## Thursday, 29 July 2021

### 奧運/麥兜/臘鴨

－－麥兜故事《黎根之歌》

1996年亞特蘭大奧運，風后李麗珊在滑浪風帆項目奪得金牌。你還記得嗎？

2021.7.29

## Sunday, 25 July 2021

### Thoughts on IMO2021 Q2

IMO 2021 is finally over. As usual I skimmed through the question and focus on the inequality problem where I usually perform the best. It was Q2 this year, but it is not an average Q2.

It turned out to be as hard as Q3 (mean score 0.375 against Q3's 0.372, but Q2 gives out more partial scores and more zeros), both harder than this year's Q6 (mean 0.481). Possibly one of the most controversial Q2 in IMO history. What makes it so interesting? Here are my thoughts.

First let us look at the question.

Problem (IMO2021 Q2). Show that
$\sum _{i,j} \sqrt{|x_i-x_j|} \leq \sum _{i,j} \sqrt{|x_i+x_j|}$
for all real numbers $x_1,...,x_n$.

If you have already spent time onto this problem, here is the solution. This is similar to the solution mentioned by TheUltimate123 in the AoPS thread.

Claim 1. The case $n = 0,1$ is trivial.

Claim 2. LHS is invariant upon shifting all $x_i$ by a constant (let us simply call that "shifting" for the rest of the solution)

Claim 3. If $x_k = 0$ for some $k$ then the $x_i$-related terms on both sides are the same which is
$2\sum _{i\neq k} \sqrt{|x_i|}$. Therefore we now WLOG assume that all terms are non-zero.

Claim 4. If $x_k+x_l=0$ for some $k\neq l$ then the $x_k$ or $x_l$-related terms on both sides are also the same, which is $2\sqrt{2|x_k|} + \sum _{i\neq k,l}(\sqrt{|x_i-x_k|}+\sqrt{|x_i+x_k|})$.

Claim 5. RHS is not minimal upon shifting unless $x_k+x_l=0$ for some $k,l$.

Proof: Jensen gives that
$\sqrt{|x+y+\varepsilon|} + \sqrt{|x+y-\varepsilon|} \leq 2\sqrt{|x+y|}$
for $x+y\neq 0$ and $\varepsilon \leq |x+y|$.
Summing over all $x_i$ gives
$\sum _{i,j} (\sqrt{|x_i+x_j+\varepsilon|} + \sqrt{|x_i+x_j-\varepsilon|}) \leq 2\sqrt{|x_i+x_j|}$.
for sufficiently small $\varepsilon$. Therefore a shift of either $\varepsilon /2$ or $-\varepsilon /2$ would give a smaller RHS while having LHS fixed.

Note that $x_k+x_l\neq 0$ is required to maintain convexity of $\sqrt{|x+y|}$.

Therefore the induction is completed. If $x_k=0$ for some $k$ then we reduce to the $n-1$ case by claim 3. Otherwise shift until $x_k+x_l=0$ for some $k,l$, then reduce to the $n-2$ case by claim 4.

The above also gives the equality case in full. Firstly adding arbitrarily many zero terms does not affect the sum. If we assume non-zero terms then the only instance where RHS decreases is by shifting, so in order to obtain equality the terms must be canceled pair by pair without the need of shifting. We therefore conclude that equality holds if and only if the terms are symmetric along zero.

*

There are alternative solutions using completely different approaches like the use of integral to prove the more generalized case $E|X+Y|^{1/2} \geq E|X-Y|^{1/2}$ for random variables $X,Y$. Even more fascinating, one used the integral
$\sqrt{|a+b|}-\sqrt{|a-b|} = \frac{1}{\sqrt{2\pi}}\int ^{\infty}_0 x^{-3/2}(\cos (a-b)x-\cos (a+b)x)dx$
$= \frac{1}{\sqrt{2\pi}}\int ^{\infty}_0 2 x^{-3/2} \sin ax \sin bx dx$.

It is also note worthy that the original problem uses the exponential $\alpha = 1/2$. The statement holds for all $\alpha \in (0,2]$. Convexity works all the way up to $\alpha = 1$ but college techniques is absolutely necessary for higher $\alpha$. One approach is to use linear algebra and binomial theorem for non-integer exponents(notice that when $\alpha = 2$ it simply reduces to prove that $4\sum x_i x_j \geq 0$ which is obvious. But then for $\alpha \in (1,2)$ you can also write $|x_i-x_j|^{\alpha} = (x_i^2+x_j^2-2x_ix_j)^{\alpha /2}$, then you may expand and compare both sides using positive-definiteness), and the other is a generalization of that integral identity (with the help of gamma function).

You might have noticed something here already: the abundance of calculus solutions, the clear generalization to something famous and common in undergraduate maths, and the flexibility in exponential (cf. most $a,b,c$ type inequalities that you saw in 2000s-2010s IMOs), all suggests that this is a very good question for undergraduate contests.

However if we consider IMO to be the pool of the most talented math students in the world (even better than those participating undergraduate contests on average), why is this a hard problem to them? And why did they perform poorly at the end?

I figured out the solution almost immediately, like within 15 minutes, and the flow of thoughts is as follows: the shift is clear immediately. Then I tried to solve the case $n=2$, which is not easy already. By using the parametrization $x_2 = rx_1$, it is found that the equality is tight if $r=-1$. Putting back to the inequality with $x_1$ and $x_2$, it can be found that when $x_1=-x_2$ the terms on both sides are equal. The cancellation trick is soon found to be working with larger $n$. We now know that induction may work as long as we can reduce to these cases. (I missed the case $x_k=0$ when I first solved the question, so I didn't get a 7 in that sense...)

By reducing to these case, we must show that shifting to these case gives a smaller RHS, and this is what we need to prove. We applied Jensen and showed that we can always get smaller as long as $x_k+x_l \neq 0$ so we are done.

...or not?

You might have spotted the problem already. The hard bit of the question is the indirect relation between the fact that you can shift to reduce RHS and the goal that you can shift to a smaller RHS that $x_k+x_l=0$.

It goes through only if you have a calculus sense, or else it could trap you forever.

For starter let us consider to shift by the maximum possible distance $\pm\min |x_i+x_j|/2 = \pm t$ so that we reached a singular point (where $x_k+x_l=0$...). The problem is: we only know that a either a shift of $+t$ or $-t$ gives a smaller RHS but not necessarily both. If you shift to the point where $x_k+x_l=0$ you are not guaranteed a smaller RHS.

Or, if you shift little by little to reduce RHS, can you guarantee that you end up with a singular point?

These are never a problem if you know calculus argument, and they can only be understood via calculus argument.

From a global point of view, the sum is definitely increasing with large enough shifts. Since the sum is also bounded below and is continuous, there must be a global minimum. Since the minimum does not occur at the smooth part it must be at one of the singular point. We can shift directly to that point to complete the induction.

To shift gradually you may argue that the derivative between two singular points do not change signs so one of the two neighboring singular point must be a local minimum [or even weaker, a local minimum with respect to the closed interval bounded by the two singular points]. Alternatively, since we know that $(\sqrt{x})' = 1/(2\sqrt{x})$ which rockets as $x\to 0$, we argue that the derivative of $\sqrt{|x_k+x_l|}$ overwhelms all other terms as we shift $x_k+x_l\to 0$, so that must incur a minimum [hence even singular point is a local minimum].

And how can you argue that by "elementary means"? Probably not, unless the official solution suggests something spectacular.

If you check everything else in the solution they are pretty standard. induction of course, simple special case ($x_k=0$) and a more complicated special case ($x_k+x_l=0$), substitution [in fact, instead of a shift, we may apply the substitution $x_i = y_i+t$ to achieve the same] to hold one side invariant, Jensen on convex functions --- these are all standard tricks in IMO. It is the final link that completes the solution, is hard to realize and that link is, regrettably unavoidable.

It is a beautiful and insightful inequality. Again quoting someone in the thread -- the beauty of this problem is that $n=2$ provides everything you need: minimal condition and the shifting. (Although to be fair, $n=2$ hides the technical problem as described above, and you will only meet that at $n\geq 3$.) It simply requires Jensen (or even just elementary comparison because it is just square root) in a two line calculation. We do not need 30 lines of machinery like other IMO inequalities did.

We can continue the list and praise the problem in many more different ways. The only criticism is that the problem unfortunately does not belong to IMO.

*

I last wrote about IMO problems in 2006, before this blog opened. I wrote that in my previous blog, a site that is long abolished (at least I do not have to worry about Blogger closing soon...yet).

I was of course, too immature to understand what happened -- I copied someone else's solution and put it on my blog. For whatever reason that post received tons of comments out of nowhere: recalling that discussion now I believe that those people are truly qualified to do these IMO problems. It is still a mystery up to now how did they found my post (out of a blog that never talked about advanced mathematics otherwise) and open a contextful discussion there, but this is the beauty of Internet in the 2000s.

Without running into a nostalgic loop, let me finish with that particular IMO problem that I discussed back in 2006. That problem was 2001 Q2 -- another Q2, another inequality, and another problem that allows elegant solution using Jensen's inequality.

Problem (IMO2001 Q2). Prove that for all positive real $a,b,c$ the inequality $\sum _{cyc} \frac{a}{\sqrt{a^2+8bc}} \geq 1$ holds.

## Friday, 23 July 2021

### 被青梅竹馬抓來(略) (2)：別人十年磨一劍，你一年磨一劍已經非常快了

"優雅な午後"
Character design: @kuonyuu, Illust: @茶桜みゅ commissioned by forretrio. Pixiv
Editing and re-posting are prohibited // 無断転載、無断使用禁止です

*

「大家早安～歡迎來到國立魔法學園，也歡迎大家成為魔法科的一分子。再次自我介紹一次，我是庫里斯，將會擔任魔法科的導師一職。特長是風魔法－－嗯，相信在座各位已經充分理解這一點了吧？」台下立刻傳來殺人般的目光。

「哎呀，真是危險～」庫里斯從講台上跳下來，水桶在他的操控下飛到他的面前：「我看看……裡面裝的居然是冰？這樣砸到頭肯定要進醫院呢。老師我可以接受水桶，但是冰桶有點過份了喔，克萊伊同學。」他特地把水桶裡面展示給同學看，只見裡面的水已經結成一整塊冰。

「嘛，老師的確不知道誰把水桶放上去，不過我看到你嘗試用魔法把水桶推下來呢。這樣遠距離操作其實很顯眼，我建議你下次準備好觸發性的魔法陣喔。」

「在此之前，我希望你們想一下－－你們來到這裡，是希望得到甚麼呢？既然你們進入了魔法科，那就代表你們的能力已經遠超常人，甚至已經可以成為即戰力了。這樣的話為甚麼要浪費三年在這裡再聽一次你們以前很可能已經聽過的知識呢？」

「如果你們是貴族的話，在這裡就讀就只是一個過程而已。身為貴族擁有一定的魔法能力就足夠了，過份強大只會讓人懼怕。你們大部分人畢業都會接過家業，在帝都舒適地生活著，這樣魔法再好又有甚麼用呢？」

「對於平民入學的你們，我要恭喜你們：進入魔法科代表通往帝都頂層的大門已經為你們躺開，你們是受益最大的一群。不過你們也可以想一下，進了這裡就一定要從事魔法相關的職業嗎？很多人都覺得身為魔法師可以當冒險者發大財，可是你們知道正是因為魔法師的折損率是冒險者之間最高的一群所以才這麼值錢的嗎？」

「我沒有要勸大家別當冒險者的意思，相反我很樂意以過來人的身份幫助你們。不過我想你們既然有能力進入這個班級，自然也有餘裕考慮標準課程以外的興趣－－你們應該不會只對冒險有興趣吧？」

「我的想法是讓你們花時間找出你們希望學習的東西，然後由我或者其他人來指導。我會花三分一的時間講解標準課程內容裡面比較有用的東西，剩下的就作為課題交給你們了。」

「至於另外三分二的時間就是個別指導的時間了。唔－－具體時間要怎樣分配主要還是看你們的興趣在哪個方面。對魔法或者戰鬥有興趣的話可以一起去地下城冒險；就算是鍊金術或者商業這些東西我也可以找我朋友幫忙。當然你們想要找我過招的話只要別偷襲，我隨時都歡迎哦。畢竟－－」

「為師我，不討厭打架呢。」

*

「庫里斯老師你好。」公主殿下首先打了個招呼她的名字是菲莉，是帝國正室的次公主。本人是頭腦派，體質一般所以成為了唯一一個沒有參加跟老師對戰的魔法科學生。不過她也繼承了優秀的血統，可以使出強力的水跟光魔法。

「菲莉公主您好，」他稍鞠一躬：「雖然公主您沒有參與比試，不過我早就聽說過您是一位優秀的光魔法師。有機會讓我指導您是我的榮幸才對。」

「老師過獎了。聽說老師作為冒險者十分活躍，這些實戰經驗對我們學生來說是非常寶貴的呢。不過－－」她稍微頓了一下：「我只是有一丁點擔心，那件事會影響老師你與貴族的關係嗎？」

「嗯。」她不置可否地點了下頭然後呼喚伊雅娜道：「其實今天想找老師你的是她才對。伊雅娜～你有東西想問吧？」

「嗯？」

「我問過父親，他也認同你的看法。不過他也說了只有對上很強的對手時才有犧牲威力而加強隱蔽性和速度的價值。即便是這樣，老師也會建議我這樣朝這個方向鍛練嗎？」

「非常好的問題呢。這樣說好了，對你來說電光劍到底是常規招式，還是保命的底牌呢？」

「嗯……算是底牌？我的魔力本來就沒法一直在劍上附魔，上次我也是最後才把這招掏出來。」

「作為底牌的招式沒有一擊必殺的能力就沒有意義了。在高階的戰鬥中，所有常規套路都是為了那一兩招殺著而準備的。你想一想，作為殺著到底是可能打不到高階對手的超強力攻擊，還是稍弱一點但是很穩的攻擊更好呢？」

「老師的意思是，對上一般的對手不用全力的電光劍也能贏，所以練習加速版電光劍應付高手更好嗎？」

「這個思考方向沒錯，不過讓我來選的話我覺得兩招同樣重要。全力的電光劍並不是一無用處－－跟不上那速度但擁有優秀防御能力的人和魔物多的是，那招作為破防非常實用。反正底牌又沒人規定只能有一張，都一起練就好了嘛。」

「……」伊雅娜同學有點無言。他是把底牌招式當成甚麼了，以為這是回家練一個星期就能熟練地使出來的招式嗎？為了讓魔法效力與她往前突刺的動作同步，她花了半年多的時間才算小成，期間魔力失控電到自己等意外更是不計其數。

「要我說的話，你甚至可以考慮再加練一招－－將輸出的魔力壓制到可以在格劍時隨時使出來的程度。我保證這招由主修物理劍術的妳使出來的話絕對能嚇到對手的。」

「老師……你是不是忘記了一件事……」

「嗯？怎樣了？」庫里斯擺出燦爛的笑容。

「咳唔，所以你這個學期的第一個課題已經決定好了－－」他清了一下喉嚨道：「我希望你花點時間研究一下在劍上附魔的不同方法。在你對兵器附魔有一定了解以後我們再來討論吧。

「好……好吧。」伊雅娜並不相信老師真的可以幫她練出電光劍的變種，但是作為學習的課題的話還是可以接受。圖書館應該有相關的書籍吧？

……才怪。

「別人十年磨一劍，你一年磨一劍已經非常快了呢！」老師帶著一貫的笑容這樣應道。當下她真的想一拳灌在老師的帥臉上，不過她覺得這一拳打在風盾上使她骨折的機會更大，所以她才忍住沒有出手。

=============

## Sunday, 4 July 2021

### 04/07/2021: Game optimization patterns

Two notable events in the world of competitive gaming.

Rolling has been a viable style in classic Tetris and we found Cheez breezing through loads of world records. The way he broke the 100 line speedrun so casually when the previously newly set record was made upon multiple extreme optimization is simply spectacular. Rolling allows one to input above 20Hz edging close to the hardware limit (30Hz for the on-off cycle). The problem now is about consistency and flexibility, and we will know more in upcoming CTM and of course CTWC.

The SMB1 Any% record has been broken once again, this time entering the last possible second at 4:54. The human sum of best is merely a few frames slower than TAS. This is probably the closest against TAS in gaming history after dragster.

They represent the frontline of game optimization: it reaches the realm where inputs are almost impossible to humans, which is a mechanical problem rather than a software problem. This matches the spiral theory where software and mechanical optimizations take the role in turns.

Fortunately we also developed more tools during the years using science. Taking SMB1 as an example, in early days people experimented different tricks and do whatever is the fastest. We knew that there are shortcuts (e.g. 4-2 clips) but we didn't know how that worked but to take that as pure luck. As time passed, we started to use TAS first to get a route that works. We then refine that to "humanly possible" tricks -- after that it's speedrunners' job to execute those "humanly possible" tricks. And how refining works? Well the science of NES games and SMB1 itself is very well established, so people can calculate all the subpixels bit by bit for the possible route. If we take subpixels as a chaotic parameter as affected by the dynamics of the character, we may perform exhaustive searches around the possible routes until we found one that works for players.

Since I mentioned the science behind speedrunning, I should mention the Youtuber that does a great job on the matter. In fact, everything I wrote above is just a set up for one of my favourite comment that mentions 4 great youtubers, three of which make videos that I would watch at highest priority. Here is the comment from this video...

Remember people:
* Summoning Salt for History
* Bismuth for Science
* FlibidyDibidy for Technology
* Karl Jobst for Social Sciences

## Friday, 2 July 2021

### TWEWY the Animation: Beat week and general thoughts

Joshua the smol character in TWEWY.

I expected Beat's week to last 4 episodes where the last episode can be spared for something else. It turns out that they spent 4.5 episodes on the week. This is not a bad choice as spending a complete episode decoding (unless Vivy ep.12.5 just trying to reiterate from another POV) will surely scare the pure anime audiences away.

At the same time, problem arises: there is not enough time to decode everything that is necessary to give a full story. Well that even happened in the original game: the secret reports contained a vast amount of information yet they are not enough to tell the full story that an average player/reader could understand. What they did in the anime is actually a wise move -- instead of covering everything essential, they highlighted the difference between OG and the anime which is considered the "true storyline" that leads to neo. There are already too many unanswered questions throughout the episodes and it is a waste of effort to answer them all in 12 minutes -- just leave them to neo instead.

I think I have criticized enough about the choice of pins (and their effect) and monsters so I do not plan to do that in detail again here. That is again disappointing in ep.8-12 as we observe loads of week 1 monsters and more boring (and wrong) presentation of pin psychs. So instead I would highlight the differences between OG and the anime here. Some of them are quite influential and some are not (and probably also true in OG just not explicitly shown), but most of them are worth mentioning.

Episode 8:
- Reapers had their eyes hidden in OG, but as shown in the anime although they still their consciousness (for now), they are probably under partial control of the pin already.
- I said that I don't want to criticize the choice of pins anymore but here's something that I actually appreciate. They used a few Pegaso pins in the late episodes but I couldn't recognize them easily. The only thing that I noticed is the Pegaso icon on the pin -- but then when I stared at the pin list, every Pegaso pin has the same logo on it! This is a funny little observation from me.
- We can see Kitaniji's hand timer in the anime which the time running out -- this is not explicitly shown in OG although he said something like that.

Episode 9:
- Makoto finally appeared! Well there is none other than him who can do a better job to distribute the red skull badges.

Episode 10:
- Eiji's truck appeared in the anime, but it is clear that the Shibuya sheeps didn't care anymore. They were already under unification of consciousness.
- Starting from the last episode it feels like the battle lags quite a lot which is surely disappointing.
- But Sho's teleportation is so cool! I wish that is shown in the week 2 final fight because every OG player would agree how painful it was to hit him in that fight...

Episode 11:
- Konishi has the potential to do ASMR on YouTube.
- Kind of interesting to see that Shibuya river is not linear -- where do the other branches lead to?
- The battle is horribly simplified and even more it's not happening in the Dead God's pad. If you do not plan to show the fight there, why putting that in previous cutscenes and also in OP?
- In OG the formation of the special space created in the Konishi fight was explained and that the pact were explicitly sent back to ambient space before the fight concludes. Instead in the anime the space simply crumbled without further explanation. The same problem happened in the Sho fight already.
- Shiki reappeared but Joshua didn't as in OG. This is actually making sense because it is said that their return was due to the return of the entry fee. Shiki's part of the entry fee, but Joshua isn't. It is possible that he just showed up randomly after playing enough Tin Pins (see secret reports week 3) and try to get into the team.
- Sho was not found dead under garbage heap. Instead he was able to talk to Neku in the anime. In fact if you think really carefully almost all characters were erased at the spot after being found severely injured, and Sho was the only exception. That could be a hint that he's not dead even in OG.
- They combined the two fights (Kitaniji phase 1 and berserk Shiki) into one. Other than my complain that it didn't happen in the pad, it's sad that the badge cutting scene was not replicated in the anime.
- Golden chain psych...that is not in the game, and no pins were specified there. The Mr.Mew ball is utterly nonsense but still made me smiled right there.
- It's still a mystery what happened when Kitaniji trapped Shiki, Beat and Joshua and try to drain energy from them. Nice to see that to be removed in the anime.

Episode 12:
- First of all, there was no indication that day 6 was gone. Were they still in day 6 or in day 7? If it's day 7 it would be cool to see the "LAST DAY" as shown in OG title.
- In OG the ultimate fusion pin just somehow appeared for you to finish the boss. In the anime this is due to the combination of power also from Shiki and Beat. But the representation is quite similar: the quintuple laser beams with a big cross.
- A door appeared when after the final fight -- the same door when Kitaniji was defeated in the pad in OG. Behind the door is a graffiti room that we have never seen before.
- Josh should have banged right after Neku's final words, just as in OG...
- Tin pin bro (and Makoto) appeared as cameo. Naomi's team as well as the 777 band are also revived.
- Half the face of RG Shiki, will she appear in neo?
- Mr.H tried to take an extra sip from the canned coffee after flipping it to confirm nothing's left
- Beat raised the question "why are they able to use such psych" and Neku guessed it's Mr.H's support. In some sense this is correct because that's in fact upon "imagination" according to OG secret report.
- Josh revealed that the player pin given to Neku is special and not the ordinary one. That explained why Kitaniji couldn't destroy that particular pin and shout "so that's why...!". He thought he couldn't gain control over Neku because Neku got an extra player pin, but it's in fact a special player pin that cannot be destroyed by him.
- It's a shame that the anime did not end with the OG ending song "A Lullaby for You"...it's such a fantastic song. I know something's wrong with the artist but still it's worthy to be put here.

As you can see most reactions came from the last 2 episode as the 2 episodes contain the core information of the story. Regarding the conversation at the end there are a few things that are worth mentioning:
- The golden key pin was an actual master key pin in OG (also in secret report), but here it become the pin that allowed them to do the final attack. We know that the harmonizer pin/fusion pin setting was abolished in the anime where the pact just use the fusion attack somehow, but are there any further implication about that?
- Mr. H explained about his graffiti. This may not be obvious but the exact same's in secret report 12.
- Joshua said that composers cannot be defeated by reapers even in RG. According to secret reports this is because the frequency of the repeaers are also lowered in RG that composer being vulnerable to reapers is invalid. However Mr. H somehow thought that Sho can do that with with the help of Taboo sigil?

The largest difference between the anime and the OG is perhaps the treatment on Rhyme. As much as the treatment being a lot less cruel it also leaves loads of extra question or even potential conflicts with the OG setting.
- In what form does Rhyme exist in UG in week 3? As some sort of energy (or soul -- see explanation from secret reports), something summoned upon psychs/pins (as in OG), noise a creature form?
- Human beings if not repeats are not supposed to be in UG during the game. In what way Rhyme was allowed to stay with the team without any problems?
- Rhyme probably did not turn into pin in the anime, but if we do not know the nature of existence of Rhyme, how do we now the nature of reincarnation that "composers can do"? Of course what Beat refereed to is to revive Rhyme as a human, but this is not necessarily what others meant.
- The most interesting thing about Rhyme is how she managed to damage Konishi. As mush as that being a reasonable answer, it is also extraordinarily funny.

*

It is not hard to find games that has a similar setup/plot as TWEWY: teen based; "game" or more generally being tagged into unexpected events; confined in a specialized area in Tokyo; the growth of character along the story: Shin Megami Tensei (SMT), Persona or even Trip in Akiba.

Comparing with these games, the story in TWEWY looks incredibly shallow. This is kind of an inherited problem due to the nature of the 7 days game where everything has to be done quick. The story maker had taken that into consideration and made reasonable stories for the three weeks. Shiki week is kind of relaxing so they can spare the time to reveal the background of Shiki and allowing more interactions. Joshua from Neku's point of view is a complete enigma and hence doesn't need much story. Beat's week is more of a decode and the big story unfolding, so the attention is shifted to the overworld as well as Rhyme. That part of the story is shallow but fitted into the game properly.

What I wanted to know more is the big story behind. It's ok for me not to dig into the discussion of angels, vibes or anything that is of "higher existence", but what happened around UG and RG should be explained clearly at the end.

For example, it was said that Neku changed (in a desirable way) so that Shibuya is saved, but how? One may argue that Neku's personality changed positively given the social norm, but these are never objective. Why was Shibuya considered boring that needs to be destroyed? Such judgement could not be made just by looking at Neku or people around him -- unless this is commonly observed in Shibuya. There are many dreading regions in a city around the world, so why Shibuya? Is it about the imagination (which, was never properly defined other than its literal meaning) declining in Shibuya? That is never addressed in the story unfortunately. In the worst case scenario everything was set up just because Joshua wanted to without any reason, then the whole thing is no more than a child's play...

At the end of the day, TWEWY is still my favourite game by a long shot. The anime may not be the best reiteration of the game, but it is good to have that as a refresher or teaser before we meet NEO. I have already order the game and is ready to stream my gameplay (for the first time). Hopefully I will see you there in 25 days :)

## Saturday, 19 June 2021

### FEH VG predictor continued: wave pattern and early estimation

Building a model for VG is something that I wanted to do for a long time. In the previous article I wrote about the basics of a VG model and the article concluded with the chart below:

The perfect wave in the first 12 hours caught my eyes -- is that a coincidence or is that a general phenomenon? The aim of this article is to look into further patterns that help us to build the model. Before we start recall the terms that I used in the previous article -- please refer to the previous article for further details.

- Three examples all extracted from VG June 2021. Please refer to the previous article for further details. You can extract the raw data from the Japanese predictor made by @rammtiger_n.

Example 1: Final (Popularity ratio >4)
Example 2: Quarterfinal 4 (Popularity ratio 1~2)
Example 3: Semifinal 2 (Popularity ratio close to 1)

- Parameter $k$: the parameter so that the accumulated score is of order $O(t^{2+k})$, or that the team's activity is of order $O(t^{1+k})$. To be more precise, for team i (i = 1,2) define $c_i(t)$ to be the constant factor which scales upon team size, and switch between three values according to the state of the hour, and $f_i(t)$ is the corresponding hour multiplier (which can either be $1.05+0.05t$ or $3.2+0.2t$). Ignoring intraday variation we assume that the team activity $A_i(t)$ is approximated by $c_i(t) f_i(t)^{1+k}$.

One should note that this parameter for the two teams are not necessarily the same, but they are close enough for most of the time. Let us assume that parameter $k$ is uniform across the two teams first.

*

The chart showed at the beginning is what happened in example 3. The curves are easily spotted because it is a perfect ping-pong where activity of the two teams are almost equal. At the same time when a team is in the excited state the other must be in the post-excitement state as it is exhausted due to bonus multiplier at the previous hour. As a result we find two perfect curves with alternating dots, one for the activity at excited state, another one for the activity at post-excitement state.

We do not have a perfect ping-pong most of the time, so are there any ways to extract such trend if it exists? One approach is to assign a factor to the three states: we may assume that the normalized activity in the excited state is 10 times the normal activity and 100 times of the post-excitement activity. Although we can explain this by the fact that flags comes in a multiplier of 100, such ratio is still affected by the parameter $k$, which we do not want to fix.

There is a smarter way to get around this: observe that the state of the two teams are almost always excited + post-excitement or normal + normal. On rare occasions it could be normal + excited or normal + post-excitement but they always cancel out. Therefore we can simply take the (geometric) average of the (normalized) activity to retrieve the trend!

Mathematically, we first guessed that the parameter to be $k_0$. We then normalize the activity by considering $A_i(t)/(f_i(t))^{1+k_0}$. By taking the geometric mean we have that
$GM = (c_1(t)c_2(t))^{1/2}(f_1(t)f_2(t))^{(k-k_0)/2}$.
If we are either in the excited + post-excitement or normal + normal states, then $\sqrt{c_1(t)c_2(t)} = c_S$ is a constant. Since $f_1(t)f_2(t)$ is always $\Theta (t^2)$, we know that the geometric mean is constant (or regressed to be constant) if and only if $k=k_0$, i.e., if the estimated parameter $k_0$ meets the true parameter. We take log GM instead of GM to even out the impact of normal + excited states against normal + post-excitement states.

As a demonstration we calculate the log-geometric mean team activity for example 1 we get the following chart (with the guess of $k=1$):

We can see a downward trend starting from hour number 8, indicating that $k=1$ is an overestimate here.

Again we retrieved the same early wavy pattern as in the first chart. It has a simple explanation: in FEH there are quests to clear. You need to clear these simple quests to get the (maximum number of) flags. The quests are mostly "clear VG with red/blue/green/colorless unit", but they require you to enter VG actually. On the other hand, you start the event with zero vote so you cannot do these quests right away. Most people do these quests with votes almost fully restored, which is exactly 4-8 hours into the event.

Now we can estimate $k$ by removing the first 4 hours as outliers and search for $k_0$ such that the linear regression returns a zero slope. Since the regressed slope is strictly decreasing with $k_0$ we can always find such $k_0$.

If we apply that on example 1 we estimate $k$ to be 0.85:

And if we apply that on example 2 we estimate $k$ to be 1.17:

The wavy pattern seems to be very consistent among all situations: we always observe two peaks, one at hour number 4 (which corresponds to 8 hours into the event since we removed the first four) and another one at hour number 12 (16 hours into the game). We may interpret these as the activity peak from players in different part of the world. Computationally the peak and troughs helps us greatly in the sense that we can do the same linear regression using the first two peaks and troughs, i.e., the data of the first 20 hours, and the result is highly correlated to the estimate using all 44 hours of data.

Example 1: $k$ estimated to be 0.8 with the data of hour number 5~20 vs 0.8 on global data

Example 2: $k$ estimated to be 0.92 with the data of hour number 5~20 vs 0.92 on global data

It seems that such estimation is always an underestimate due to (out-of-correlation) increased activity at the far end, but we can always add a little bit to our estimate.

*

So, what can we do with the predictor now? This is a purposed way of creating a prediction:

- Use the early data to estimate the constant factor for teams' activity with $k_0=1$
- Predict by combining team activity and states guessing
- Analyze team composition by wave decomposition at hour number 20 and modify $c_i(t)$ accordingly
- Update $k_0$ by linear regression every time before iterating through the prediction after hour number 12 or 20
- ???
- Feathers!

As much as the above being a big and serious discussion, I still prefer participating the event in a simple way by guessing frequency of the bonus hours linearly. A 99% accurate predictor? Sure but no thanks if I am the one to write the codes. Not to mention that it is actually quite hard to measure the error in a dynamic system and we just can't tell in a mathematically rigorously way that how accurate our predictor could be...

The charts were not properly imported onto google drive, but you can plot them easily. Column L-N are time-normalized difference and bonus boundaries with $k=1$. Column T is the log-geometric mean of team activities. You can change $k$ as you like at W4 and W5, but the $k$ for the two teams are by default equal. The three labels are SF2, F and QF4 which correspond to examples 3, 1 and 2 respectively.

## Friday, 11 June 2021

### Some optimization on the FEH VG predictors

Voting Gauntlets in FEH is always controversial in many ways. In terms of outcome some hates to see popular characters always triumph over ordinary characters while the rest complains how the result is unpredictable and favors the chasing side by so much. In terms of reward some players are unhappy about the lack of rewards -- well actually 12 orbs is a lot, but the feathers are also very friendly to new players (and even me back in the days). In terms of difficulty, some found that playing with 3 random characters is quite fun, but some say they have terrible luck and facing 3 fallen Edgelord is bullshit.

But today I want to talk about mathematics and not the game mode itself. How can we predict the outcome given the first few/12/24 hours of data? Certainly there are a few attempts already: on Reddit there are a few predictors on the West and also one from Japan. I found that the interface of the Japan predictor is pretty nice, despite that the prediction is sometimes off.

In the past I have talked about VG in the sense of a multiplayer game -- in the game theory sense, but this article is doing the complete opposite. We assume that the reaction is fixed under some unknown parameters, and the goal is to build a model out of that.

I do not plan to build a predictor by myself. It takes lots of time and does not benefit one so much ingame: in terms of ranks it makes no difference if the bonus hour shifts as everyone has the same bonus time. You can almost always get the highest reward by not missing the bonus hours in the last 20-24 hours, which can be done using VG bots. The prediction for the last few games where final result matters, can be predicted fairly accurately by most models anyway.

The reason I wanted to write this is because there are a few things that I spotted that are relevant but they were not accounted in existing models, so it serves more as an investigation.

For starters, these are what you need to know: (FEH and negligible details are cropped, just to give a sufficient model here)
- Two teams undergo a head-to-head battle over 44 hours.
- Every player has a voting gauge which recovers by 1 vote per hour and is capped by 8.
- Every player has 2000 flags which they can spend over the battles. One may spend a maximum of 100 flags per vote that they applied. With N flags applied the score is multiplied by N. For example if one spends 8 votes with 800 flags then the score is multiplied by 800. If no flags were spent the multiplier is 1 per vote.
- There are two multipliers: the normal multiplier starts from 1.1x and increase by 0.05x per hour. The bonus multiplier starts from 3.4x and increase by 0.2x per hour.
- The score is updated every hour. If a team is 1% more than the other, bonus multiplier will be triggered for the weaker team during the hour.
- The team with higher score at the end wins.

Score normalization

First of all, we know that the score is not growing linearly and we need a way to normalize them for a time-invariant comparison. A clear choice would be the direct score ratio between the two teams. This is a very intuitive choice which also hooks with the bonus trigger, but the accumulated score certainly affects the velocity of this indicator over time.

Another choice is to divide the score by the multiplier, whether it's the ordinary or bonus multiplier does not matter too much because that is just a constant scaling (almost). The problem is that player's VG activity isn't constant either: higher multipliers are expected at the end so they prefer to spend flags towards the end. With flags around the score obtained by not spending flag is basically negligible. We need to capture when people spend flags.

With everything being non-linear it is so hard to decide the right exponent, so I decided to look at the accumulated score instead. It is natural to assume that players' activity is non-decreasing in general, then their points gained per hour, after enlarged by the linearly growing multiplier, is at least linear. As a result, the accumulated points are at least quadratic, i.e., $\Omega (t^2)$.

Assume that the players' activity -- or the teams' activity as a whole, is of order $O(t^{1+k})$ then the accumulated score will be of order $O(t^{2+k})$. It is not hard to find that the accumulated score is indeed at the order $O(t^{2+k})$ for some small $k$ -- so let us just divide everything by $t^2$ before we look into the parameter $k$.

Here are the two typical examples taken from VG 2021 June.

Example 1: VG final (F!Corrin vs Klein)

Example 2: VG quarterfinal 4

These are two typical matches in VG: example 1 is when the popularity of one clearly overwhelms the other, while example 2 happens when a team is of significantly higher popularly but not as extreme as example 1.

All the charts are time-normalized by $t^2$ where $t$ is the average of the two multipliers.

In the first chart, the orange and yellow line indicates the time-normalized boundary for bonus multipliers, while the blue line shows the normalized score difference. The second chart indicates the normalized score activity of the first team (a positive score difference means that the first team is leading).

We expect the normalized player activity should be of order $O(t^{k-1})$, and from here we can estimate $k$. The spikes are when bonus multiplier happens.

We can see that the parameter $k$ clearly varies in different situation. We can assume that $k$ is close to 1 in example 1 while $k$ is clearly much smaller than 1 in example 2. In fact, $k=0.2$ is a pretty good estimate. The $k$ value can also be verified by checking the growth rate of the bonus boundary curve.

We can explain the correlation by how player anticipate the battle towards the end instead of casually spending their flags in the middle. Very interestingly the parameter $k$ seems to be independent of the ratio of player base size: the dominant a team is, the more bonus hours the opposite team will get. So in theory if the parameter is decided by the frequency of bonus hours, then the parameter for the two teams should be different, but that is not the case here. If we plot the activity of both teams on the same chart for example 1, we can see that the parameter for the two teams are more or less equal.

Calculating the parameter $k$ would be extremely helpful because we can then get a normalized data. (And we will cover that in the sequel of this article!)

The three states of players

With the above graph we see that the activity of the players divide into three categories, or three states that we call.

A team is in an excited state if it receives a long-waited bonus hour, or a bonus hours that shoots them to the leading position. The activity clearly spikes for this hour.

A team is in a post-excitement state if the bonus hour in excited state shoots them into leading position with bonus triggered on the opponent then the team enters the post-excitement state where activity is abnormally low because they run out of votes and flags.

A team is in a normal state otherwise. There are more fluctuation within this state depending on the score situation.

We can plot the same for example 2 which is a lot more chaotic, but the activity is still clearly divided into the three states as described.

The pattern shows that most players aren't playing to optimize the chance to win as a team. Rushing to overtake the opponent early has little to no effect on the final result but looks good in a team sport, while the increased activity when a team enters bonus multiplier is natural as it maximizes score gain (for those who urgently need to spend flags).

I am actually quite surprised that this was not taken into account by most predictors, as it plays an important role at the end where people react to the hourly updated results vigorously. For example the JP predictor predicted a constant downward ping-pong at the end for example 1 (i.e., bonus for second team for 1 hour + no bonus for both teams for 1 hour alternately and these two together results in favor of the second team), but the reaction from the second team at hour 42 should be much more violent as it sends the team into leading position -- and of course the first team hit them back with 10x the power. This is the nature of VG predictor: players react to score but not team result.

Daily variation on player activity

Clearly players do not stay awake 24/7 for these shitty reward (well even if orbs are worth its monetary cost that is merely a burger meal, so it's not worth the time to stay awake overnight), so they stop playing when they are asleep. But people over the world situates in different time zone and they sleep over different time. More importantly, the taste from different part of the world seems to be different. As a result, we may observe higher activity from one team during daytime then higher activity from the other team during nighttime (which could then be daytime for that part of the world).

Assuming that the main division would be Japanese (or Asian) players vs the West, we can divide the players into 3 groups: time-invariant players, Asian players and Western players. We estimate the portion of the three and how supportive they are to each team. This is taken as a scaling factor when we predict the future outcomes.

And how do we do that? Well this is simple linear algebra -- these three groups of players can be modelled into 3 kinds of waves: constant, sine and cosine waves, orthogonal to each other. We can then apply orthogonal projection to estimate the portion from each of the three.

*

I really believe that the three factors together with what we already have around, builds a very accurate VG predictor, but surely no one would waste the time doing that.

To conclude this article let me show the graph plotted for the same VG but semifinal 2, which is ping-pong all the way. I stacked (apology for the poor stacking) the two graphs together so that you can observe the interaction more clearly. Since it's a perfect ping-pong first the first 20 hours or so, you can see that the activities alternates from excited and post-excitement states but not the normal state. Although the waving pattern in the first 20 hours which in fact, also occurred in the first two examples, seems to raise more questions from here...

(To be continued)

Acknowledgements: raw data extracted from the Japanese predictor by @rammtiger_n

## Wednesday, 2 June 2021

### 夢醒

The pride of London. Captured from ChelseaTV.

## Thursday, 27 May 2021

### 27/5/2021: Thoughts on nos techniques

My second S on real lv.3. Stamina maps are my favourite.

Today I realized that my e-amusement album on facebook was gone. I knew that the e-amusement app removed its linkage with facebook but the album should still be there, but that isn't the case anymore. It took me sometime to retrieve all my DDR/beatsream captures, but they are worth the time. I guess it's also a good time to take record of my MUG progress.

Since the lockdown I have been playing ring fit or doing exercises at home a lot more, so I simply play DDR less and less frequent. Playing 18 is no longer a regular thing to me again, although Konami didn't produce enough sensible 18s in the past year...び is pure shit in my view.

So I basically play Nos all the time, and the progress is clear. By the end of Op.2 my recital grade was 5300, it was then 5500 by the end of 2019, and now my grade is 5760. In other words, I can obtain S rank consistently across level 11 by the end of 2019, but I can S consistently across level 12.5 and a few 13.0 right now.

What's hard to me right now are basically nonsense streams (like True Blue) or hardcore trills (like Appassionata 3rd movement, Etude Op. 25-11 "Winter Wind" or the Ballade No.1 in G minor). It takes time for me to recall my piano muscle memory...but what's interesting to me is how me or other nos player play maps at 12.5-13.0, start to differ from other MUGs. We need to deal with streams that are practically impossible to hit in a discrete way, so we need some other approaches, and the choice would be to sweep or to play like a pianist?

There are other MUGs in the arcade where you can sweep the keys instead of hitting independent to be count as a hit. Chunithm is the prime example but people also do that sometimes in jubeat, but the technique is widely adopted in Chunithm while restricted to a small group of players for jubeat. Why?

Chunithm encourages sweeping in nature: a flat input device and sliding notes exist for sweeping already, so sweeping ordinary notes is just something natural to them. The notes are usually 4 keys wide and the timing is loose -- that makes sweeping much easier.

On the other hand, jubeat consists of a 4x4 pad, so sweeping for longer streams implies a change in direction during the sweep, and to keep a consistent pace during the sweep is definitely hard. The streams are not for sweeping most of the time where notes appear alternately on the left and right half: sweeping anything more broken than 2-7-10-15 would be extremely hard. At top level (the only instance where sweeping could be relevant) people always go for the hard mode, so positioning plus timing would be too demanding for this technique.

Nos interestingly situates between the two. Its panel is linear (although not flat) and sweeping is easy. There are also maps where sweeping is clearly desired (e.g.
ピアノ体操第一). The timing is not as loose as Chunithm but a lot easier than jubeat on hard mode. On the other hand, sweeping is quite hard when the notes are not spaced apart enough (like Etude 10-4), and bad sweeping causes a bunch of good or misses. As a result, both sweeping or piano style are equally popular among high end players. We can even see both styles at KAC final level.

I have been on piano style since the beginning, but sweeping really helps me to master many level 12.5 maps especially on the left hand. Typical examples include 蒼氷のフラグメント and Moonstone. For バルベリア・タンゴ I have been trying to trill in the pianistic way but it's simply too hard on the left hand. I gave up and swept with one finger instead...and that's an instant S. I felt like I have fallen somehow...

I still think for more complicated streams the pianistic style would trump against sweeping, but to do that smoothly would require much more practice to me or anyone else. Maybe if I play like now (every Thursday for the new songs basically) for another year I will be able to S half of the real level 3...?

## Sunday, 23 May 2021

### TWEWY the Animation: Joshua week impression

Among all of Josh's smol faces, his T-pose still impresses me the most.

Week 2 occupied 4 episodes instead of 3 or even 3.5 episodes. This is making sense because we see the most developments and basically all characters in the week. Considering the nature of week 3 which is more like a team solo that part it should be easily covered in 3 episodes (with one episode solely for the final fights), unless they want to introduce more NEO hypes (which they did also this week).

Again I will focus on the difference between OG (the original DS game) and the anime. It is apparent at this point that covering the story with like, half the detail is already impossible (when you also want to display the fights). One particular example this week is that Sho's math meme is really weird in contrast to what we saw in the OG dialogues because it's simply coming out of nowhere. The story focused on interactions among people and their feelings which is so delicate that most dialogues are relevant to the main plot. As such the producer have already done a satisfactory job to cover what should have been covered. My critics are again mainly on how the anime could have been closer to OG in terms of game mechanics.

Let's start again with pins and monsters.

On Ep.4, they finally got the monsters right: Carcinofolk appears on W2D1. Neku used Burning Melon here which is effective against crabs, but not much use against the rhinos. Instead of a fire sphere bouncing on the screen it was shown as ordinary fireballs only. He also used Pop Pendulum against kangaroos, which of course failed because this pin is too damn slow to hit them. I always preferred JoM slashing attacks as it stuns the enemy, and I have no idea why they never used those pins.

Nothing too much about Ep.5, but they have once used the ordinary Pyrokinesis pin again. Starter pin should not be used at this stage anymore! And since your Burning Melon was shown as fireball last episode, the Pyrokinesis pin can easily be replaced by some other random fire pin anyhow. Oh and taboo noises firing cannons...

On Ep.6 the game started with Garage Wolf and Doom Metal Drake...the former only appears on week 1 and the latter only on W3D7. This is just inappropriate. They finally used Frozen Cool instead of the Ice- pins, but this is the unevolved pin (cf. Blizzard Cool) so we should see one icicle instead of three. Swift Storm, Swift End is another new pin introduced and it seems like they really like Dragon Coulture. The other new pin introduced was Crackle Pop Barrier, which was shown as a weak barrier instead of time-limited invincibility. Joshua also exhibits his levitation + angel beam ability on day 6 (instead of day 5 in OG) this episode...his attack is always my favourite.

Ep.7 started with the scramble crossing mini-boss fight. In the anime Neku faced Jungle Boomer and Grindcore Minks, but there were two red minks instead of the red-gold duo! There were Mosh Grizzly around which is also untrue to OG as it should have been Wall of Grizzly (black wolf) instead. In the boss fight Neku also (explicitly) used the starter pin Thunderbolt pin for the first time...again why? Storm Warning and the Spider series are similar but much better. Thunderbolt-type attacks are effective against Sho since he evade attacks so swiftly though.

*Edit: it seems like Superfine beam were used in Ep.7 but I couldn't recall any attach like that...anyway.

Other OG related

Sho's missions started with [Game I(/II/III)] which is the same as OG. I now realized it looks like [Maths I/II/III] for Japanese's high school maths curriculum, but I am not sure if there are any relation between the two. The stupid 30+74 was also modified into sqrt(10816).

Even before the anime we already know that many regions are renamed into their reality name as SE got the consent from the property owners. We saw the Tsutaya O-East last time, and this time it's Shibu Q-front (instead of Q-head). Then later in Ep.7 Pork City was renamed to Mark City. Back in 2008 I have not visited Shibuya so I wouldn't have react to Mark City even if that was the name in OG, but since now I have been there many time and also that the Mark City was shown explicitly (Pork City was never shown in OG -- you just enter a mall from Dogenzaka) I immediately understand the way the game describe Pork/Mark City as "where emotion gathers" -- for those who don't know, Mark City is a large complex above the Tokyu-Tokyo Metro Shibuya station...and one of the shopping malls loaded with top restaurants in Shibuya.

At a point Mr.H helped Joshua and Neku by adding new functions to their phones. This is making less sense on smartphones as they have a much more compact designs comparing to phones in the 00's. That reminds me of the time when there were shops specializing in amending the hardware of phones and GBAs adding weird functions on it.

In OG we never saw the movement of characters other than the protagonist team. Apparently the reapers can fly. Another difference in Ep.6 is that Joshua explicitly mentioning the isolated UG area, in particular the existence of Shinjuku UG.

A map was shown when Joshua talked about route 5 and that is the exact same map in OG and I loved that.

The mission on day 7 was not issued by Sho but by the conductor himself. In OG this is reckoned by the way the mission was written. The mission started with "Game 7" instead of "Game IV/VII", but this was not delivered in the anime which is slightly disappointing.

The representation as in Ep.7 clearly deviates from OG more than the previous episodes. Firstly we have spider noises that never appeared in OG, then we have imaginary space appearing much earlier than Sho's fight (and visually it is so different -- I wonder now how Konishi's fight would be). The fight is a bit weird though, Joshua and Neku never managed to hit Sho effectively at least on screen but the fusion attack somehow did the job...

Impressions

- Shiki's emotion was close to the OG emotions most of the time, but Joshua's smug-ness is much clearer in the anime than in OG. He simply has so many different smol faces in the four episodes.
- Tin Pin's gone. Natural choice but I hope it will at least be in NEO.
- Mr. H calling Neku correctly in anime.
- Joshua talking about they need to attack the taboo noises to deal effective damage: a possible reference to puck relay.
- The N---d Prey shown on Ep.6 after Neku's flashback

*

I can hardly say the anime met my full satisfaction so far, but I will rate it a 9/10 regardless. Let's see what would happen on the Beat week!

## Friday, 14 May 2021

### After Dark 4.0 in Win 10: screensaver being reset to "None"

Installing After Dark 4.0 in Win 7/10

On my new computer (win 10 of course) I frequently encounter the following problem: the screensaver, after a while, resets to "None" (no screensaver) in the screensaver setting. You can explicitly go to the setting and set the AD screensaver again, but that is a waste of time.

After some investigating I found that this is again something that is lost in compatibility mode.

If you go to HKEY_CURRENT_USER/Control Panel/Desktop, you will see the default path for the current screensaver. If your screensaver was reset to "None", then the path is likely

C:\WINDOWS\system32\AFTERD~1.SCR

but it turns out that the screensaver was installed in SysWOW64!

The simplest solution is then to copy the .scr file also into system32, and that worked immediately for me without further efforts required.

Edit (16/7/2021): I found that leaving the screensaver in SysWOW64 causes all sorts of weird stuffs: the screensaver automatically triggers when you move the mouse to the right-top corner (while Windows does not have hot corners by default), and the screensaver triggers somehow when you are typing password in the lock screen blocking you from logging on. Just make sure the screensaver only stays in System32 and everything will be fine.

## Friday, 7 May 2021

### 07/05/2021: Chelsea's road to Istanbul

There are quite some aftermaths since ESL collapsed: protests at Old Trafford, and Chelsea allowing fans representatives to join some board meetings. But none of them matches Chelsea's competitive achievement just a few hours ago.

That is, a magnificent performance outclassing Real Madrid over two legs which wins them a ticket to the Champions League final in Istanbul -- somewhere we could have been to 16 years ago if not the phantom goal from Liverpool.

I couldn't help but to shed tears with semifinal scenes flashing through my head...

In 2008 Ivanovic scored two headers out of nowhere marking the beginning of his era in Chelsea; and of course the 4-4 is one of the most entertaining match ever.

In 2009 that was the night of fucking disgrace which I don't want to talk about that ever again. But the 1-1 game is absolutely filled with quality: Essien scored a stunner and so did Iniesta. The Barca team went on and become one of the strongest team in history (despite multiple controversies).

And 2012? I can still recite every single quote and every goalgasm from Neville for the two miraculous goals.

The semifinals are never boring, and so was today's game. Real Madrid's midfield were so exhausted and they simply failed to pressurize Jorginho -- who always play like a master whenever unmarked.

Kante was of course the star above all, he was too good on both ends of the field. It was his change in tempo that created the first goal. His contribution to the second goal is even more direct: a clear interception before sending to Pulisic who fed Mount. He is almost flawless in the tie. He even won all aerial challenges which is amazing considering he is just 5'7!

The other players should receive their deserved praise too. Mendy's two crucial saves saved Chelsea in the first half; AZP playing the best he possibly can with his declining physicality; Rudiger who is solid as rock...it is truly a team effort.

I don't blame Hazard for being sloppy in the game. He did he part, just that he simply lost his bursting power he had back in the days. He was thoroughly neutralized under the effort of Christensen and AZP, and the rest of Real's midfield were even less creative. Tuchel once again showed his tactical geniusness, and Zidine meanwhile, showed that his team is so awful under 3-back against top teams.

Back in the days I fell into love with Chelsea because of Mourinho and his 1-0ism. I thoroughly believe in 1-0ism works in sports in particular football (I like 1-0ism in baseball too, though I have to admit it may not be the best way there). Being good defensively however sometimes implies the lack of sparks on the other end and the lack of consistency to win against lower teams. We have seen enough of that from Chelsea during these years.

Some like Abramovich of course prefer the complete opposite: he wanted beautiful football. He bought many good attackers but most of them didn't perform as expected. On the other hand, coaches/managers that played attacking football -- like Ancelotti's team with 103 seasonal goals in 2009-10, AVB's high pressing team in 2011-12 and Sarri's Sarriball in 2018-19 -- never reached the same level of success in terms of European performance.

At a point we thought that it is impossible to play beautiful and solid football at the same time. But now with Chelsea and City's performance this year we see a glimpse of hope that this is actually possible. The key of playing solid and beautiful football simultaneously is not all about stacking world class players (like PSG). It's being able to run, to play physically, and to execute tactics thoroughly. This is the common feature we observed in both the City vs PSG and Chelsea vs Real Madrid ties. These are the two teams that overwhelmed their opponents in all aspects, and they deserve the spots in Istanbul.

To be honest I expect a dull display in the final because the two teams will probably play very conservatively, but that does not mean it is dull tactically, it is the right tactics adopted by both teams that prevent the talents from making comfortable decisions. This may not be a match that is enjoyed the most by general public, but I certainly have very high expectation on it.

Most of Chelsea's players actually never played in the Champions League semifinal let alone final, but that's not something we should worry. There were numerous players in the 2012 Chelsea team who just joined. Cahill even just joined the club during the winter transfer window! The newcomers in 2021  -- Werner, Havertz, Chilwell and Silva -- have shown their quality and we should be faithful that they can do it again on May 29.

## Friday, 30 April 2021

### 被青梅竹馬抓來當教魔法的我想要教出最強學生(仮)(1)：感覺進了火坑但擅長的不是水屬性可以怎辦

"Wizard in the magic sunset"
Character design: @kuonyuu, Illust: @mihayuuno commissioned by forretrio. Pixiv
Editing and re-posting are prohibited // 無断転載・自作発言禁止

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「為甚麼要把我留在場上？」少女擺出生氣的樣子問道。一方面她在風暴裡目睹了青年出手的瞬間，她很清楚如果青年在那個時候朝她攻擊的話她是肯定會像其他同學們那樣倒下的。另一方面，她雙腿似乎還不怎樣聽使喚，拖一下時間她才有反攻的希望。

「因為你劍術很強吧？沒記住學生們的資料可是教師失格的呢。家中劍術流派第四代傳人、十三歲時隨行魔物討伐隊擔任前衛、一年前成為公主殿下的貼身侍衛。我沒說錯吧？伊．雅．娜．同．學？」青年一字一字將少女的名字唸出來，似乎從少女閃過驚訝的眼神中獲得了一絲愉悅。

「……」少女也沒有回應他。既然自己被盯上了肯定要跟他打一場，那還是等自己多恢復一點比較好。

「吃我這招！－－」少女改揮為刺，手臂直直地向前刺去。之前一直沒用魔法的她此刻將雷電魔法附在劍上，雷電到最後一刻都可以改變攻擊的目標，如果說她有辦法對青年造成傷害的話應該就是這招了吧。附魔的一劍比之前的攻勢都更快更狠。

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「我說啊……大小姐你不要擅自闖進來喝我的茶好嗎？這些茶都是我找遍全國收回來的珍品，有錢都買不到耶。」

「我只是來看一下你有沒有辦法搞定那幫學生而已，看過試煉之後我就安心了。」

「難道你覺得我打不過他們嗎？還在讀書時我就能單挑一整班了，現在要贏根本毫無難度吧。」

「立場不同，要採取的打法也不同。你這次做得很好，在他們心中留下了強大的形象又不至於讓他們害怕或厭惡你。要是你下狠手打傷他們的話麻煩才大呢。」

「雖然我是冒險者，但出手輕重我還是懂的。我過來是真的希望可以好好教書，不但因為是妳讓我過來，也是為了給別人還一個人情。」

「你的意思是說，我的邀請份量不夠嗎？」女性把茶杯放下，俏冷的臉龐露出了戲虐的笑容。

「呵呵，你這幾年也混得不錯嘛。幾個在邊疆活躍的大公會都搶著指名要你，不知道如果我把你的使用權賣給他們的話可以值多少錢呢。」

「……還請您高抬貴手，我一定會安份地在這裡教書的。」

「嗯，你明白就好。雖然你有點鬼畜，但我相信你不會隨便坑學生的。我只是來提醒你一下，這一屆學生的實力和身份非比尋常，除了我以外還有很多人在背後關注你。萬一出了甚麼事的話，我可沒有百分百的信心可以把你保住。」

「那我現在辭職還來得及嗎？」原來是屎缺啊。

「你可以像以前的教師那樣落跑啊，不過我保証以後不會有半個公會雇用你而已，最差的狀況是冒險者的身份被註銷吧？」她平淡地描述著可怕的後果。當初把庫里斯弄出王都當冒險者的人是她，他毫不懷疑只要她一聲令下，自己所累積的一切瞬間就會灰飛煙滅。

「呃……」

「不想死的話，我建議你盡快跟學生們混熟。這樣你才有辦法駕馭那堆小怪物。」

「謝謝你的好意，我會用自己的辦法把跟他們打好關係的。先從公主殿下的侍衛入手也是我事先計劃好的。」

「好吧，你加油。那麼我也差不多該回去了，晚上我還要回去宰相府接見其他客人。」

「啊……」似乎自己與她的距離變得更遙遠了：「現在做到甚麼位置了？」

「算是副手吧。每天都從早忙到晚，至少當冒險者可以給自己放假呢。」

「我們的進行任務的時候的生活也不是大小姐能忍受的，算是彼此彼此吧。」

「嗯，我到時會聯絡你的。還有－－」他頓了一下問道：「改天我可以約你出來吃頓飯嗎？」

「嘛，算了」他伸了個懶腰，心念一轉一股風便把門關上。與她的重逢來得比自己預料中來得早，不過至少她接受了自己的邀請，那麼以後再相處的時間肯定不會少吧。把目光投向辦公桌上那疊厚厚的資料，他自言自語說道：「現在該想想怎樣料理我可愛的學生們了－－」

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－－以上，就是我想寫的故事。