dreamcatcher got fewer points than they should have, though arguably giving them even just those would have earned them the win, so this is understandably the hill on which most InSomnia are willing to die.

The problem is also that both CLC and Everglow appear to have gotten full expert scores, with Dreamcatcher getting even less than the minimum of 0 points. Given Deja Vu’s modestly successful first week (at least by mid-tier girl group standards) and complete lack of anything to suggest that it is utterly bad (full disclosure: it is my favorite song of 2019, and possibly my favorite Dreamcatcher song, period), this is difficult to read as anything other than an act of sheer malice, meant to humiliate Dreamcatcher.

The prevalent narrative, at least shortly after that episode ended, is that there was a rectifiable miscalculation. This was no miscalculation. This was a deliberate act of sabotage. This was highway robbery, plain and simple.

Can expert scores be negative?

The whole expert preferences part of this competition is cloaked in mystery. All we know is that it takes up 15% of the score, that is, a maximum of 1500 points. Our objection hinges on one assumption: experts do not give negative scores. This is, to my knowledge, not explicitly stated.

However, there is a good reason that indeed, experts should not be allowed to give negative scores. If an expert is allowed to give arbitrarily negative scores, then they can clearly have their pick by setting the other two scores as low as they please, even taking away points earned in other categories. Obviously, this would be unfair.

But even if they were allowed to give negative scores, albeit with a limit, this would still be unfair. For example, if the expert scores were from -1500 to 1500, then by adding 1500 points to all contestants, we can see that this has the same effect on the final outcome as their scores being from 0 to 3000. The maximum score in this hypothetical scenario is now 11500 instead of 10000, and expert opinion would count for 3000 out of 11500 which rounds out to about 26% (an outrageous percentage), not 15%.

For the expert score to be at its stated 15%, no expert should be allowed to give a score lower than zero. Yet this appears to be, among many other things, exactly what happened.

Conclusion

While it is easy and somewhat comfortable to assume that what had happened was a simple lapse on The Show’s part, this is unlikely to be the case. The suspiciously close digital score and bewildering expert marks that fail to reflect any tangible realities suggest deliberate, calculated tampering with the final scores to Dreamcatcher’s detriment.

It is even harder to assume good faith when, in the face of numerous social media posts, news articles, and videos pointing out the discrepancies, SBS MTV The Show have not only failed to acknowledge wrongdoing, they also refused to award a winner for today’s episode and continue to mock angry InSomnia on their Twitter account. (You can reasonably make the case that this is a special performance, but in light of their announcement of X1’s victory during a similar special episode, this does not sit well with many.) The inclusion of a first win montage right after Dreamcatcher’s performance on today’s episode does not help things.

In conclusion, SBS MTV The Show have not only failed to give Dreamcatcher the first win they earned, they continue to refuse to rectify this mistake and make amends, opting instead to continue mocking and insulting fans justifiably upset by this. Dreamcatcher and their fans deserve so much better.

이 세 분 성대에 입주를 하고 싶은데 주택청약 좀 들 수 있을까 해서요. 1순위 조건이 드캐 향한 마음 꾸준하게 저축하기라고 들었는데 그거는 걱정 하덜 마시고 매달 1일 드캐 사랑하는 마음 자동이체 신청도 부탁드려요😉#드림캐쳐_시연_한동_가현 #모든_날_모든_순간 #THESHOW @hf_dreamcatcher pic.twitter.com/kFY7GHs8Rw

— THE SHOW (@sbsmtvtheshow) October 1, 2019

The Music Log, Ep. 2: On the lack of content + Dreamcatcher, “Deja Vu” review

As some people know, the reason I started this blog is so that I could write full-length posts about music without the length and formatting restrictions that characterize Twitter. While the Music Log thread was a nice exercise in brevity, I sometimes felt that I undersold some truly great songs because 500-ish characters was not quite enough to convey what I liked or disliked about certain pieces of music. Sadly, due both to increasing obligations and a sense of disenchantment with the state of music in general, I have not written as much here as I intended. While as a whole 2019 has felt like a bit of a down year for music everywhere, the summer doldrums have brought out plenty of music I felt apathy if not antipathy towards. Still, there have been highlights, and I would be remiss in my duties as an aspiring music writer if I didn’t cover them, even if late.

Digging through my drafts, I found a half-finished list of my favorite title tracks as of July 2, 2019. It’s a bit too late to publish it, but oh well. For the record, at the time it was:
Dreamcatcher, “Piri”
A.C.E, “Under Cover”
WJSN, “La La Love”
and then some revolving door of Lovelyz’s “When We Were Us (Beautiful Days),” NCT 127’s “Superhuman,” Twice’s “Fancy,” and DreamNote’s “Hakuna Matata,” among others.

(Also, my favorite album at the time was Panihida, by Krzysztof Drabikowski’s version of Batushka. I still owe this one and several full-length album releases at least a short review.)

More than two months have passed and this list is mostly the same, except for one major change. I will elaborate on this shortly.

—

(Before we proceed, let me point out how awesomely self-referential this post is. Déjà vu is French for “already seen,” and this is a re-view. Hehe.)

While no two Dreamcatcher title tracks really sound the same, except perhaps for the presence of distorted guitar and heavy percussion, “Deja Vu” stands out as something a bit different. I don’t think I’d call it a ballad, but it is definitely a lot more downbeat than their usual explosive blend of dance pop and rock. Even the chorus, which is usually where the song explodes, is a touch more restrained than usual. The song’s real payoff is the final iteration of the said chorus, which kicks up the tempo (and, briefly, the key) into more familiar territory.

This is an extremely risky gamble (especially for me, a stickler for good choruses), and I’ve seen this kind of thing attempted a couple of times this year, but this is one of the very few times I feel that it succeeds. This is due, in part, to the smartly-written two-part chorus. The first part is a repetition of the last line of the prechorus. It’s a much sparser, more repetitive hook than I’m used to from Dreamcatcher, who normally go full throttle in their choruses. That said, it’s extremely effective. The moment I watched the highlight medley and the trailer for the fantastic animated MV, it got stuck in my head, and hasn’t really left. However, taking it to the next level is the second half of the chorus, where Siyeon belts out a somewhat more rousing melody. This is both an effective way to keep the song from getting stale and a brilliant setup for the tantalizing finish.

More importantly, the whole song, not just the choruses, feels worthwhile. This is mainly because of the girls’ reliably top-notch performances. Siyeon is earning much-deserved recognition as a top-tier main vocal (listen to her cover of EXO’s “Overdose”), but everyone shines with what they’ve been given. Handong’s velvety voice finally gets significant time on a featured track, and while SuA/Gahyeon fans might have wanted rap verses from those two (especially with what Gahyeon showed on “Piri”) their vocals are no joke either. I need, however, to talk about Dami’s performance. Despite being the group’s main rapper, she doesn’t really get a rap verse per se, instead getting a few lines to sing. Anyone who’s listened to their B-sides or her cover of Adele’s “Someone Like You” knows that she can sing her heart out, and that is exactly what she does here. Her low, quivering voice here is perfect for the frailty conveyed in the verses, and adds yet another layer of texture to the gorgeous blend of vocals here. These verses could easily have been boring or throwaway, but the sheer emotion the girls convey in their performances elevates them and has me totally on board.

The production is also on point. Not that any of their material is badly mixed or mastered, but in the past, I’ve occasionally found myself wondering how much harder the song would hit with a more bottom-heavy bass or a gnarlier, thicker guitar tone. “Deja Vu” delivers on both counts. The spine-rattling bass pluck that precedes the chorus, best enjoyed on earphones, is probably the heaviest I’ve heard in any of their titles. As for the guitar tone, I don’t know if there’s really an appreciable change to it, but the absence of palm mutes usually used to give the song heaviness means that the guitar always rings out fully, fleshing out the symphonic chorus admirably. Add some beautiful vocal harmonies (especially from JiU/Yoohyeon) and some thunderous tom work, and you get a satisfying aural experience.

Based on the teasers, I worried that this would be just a little too languid or repetitive for me to really love, but as I should’ve learned from the “Dalla Dalla” teaser, you cannot judge a song until you have heard it in its entirety. This may be premature as hell, but right now, “Deja Vu” is my favorite song of 2019, and quite possibly the best song in Dreamcatcher’s unimpeachable discography.

Rating (based on the old Music Log format): 5/5

Name the members! What are the odds of getting at least one right? (an actually interesting math problem for stan Twitter)

This game has circulated K-Pop stan Twitter:

I’ve never actually attempted it myself, as I know that 99.999999% of the time I will screw this up. Badly. However, I’ve seen some pretty amusing results and heard some pretty funny stories about it.

Sometimes, people would play this game by getting a list of member names first to match them to their faces. This ostensibly increases their odds of getting everyone’s names right from practically zero to merely something undetectable by 10-digit calculators, and perhaps gives them something of a fighting chance of getting at least one. Anyway, let’s try this variant of this game, as described in a puzzle I posed to my followers:

with larger groups, the probability of getting at least one name correctly

[for example, a small group like mamamoo (4 members) vs. a large group like nct (21 members)]

— 🐼 GFRIEND LOCKDOWN | 1 year with 드림캐쳐 | #시연_v1 🐺 (@damivariable) September 1, 2019

It was pretty close, but a plurality said that the odds of getting at least one correct get better as the number of members gets larger. Let’s see how correct this is.

(Before we proceed, of course I’m making the following assumptions: we play fairly, i.e. we assign distinct names to every member.)

A Preliminary Approach

Let’s start by looking at small cases.

Say you have only one member. This isn’t particularly interesting, because there’s literally no way you can mess this up. This case also pretty much never pops up, so we can safely ignore it.

Now suppose we’re talking two members. Still not particularly interesting–there are only two possibilities. Either you get them both right, or you get them both wrong. So the odds are now 50%.

If we stop here, then our conclusion is that the odds are worse the more members you have.

Let’s move on to three members, the first actually “interesting” case here.

For convenience, let’s say that the members are 1, 2, and 3. Then we can think of each assignment of “names” as an arrangement of the three numbers: the first number is the “name” we assign to Member 1, the second number is the name assigned to Member 2, and the third member is the name assigned to Member 3. A number in the correct place is a member correctly named. Basically, getting them all right would be the arrangement 1 2 3.

There aren’t too many possible arrangements. In fact, there are exactly 6, and it’s not hard to list them all (red numbers denote numbers in their places):

1 2 3 ✔️
1 3 2 ✔️
2 1 3 ✔️
2 3 1
3 1 2
3 2 1 ✔️

Thus, the odds of getting at least one member are $\dfrac{4}{6} \approx 67\%$ .

In a similar vein, we can attack the case of four members. The list is a bit longer, but still manageable:

1 2 3 4 ✔️ 2 1 3 4 ✔️ 3 1 2 4 ✔️ 4 1 2 3
1 2 4 3 ✔️ 2 1 4 3 3 1 4 2 4 1 3 2 ✔️
1 3 2 4 ✔️ 2 3 1 4 ✔️ 3 2 1 4 ✔️ 4 2 1 3 ✔️
1 3 4 2 ✔️ 2 3 4 1 3 2 4 1 ✔️ 4 2 3 1 ✔️
1 4 2 3 ✔️ 2 4 1 3 3 4 1 2 4 3 1 2
1 4 3 2 ✔️ 2 4 3 1 ✔️ 3 4 2 1 4 3 2 1

This gives us odds of $\dfrac{15}{24} \approx 63\%$ .

Listing all arrangements isn’t exactly the best way to study this though. The reason is that the number of arrangements grows extremely rapidly as the number of members does. For five members, we have 120 arrangements; for seven members, we have 5,040 arrangements; for the 21-member NCT, there are more than 51 quintillion arrangements (51,090,942,171,709,440,000 to be exact)!

We thus need to attack this more systematically. In the succeeding sections, we will cover some basic counting concepts, then apply them to this problem.

Factorials

Again and again, we will be considering products of the form $n \times (n - 1) \times \dots \times 3 \times 2 \times 1$ where $n$ is a positive integer. We denote this product by $n!$ . Before you break your roommate’s ear yelling out numbers, know that this is actually read as “n factorial.” For example, $3! = 3 \times 2 \times 1 = 6$ ; $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ , and $21! = 51,090,942,171,709,440,000$ . As a matter of convention, it is useful to define $0! = 1$ . We will see why in a bit.

In fact, the number of possibilities for $n$ members is exactly $n!$ . To see why this is so, let’s look at the case for three members. For member 1, there are exactly three possible names you can give them. Then for member 2, since one of the names has been assigned to member 1, there are exactly two more names. Finally, for member 3, there is exactly one name remaining.

Thus the total number of possibilities we have for three members is $3 \times 2 \times 1 = 3! = 6$ . More generally, if you have $n$ members, then there are $n$ possible names for the first member, $n - 1$ possible names for the second member, and so on and so forth until you have to assign the last name to the last member, and the total number of possibilities is $n \times (n - 1) \times \dots \times 1$ which is exactly how we defined $n!$ .

Permutations and Combinations

A permutation of $k$ distinct objects out of $n$ distinct ones is an ordered selection of $k$ objects out of $n$ , where $k \le n$ . We denote by $P(n, k)$ the number of permutations of $k$ objects out of $n$ . For example, suppose we want $P(n, n)$ . That is, we choose all $n$ objects, and order them. Basically, we are just counting the number of ways to arrange $n$ objects. Thus, from the above, $P(n, n) = n!$ .

For another example, we want $P(3, 2)$ . We have exactly six such permutations: 1 2, 1 3, 2 1, 2 3, 3 1, and 3 2. There is another way to arrive at that answer without listing everything down. First, we count the number of possible choices for our first number. There are three. Then, for each choice of the first number the number of choices for the second number is two, and so we have $3 \times 2 = 6$ permutations, as desired.

More generally, suppose we want permutations of $k$ objects out of $n$ . For the first object, there are $n$ possibilities. For the second, since we already chose one object, there are $n - 1$ possibilities left. For the third, as we have chosen two objects, there are $n - 2$ possibilities. And so on and so forth. Until we get to the $k$ th object, where, since we have chosen $k - 1$ objects, we have $n - (k - 1) = n - k + 1$ possibilities. Thus,

$P(n, k) = n \times (n - 1) \times (n - 2) \times \dots \times (n - k + 1)$ .

In fact, we can rewrite this formula in terms of factorials. This is the product of all numbers from $n$ to $n - k + 1$ . However, we can think of this as the product of all numbers from $n$ to $1$ , but we remove everything from $n - k$ to $1$ . Thus, we have

$\displaystyle P(n, k) = \frac{n!}{(n-k)!}.$

(This is why we use $0! = 1$ .)

Combinations are like permutations, except that this time, order does not matter. We denote the number of combinations of $k$ objects out of $n$ by $C(n,k)$ .

For example, if we want $\displaystyle C(3,2)$ , we want to choose two out of 1, 2, 3. We choose 1 and 2, 1 and 3, or 2 and 3. (Note that choosing 2 and 1 is the same as choosing 1 and 2.) Thus $\displaystyle C(3,2) = 3$ .

Let’s find a more general pattern given $n$ and $k$ . From the above, counting the number of permutations of $k$ objects out of $n$ , we have $\displaystyle P(n, k) = \frac{n!}{(n-k)!}$ . However, for each combination of $k$ objects, we have $k!$ different arrangements, and thus there are $k!$ different permutations corresponding to any such combination. Thus, we have

$\begin{aligned}[t] C(n,k) &= \frac{P(n,k)}{k!} \\ &= \frac{1}{k!} \times \frac{n!}{(n-k)!} \\ &= \frac{n!}{k! \times (n-k)!} \end{aligned}$

The Principle of Inclusion And Exclusion

Suppose we want to count the number of items in a bunch of sets. Say we have two fandoms, A and B, and we want to count how many people are in at least one of the two fandoms. First, we count the number in A, then we count the number in B. Then we add them. Unfortunately, this approach gives us too many, because some people are in both fandoms and get counted twice. To compensate, we subtract the number of people in both fandoms. Now everyone is counted exactly once, and we are happy.

Let’s try this for the case of three fandoms. Let’s add C as a third fandom. So if we naively add the number of those in A, those in B, and those in C, we’ve counted the number of people in A and B, B and C, or C and A twice, and in fact we’ve counted the number of people in all three fandoms thrice! To compensate, we subtract the number of people in A and B, B and C, and C and A.

However, when we do this, we subtract the number of people in all three fandoms thrice. Since we added them thrice earlier, they are no longer accounted for. To finish off, we have to add this number back.

In general, if we want to count the total number of distinct elements of a union of sets, we first count the number in one of the sets, then we subtract the number in two of the sets, then add back the number in three of the sets, and so(w)on and so forth.

This formula is called the principle of inclusion and exclusion.

Let’s now try and apply this to our original problem. For the case of four members, we counted exactly 15 arrangements with at least one correctly placed number. Let’s see if we can get this number without listing everything.

First, we count the number of arrangements with one correctly placed number. If 1 is placed correctly, then we can place 2, 3, 4 wherever we want. This gives us $3! = 6$ arrangements where 1 is placed correctly. Similarly, we have 6 arrangements where 2 is placed correctly, 6 arrangements where 3 is placed correctly, and 6 arrangements where 4 is placed correctly.

Of course, if we naively add these together, we get $6 \times 4 = 24$ arrangements with at least one correctly placed number, which doesn’t make any sense at all.

The problem is that we’ve counted some arrangements more than once. For example, when we counted the arrangements with 1 placed correctly, we also counted some arrangements where 2 was placed correctly as well; these arrangements then got double-counted when we counted the number of arrangements with two placed correctly.

To compensate, we count the number of arrangements with two correctly placed numbers. For example, if we place 1 and 2 correctly, we have to place 3 and 4, and there are exactly $2! = 2$ ways to do this. Similar count for 1 and 3, 1 and 4, 2 and 3, 2 and 4, 3 and 4. Thus we subtract $2 \times 6 = 12$ from our answer.

This leaves us with an undercount now, though, for similar reasons. The arrangements where three numbers were placed correctly were subtracted too many times, so now we have to add them back in. We have exactly one arrangement where 1, 2, and 3 are placed correctly. Similarly, we have one arrangement where 1, 2, and 4 are placed correctly; one where 1, 3, and 4 are placed correctly; one where 2, 3, and 4 are placed correctly. We thus add $4$ to our answer.

And again, to compensate for the overcounting, we have to subtract the arrangement where all four are placed correctly, namely 1 2 3 4. Thus, finally we subtract $1$ , and our answer is $24 - 12 + 4 - 1 = 15$ , as we wanted.

Let’s tackle this generally. Suppose you want to count the number of arrangements of $n$ numbers with at least $k$ of them placed correctly. First, we choose which $k$ of them to place correctly. There are $C(n,k)$ ways to do this. Then, for the remaining $n - k$ objects, there are $(n - k)!$ ways to arrange them. Hence, there are $\displaystyle C(n,k)\times(n-k)!$ such arrangements.

Thus, by applying the same reasoning as in the preceding example, the number of arrangements of $n$ numbers with at least one number placed correctly, which we shall denote by $A(n)$ , is given by the formula

$\begin{aligned}[t] A(n) &= \displaystyle C(n,1)\times(n-1)! - C(n,2)\times(n-2)! + C(n,3)\times(n-3)! - \dots + (-1)^{n-1} \times C(n,n) \times (n-n)! \\ &= \frac{n!}{1!\times(n-1)!}(n-1)! - \frac{n!}{2!\times(n-2)!}(n-2)! + \frac{n!}{3!\times(n-3)!}\times(n-3)! - \dots + (-1)^{n-1}\times\frac{n!}{n!\times 0!}\times 0! \\ &= n! \times \left(1 - \frac{1}{2!} + \frac{1}{3!} - \dots + \frac{(-1)^{n-1}}{n!} \right) \end{aligned}$

and so, if we define by $P(n)$ the probability of getting at least one out of $n$ members right (let’s not confuse this with $P(n,k)$ ), we have

$\begin{aligned}[t] P(n) &= \frac{A(n)}{n!} \\ &= 1 - \frac{1}{2!} + \frac{1}{3!} - \dots + \frac{(-1)^{n-1}}{n!}. \end{aligned}$

With the above formula, let’s study the behavior of the values of $P(n)$ .

$\begin{array}{c|c} n & P(n) \\ \hline 1 & 1.000000 \\ 2 & 0.500000 \\ 3 & 0.666667 \\ 4 & 0.625000 \\ 5 & 0.633333 \\ 6 & 0.631944 \\ 7 & 0.632143 \\ 8 & 0.632118 \\ 9 & 0.632121 \\ 10 & 0.632121 \end{array}$

You can see that the values go up and down. However, they quickly get close to about 0.632121. In fact, we are sure they won’t get too far from that after some point. Note that the difference between consecutive values of $n$ is $\dfrac{1}{n!}$ . Since $n!$ gets large rapidly, $\dfrac{1}{n!}$ gets very small rapidly. Moreover, aside from the changes getting very small rapidly, they also take turns going in opposite directions, so any “attempts” to break away from this limiting value are immediately undone by the next change.

This reasoning can made just a bit more rigorous to conclude that indeed the answer is that the probability, in fact, stays roughly the same as $n$ gets large.

An Infinite Series

We can do even better than 0.63, in fact, if we want a name for this value. Euler’s number, denoted in many calculators as $e$ with approximate value $e \approx 2.71828$ , is a magical and mysterious thing, and if I were to try and explain all of its awesome properties, I would never finish this article. Anyway, one of the cool things about it is that for any number $x$ , the sum

$\displaystyle 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$

gets closer and closer to the exponential $e^x$ the more terms we add.

If we let $x = -1$ , this means that

$\displaystyle 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots$

gets closer and closer to $e^{-1} = \dfrac{1}{e} \approx 0.37$ .

Hence, from simple manipulation, we see that

$\displaystyle \frac{1}{1!} - \frac{1}{2!} + \frac{1}{3!} - \dots$

gets closer and closer to $1 - \dfrac{1}{e} \approx 0.63$ the more terms we add. But in fact the sums of the first $n$ terms here are exactly our values of $P(n)$ .

Exercise: The Case $n = 7$

e0097535b9082b6bd787e3d4a7955c5c92a32c4fc538e5d381aa6eef491b86bc.jpg

These girls’ stage names are JiU, SuA, Siyeon, Handong, Yoohyeon, Dami, and Gahyeon.

Based on the formula above, what are the odds you will get at least one of them right?
Do try and identify these members.
Bonus: try to identify them in the clip below.

The 2019 Music Log: Ep. 1 (a reboot)

(aka the real reason I launched this blog)

So, here we are, rebooting my 2019 Music Log, which I started on Twitter. For my newer readers or those who didn’t follow me from Twitter, this was a Twitter thread in which I decided to review what I was listening to. Last year, it was mostly metal album reviews; this year, it seems I’ve turned to K-Pop singles, discussing an occasional metal album here and there.

2019 MUSIC LOG: A THREAD

This year, like with last year, I will be posting reviews of full albums I listen to. Again, mostly full-lengths, but I *might* review the occasional EP/split.

This time, against my better judgment, I will also review (mostly K-Pop) singles.

— x** | lf: 1000 corviknight ⚔️🛡️ (@xdoublestar) January 2, 2019

While I enjoyed doing this thread very much, I decided that I wanted to change things up for a couple of reasons. First, navigation was a nightmare. The thread was at roughly 200 tweets long, meaning if I wanted to dig up something from March I’d have to sift through a lot of stuff. More than that, I found my chosen two-tweet format limiting. Some songs I didn’t feel like discussing for more than one tweet. Other songs I really wanted to dive into in more detail. With me deciding to be more proactive in pursuing my creative interests, I’ve decided (with some encouragement/prodding from friends) to launch this blog, and to take the Music Log here.

First major change: I’ve decided to somewhat try and deflate the ratings. Followers of my previous thread may notice that I handed out a lot of 4s. I grade like a teacher, partly because I am one, so I like to give high scores. This also goes with my philosophy of generally trying to see the good/enjoy in music, though of course with a critical ear. However, as a result I find that I’ve given a lot of songs 4s even though my enjoyment levels over these vary rather wildly. Thus, I’ve decided to move with a working average-to-good song rating of 3.5, moving everything else appropriately. This decision is effective now, but will not affect older ratings. I’m too lazy to revisit five months’ worth of reviews, and I’ve already revisited most of them anyway.

(Speaking of which: I will, however, continue doing a revisited feature, albeit slightly tweaked. I’ll focus less on amending older reviews and more on recapping what grew on me from the last month. More on this on the 15th of July.)

Anyway, without further ado, here is the rebooted 2019 Music Log!

Note: Before anyone asks, I’m not gonna cover any of the BTS World songs yet; I’ve figured I’ll do a full breakdown/review of the album once it comes out in its entirety. I’m still gonna post my Map of the Soul: Persona review (not-really-spoiler: I like it) at some point. I’m really backlogged with regards to album reviews, and I hope to be able to fix this soon.

Red Velvet, “Zimzalabim” (3.5/5)

I don’t mind messy in my pop music (I will fight to the death for songs like “Wolf” and “I Got A Boy”) but I do prefer, if not a proper chorus, some central melody amidst the chaos. That said, the chanted titular refrain is not quite it. I get the purpose and even the appeal, but within the framework of the song it serves as example #22537689 of the trap-inspired, momentum-killing chorus that has infested modern American pop music, and now K-Pop. (The background chimes sadly do not help it.) Otherwise, though, there’s a lot to like about this song. The melodic parts are reliably infectious, the electro house dance break is a nice infusion of energy, the vocals in the bridge are spectacular, and the spirited final chorus is one I wish they’d utilized more throughout the song.

Stray Kids, “Side Effects” (3.5/5)

Had I dropped a review for this one the same day I listened to it I would probably have given it a 2.5. This song is rather inscrutable, alienating even for the first few spins, but a pulsating bassline and some subtle hooks reward the persistent listener. It is still a little too stop-and-go for my liking, and I still prefer “Miroh” and its galvanizing melody, but this is a daring piece of music that has won me over.

SF9, “RPM” (3.5/5)

It seems like SF9 heard all my complaints about their last comeback and quickly fixed them. The needless Auto-Tune is mostly gone. They’ve replaced the languid reggae-esque verses with some chugging electric guitar. They do come in with a rather jarring drop that had me mouthing the chorus of J-Kwon’s “Tipsy” over it for some reason, but they do actually follow it up with a proper sung chorus, something I wish more artists would do! This isn’t quite as anthemic as “Now Or Never,” but this is on the good side of the SF9 releases I’ve heard.

Lay, “Honey” (1.5/5)

I admire Lay’s work as one of EXO’s vocalists, but his solo material just hasn’t been for me. That said, last year’s “Namanana” was pretty nice. This one throws away all the goodwill that one built up with me, somehow managing to be both forgettable and incredibly obnoxious at the same time. The repetitive and rather corny music box leitmotif is grating enough, and his disaffected delivery isn’t helping keep my interest, but the random “skrrting” is the final nail in the coffin. I’ve forgiven boy groups for doing it like once or twice in an otherwise good song, but Lay took it way too far. I don’t really see myself returning to this.

Twice, “Breakthrough” (4/5)

I’m a fan of Twice’s upbeat concepts, but this is absolutely a style I want them to explore further. Those gorgeous harmonies, that sultry chorus, that brassy instrumental that is edgy enough to be exciting without succumbing to trendiness. It vaguely reminds me of some of my favorite Mamamoo material. While I’m not sure I’d really call it a breakthrough, I really enjoy this.

Twice, “Happy Happy” (3.5/5)

This is lighthearted summer fun that thankfully doesn’t go into “BDZ”-esque cloying territory. It’s not the showstopper that last year’s “What Is Love” was, and it doesn’t have the knockout moment that defines Twice’s best work, but it’s more than adequate.

Somi, “Birthday” (2.5/5)

Congratulations to Somi for finally debuting! Now she gets to put out a disappointing drop chorus like everyone else. The rest of the song is fine, promising even (though I could do without some of the Meghan Trainor-isms) but based on this and his other recent material it seems that Teddy has forgotten how to write an actual chorus. What a shame.

Monsta X feat. French Montana, “Who Do U Love?” (4/5)

I’m really not sure that I want Monsta X to keep doing this style (as big a fan as I am of it), but do they pull it off. French Montana’s rap verse is awkward, clunky, and has the annoying skrrting noises I complained about above, to the point that I can’t help but imagine if one of the group’s very capable rappers had just taken that verse instead, but the pleasant side effect of this otherwise superfluous feature is that we get to hear I.M and Joohoney sing, and they’re not bad at it. Everyone who knows my retro instincts would figure out that this is bait for me, and I am absolutely taking it.

Ateez, “Illusion” (3.5/5)

The easygoing, sing-songy even chorus is just so ridiculously sticky. It’s the kind of slurred, Auto-Tuned singing that can get irritating, but the melody is so hooky that I forgive it that. Other than that it’s a straightforward hip-hop song, but a more than adequate one.

Ateez, “Wave” (3/5)

I feel that “Illusion” is the better track, as this is too tropical pop for my liking, but this song got them their first win, so I can’t really complain. Besides, the square wave synth in the prechorus is awesome, and they have the charisma to pull this off.

Taylor Swift, “You Need To Calm Down” (3/5)

Lorde’s “Royals” is a decent song, but I don’t think it needed to be remade, let alone repackaged into a different track, this soon.

Bonus clip:

Siyeon, “Speechless”

Speaking of remakes. I love Lee Siyeon. That’s it. Seriously, this might be my new favorite special clip of theirs. While her “Faded” cover is close to my heart and has helped me recruit a few new InSomnia, this one is a far better showcase of her vocals, especially her criminally underutilized lower range. I’ve been pretty loud about emphasizing that songwriting is more important to me than vocal talent or technical ability, but Dreamcatcher have long had both in spades anyway, and I think they’d definitely be a bit worse off without the bonafide rock star main vocal/goddess that Siyeon is, so I’ll let her flex on all of you here.

And that’s it for the first episode of the 2019 Music Log! See you all… not sure when I intend to write another one. Maybe Friday, but no promises. I don’t know yet.

First post, and a brief intro

Welcome to my little corner of the internet! I’d been hinting at this for a while now on my Twitter, but I’ve decided to start a blog. This is a space for things I’d like to have more than 280 characters for. Most of my posts here will be about music, specifically K-Pop and metal. I’ll probably end up posting a lot of other things, though, like math (I just learned WordPress has improved its $\LaTeX$ support, yay!), art, poetry, music, basketball, and miscellaneous rants. Basically, this will be my Twitter, but more verbose, and deader.

About me

I’ve had my fair share of aliases, but these days I tend to go by $X^{**}$ . In mathematics, this is the double dual space of the vector space $X$ . What that is, I might explain in greater detail if I’m not feeling too lazy.

Anyway, I’m a mathematics grad student and teacher in my late 20s. I listen to a lot of music, but these days my attention is mostly on K-Pop and most subgenres of metal (except perhaps funeral doom and drone, as I have little patience for them). Some of my favorite musical acts are Dreamcatcher, Madder Mortem, My Chemical Romance, Riverside, Spawn of Possession, Dodecahedron, Nevermore, and Symphony X, among many, many others. I mess around on the guitar and keyboards a bit, and like to do ill-advised karaoke to songs I have no business even attempting to sing. Rest assured, an actual release is probably years away. Or not.

I’ve also been playing Pokémon most of my life. My favorites are (Mega) Houndoom and Jigglypuff. Others I love, in no particular order: Mismagius, Honchkrow, Escavalier, Joltik, Roserade, Aegislash, Haxorus, Cinccino, and yes, Corviknight (despite it not having been released yet).

That’s all for now.