The identity problem of FEH as raised in reddit.

A very interesting and critical viewpoint that explains perfectly on what happened. Not going to analyse in detail (because you can see some wise comments on reddit already), but this is what I think as a FEH player since launching.

-Past-

I joined FEH because of my love on the FE series as well as the advertised low-effort gaming as compared to other well-known CCGs (FGO, Fantasica etc). Things came better than what I expected (I liked their overall money-attracting strategy as expected). Didn't really feel the salt from reddit but what started to itch me is the Ayra pool (although I wanted her much and got her eventually), and IS really underperformed before the v2.0 change.

-Present-

The first Tempest trial is interesting as it's also the hardest consider and without considering powercreeping. It was then just a matter of endless grinding and pain in selecting the right seal to give out.

The two arenas are definitely much of a pain. The two meta units B!Lyn and Rein not only destroyed the balance but more importantly diversity within the battlefield. In the past Takumi was once considered a meta but you don't see them that often and you have multiple and creative ways to counter him. B!Lyn and Rein, on the other hand, is damn annoying to counter if you meet them consecutively in the assault. As much as you don't much to give out easy defense point, it's more like an offense for me to use those annoying units to steal a defense victory.

Voting gauntlet...this is the most interesting one that I have ever seen. It worked in big teams (instead of scattered teams like Fanta) yet super balanced with extremely balanced mechanism making it tense and fun (see my previous entries on the mathematics behind). And ever better, it gives out super generous prizes with less effort required comparing with the TT mode. Judging from the tier composition as from the reddit survey reddit players definitely represents a large chunk of top players from the west --- and the tactical battle against the will from the East (i.e., Japan) is simply delicious. The only downisde is perhaps the choice of the participants knowing that there are too many popular characters. (Or bias on male characters. But well...)

The difficulty of the game is alright as they keep adding hard challenges, in which some are really challenging without using those metas/+10 units. (Got to admit that Genny is destroying every GHB/BHB recently...). Chapter 11&12 is of course the hardest and takes a lot of tactical consideration due to the turn limit -- those alternative goals can sometimes be interesting and challenging, and we might find more later.

-Future-

What do I want from the game?

It's very interesting that they asked as for that straightaway, i.e. they probably had nothing to serve right away, but that allows greater flexibility as well. Idle game is of course my favourite, but I don't mind trying out other modes as well.

Bias in characters pushed out is definitely one of the main problem to be resolved. While promoting new games, it is important for IS to realize that a major part of the fans came from the GBA era where they played the first 7 franchises of the series (by means of emulator etc) and old heroes is highly welcome. Not only that they tame those old fans, but it also solidifies the big picture of the FE history which is certainly attractive to new comers.

And more good artwork please. FE Chiper set a high standard in illustrations, and yet FEH somehow come short -- some are certainly cute/moe/lewd/sexy/pretty/... (Ayra!) but some simply failed to illustrate the character of the characters. Lachesis and Ursula are two obvious examples, and they are unfortunately two of my favourite heroes from the original series.

But well. Again I'll put my faith on IS and FEH, who won the 2017 best game award :)

And happy 2018.

## Thursday, 18 January 2018

## Friday, 8 December 2017

### Some recent maths activity

Yesterday I received the question from my engineering friend:

Let $f$ be a real function so that for all $x,y\in \mathbb{R}$, $f(x+y) = f(x)+f(y)+xy(x+y)$ and $\lim _{x \to 0} f(x)/x = 1$ hold. Find $f$.

It does not like a casual question to ordinary university students, not even for maths students...but anyway one may notice that $xy(x+y) = (x+y)^3 - x^3 - y^3$ if you know symmetric polynomials well, and the free linear term makes up the limit we have $f(x) = x^3/3 + x$.

What about uniqueness?

Well, it is easy to show that the function is continuous by exploiting the equality $f(2x) = 2f(x) + 2x^3$, but even stronger we can prove differentiability. Rearranging gives

$\frac{f(x+y)-f(y)}{x} = \frac{f(x) + xy(x+y)}{x}$

Taking limit $x\to 0$ yields $f'(y) = 1 + y^2$ - not only that the derivative exists, we also get a complete DE with an initial value $f(0) = 0$. That easily solves to $f(x) = x^3/3 + x$.

What if the limit condition is changed? Say, $\lim _{x\to 1} f(x)/(x-1) = 1$? We can rewrite the expression as the following:

$f(x+y-1) = f(x-1)+f(y)+(x-1)y(x+y-1)$

Dividing both sides by $(x-1)$ reduces the question to the original case which gives the same solution.

Let is consider the functional equation at a much generalized form: $f(x+y) = f(x)+f(y)+g(x,y)$. According to the above argument if we managed to show that

$\lim _{x\to 0} (f(x)+g(x,y))/x$ exists then we can easily reduce it back to a DE where existance or uniqueness is clear. However this is hard to work around if we do not assume the limit condition because we know pretty much nothing about $f$. It does not work by assuming continuity of $f$ or $g$, since we may come across to some very nasty functions like the Weierstrass function which makes no sense in these questions. We leave a few observations here without solving it (or even getting close):

1. $g$ must by symmetric. This is obvious by observing the rest of the term. In particular, if $g$ is a polynomial then it is in the ring generated by $\sigma _1 = x+y$ and $\sigma _2 = xy$.

2. If $g(x,y) = O(xy)$ for small $x,y$ then it is possible to recover $\lim _{x\to 0}f(x)/x = 1$ using estimates like $f(x) = 2^n f(2^{-n}x) + O(x^2)$ or $f(x) = nf(x/n) + \log n O(x^2)$.

3. If $g$ is Lipschitz we know immediately that it's differentiable a.e. but that says we could have uncountably many non-differentiable points that we not want to deal with...

But that's it. I do not want to spend more than 60 minutes on this useless (for me) problem :d

------------

The 1st Simon-Marais (aka the Pacific Putnam) was held on October 2017 and the statistics are finally out (compare the efficiency against IMO marking team...). It's very surprising that only 1~2% of the students got problem A4 (and A3). I expected the top rankers to be close to 42 (aka 6 correct answers) but it turned out that not many olympiad players participated the event as can be judged from the award list. I expected the event to be much harder next year.

Let $f$ be a real function so that for all $x,y\in \mathbb{R}$, $f(x+y) = f(x)+f(y)+xy(x+y)$ and $\lim _{x \to 0} f(x)/x = 1$ hold. Find $f$.

It does not like a casual question to ordinary university students, not even for maths students...but anyway one may notice that $xy(x+y) = (x+y)^3 - x^3 - y^3$ if you know symmetric polynomials well, and the free linear term makes up the limit we have $f(x) = x^3/3 + x$.

What about uniqueness?

Well, it is easy to show that the function is continuous by exploiting the equality $f(2x) = 2f(x) + 2x^3$, but even stronger we can prove differentiability. Rearranging gives

$\frac{f(x+y)-f(y)}{x} = \frac{f(x) + xy(x+y)}{x}$

Taking limit $x\to 0$ yields $f'(y) = 1 + y^2$ - not only that the derivative exists, we also get a complete DE with an initial value $f(0) = 0$. That easily solves to $f(x) = x^3/3 + x$.

What if the limit condition is changed? Say, $\lim _{x\to 1} f(x)/(x-1) = 1$? We can rewrite the expression as the following:

$f(x+y-1) = f(x-1)+f(y)+(x-1)y(x+y-1)$

Dividing both sides by $(x-1)$ reduces the question to the original case which gives the same solution.

Let is consider the functional equation at a much generalized form: $f(x+y) = f(x)+f(y)+g(x,y)$. According to the above argument if we managed to show that

$\lim _{x\to 0} (f(x)+g(x,y))/x$ exists then we can easily reduce it back to a DE where existance or uniqueness is clear. However this is hard to work around if we do not assume the limit condition because we know pretty much nothing about $f$. It does not work by assuming continuity of $f$ or $g$, since we may come across to some very nasty functions like the Weierstrass function which makes no sense in these questions. We leave a few observations here without solving it (or even getting close):

1. $g$ must by symmetric. This is obvious by observing the rest of the term. In particular, if $g$ is a polynomial then it is in the ring generated by $\sigma _1 = x+y$ and $\sigma _2 = xy$.

2. If $g(x,y) = O(xy)$ for small $x,y$ then it is possible to recover $\lim _{x\to 0}f(x)/x = 1$ using estimates like $f(x) = 2^n f(2^{-n}x) + O(x^2)$ or $f(x) = nf(x/n) + \log n O(x^2)$.

3. If $g$ is Lipschitz we know immediately that it's differentiable a.e. but that says we could have uncountably many non-differentiable points that we not want to deal with...

But that's it. I do not want to spend more than 60 minutes on this useless (for me) problem :d

------------

The 1st Simon-Marais (aka the Pacific Putnam) was held on October 2017 and the statistics are finally out (compare the efficiency against IMO marking team...). It's very surprising that only 1~2% of the students got problem A4 (and A3). I expected the top rankers to be close to 42 (aka 6 correct answers) but it turned out that not many olympiad players participated the event as can be judged from the award list. I expected the event to be much harder next year.

$\

## Wednesday, 6 December 2017

### 6/12/17

最近不知為何又重溫了好幾十遍惡之三部曲的系列，果然這種唱說風格的華爾滋真是太棒了。與以往不同的是，這次好像有了可以與鏡音雙子比較的東西。

金髮。

雙子。一男一女。

籠中鳥。

哎呀……知道我在說甚麼的人知道就好。我不想回憶起這個又胃痛又糟糕的故事了。

距離上次發日記已是兩個月有多，期間我大概都在做同一件事吧。

---------------------------

2017年10月30日

東京，台場。

這家希爾頓的裝潢與格調可謂完勝迪士尼裡面那一家。

明亮的雙人房外面正好能看到明月映襯著的東京灣。我與他就這樣坐在露台上，旁邊是熱乎乎的蜜汁烤肋排和薯條，加上名物響21跟裝滿冰塊的瓶子。

「想不到你剛來一天就能把這支弄到手真是厲害呢。」

「我剛好相熟的朋友在銀座經營酒舖，要弄一支來還是可以的。在外面難找倒是真的呢。」他徒手撕下一塊烤肉放進口，順手把威士忌倒進兩人的杯裡：「這個我剛讓廚房做出來的，趕快趁熱試試，我覺得烤得蠻不錯的。」

「也不能說是難搶，只是價錢問題而已。網上教人去唐吉訶德或者西武超市搶的攻略變多才是看起來難搶的原因；你不差那一萬幾千塊的話amazon也可以訂到呢。可惜我可沒你那麼有錢，上次搶到兩支帶回去望了好久最後還是決定轉手賣掉。」隨手打開amazon鍵入響21，サントリー官方售價是零售價的一倍以上，媽的。

「轉賣賺多少了？這麼說來你應該還沒試過它吧，不試的話你肯定會後悔的。」

的確是好酒。清徹的琥珀色酒液散發著華麗的和氣，雖然複雜但又完美地融合到一起，正好符合響系列對酒品平衡和諧的哲學；雖然是烈酒但流入口腔只有輕微的剌激感，涼風輕拂下我不禁浸醉於眼前的美酒－－啊，月色真美。

「賺了大概五六單FGO吧，我沒玩就是了。最近都忘著玩火紋手遊給任天堂賺錢呢。」

「微課不如無課，課個五六單還不如把錢留著去下面的游泳池玩呢。說起來你這個燻雞粉應該也去朝聖過了吧？」

「我又不是朝聖狂熱者，再說不帶個伴進去看上去就太寒酸了呢……咦不對，上次你明明說過明明不喜歡希爾頓的怎麼又來了？」

「還不是女友想來……結果一抵就逛了一整天，晚上又趕去高田馬場那邊喝酒了。聽說今晚有音遊界DJ樣子，下次讓她給你介紹一個好了。」

……

大家有默契地靜靜把肋排分掉，響21也所剩無幾，我帶著搖搖晃晃的腳步走回房間把我帶來的五年熟成加賀梅酒拿出來。雖然價格和品位比不上響21，不過我對我對梅酒的眼光還是有信心的。

他率先打破沈默：「品完酒我們該談一下正事囉。」

「說實話我實在不太想接手你家的公司……股東成份有點複雜，我又沒有多餘的票灌下去。在這個節點惹到別的勢力的話可是不太妙呢。」

「票也不用太多每季10張就好了啦，反正中價區價格彈性比較大，要調節也比較不會被投訴。」

「我就真的沒有餘裕能把票分出來了……按照我的模型來看如果我家公司的票再減三張左右的話高價循環就會崩潰，能套錢的地方少了一個的話會十分不方便。對面也盯票流盯得很緊，我換過來應該很快就會被發現了。」

「嗯……反正我這邊有的是人，來交易一下如何？我把你加進換票網絡裡，這樣對手會比較難鎖定你的動向；相應地我希望你能放5-8張票在我們這邊公司，這方面同樣可以透過網絡進行。」要比喻的話換票網絡大概就像洋蔥網絡(TOR)一樣，將點對點的連結遮蔽起來。

「這個可以。不過我在明處不需要太多小動作，放20票進網絡就夠了。你公司需要的票數我會從我手下分出來。」

「真夠精打細算呢。你的目標定好了嗎？」

「其實我沒有，不過手下們可是有一大串目想炸的人呢。有一位想搶新的碧藍艦娘經理不但被守下來，對方還真接把公司炸爛，每次提到那公司他都氣得牙癢癢的呢。反正現在是第一個賽季，多收攏人心才是最重要的。」

「多收下線準備下季再大幹一場嗎？」

「老了肝不起來了啦。有互助圈子才能確保大家不會餓死也不會輕易被攻擊，玩下去就當多認識朋友好了……總之你幫我將票吃掉，我幫你經營集火，這樣沒問題吧。」

「沒問題，那就拜托你囉。」

「合作愉快。」

「Cheers」

---------------------------

忙完手上的事情，聖誕大概就有空能寫作了。到時看上面能不能拓展成外傳囉。

另外好老婆不炒嗎？關鍵字： ACGN-stock

金髮。

雙子。一男一女。

籠中鳥。

哎呀……知道我在說甚麼的人知道就好。我不想回憶起這個又胃痛又糟糕的故事了。

距離上次發日記已是兩個月有多，期間我大概都在做同一件事吧。

---------------------------

2017年10月30日

東京，台場。

這家希爾頓的裝潢與格調可謂完勝迪士尼裡面那一家。

明亮的雙人房外面正好能看到明月映襯著的東京灣。我與他就這樣坐在露台上，旁邊是熱乎乎的蜜汁烤肋排和薯條，加上名物響21跟裝滿冰塊的瓶子。

「想不到你剛來一天就能把這支弄到手真是厲害呢。」

「我剛好相熟的朋友在銀座經營酒舖，要弄一支來還是可以的。在外面難找倒是真的呢。」他徒手撕下一塊烤肉放進口，順手把威士忌倒進兩人的杯裡：「這個我剛讓廚房做出來的，趕快趁熱試試，我覺得烤得蠻不錯的。」

「也不能說是難搶，只是價錢問題而已。網上教人去唐吉訶德或者西武超市搶的攻略變多才是看起來難搶的原因；你不差那一萬幾千塊的話amazon也可以訂到呢。可惜我可沒你那麼有錢，上次搶到兩支帶回去望了好久最後還是決定轉手賣掉。」隨手打開amazon鍵入響21，サントリー官方售價是零售價的一倍以上，媽的。

「轉賣賺多少了？這麼說來你應該還沒試過它吧，不試的話你肯定會後悔的。」

的確是好酒。清徹的琥珀色酒液散發著華麗的和氣，雖然複雜但又完美地融合到一起，正好符合響系列對酒品平衡和諧的哲學；雖然是烈酒但流入口腔只有輕微的剌激感，涼風輕拂下我不禁浸醉於眼前的美酒－－啊，月色真美。

「賺了大概五六單FGO吧，我沒玩就是了。最近都忘著玩火紋手遊給任天堂賺錢呢。」

「微課不如無課，課個五六單還不如把錢留著去下面的游泳池玩呢。說起來你這個燻雞粉應該也去朝聖過了吧？」

「我又不是朝聖狂熱者，再說不帶個伴進去看上去就太寒酸了呢……咦不對，上次你明明說過明明不喜歡希爾頓的怎麼又來了？」

「還不是女友想來……結果一抵就逛了一整天，晚上又趕去高田馬場那邊喝酒了。聽說今晚有音遊界DJ樣子，下次讓她給你介紹一個好了。」

……

大家有默契地靜靜把肋排分掉，響21也所剩無幾，我帶著搖搖晃晃的腳步走回房間把我帶來的五年熟成加賀梅酒拿出來。雖然價格和品位比不上響21，不過我對我對梅酒的眼光還是有信心的。

他率先打破沈默：「品完酒我們該談一下正事囉。」

「說實話我實在不太想接手你家的公司……股東成份有點複雜，我又沒有多餘的票灌下去。在這個節點惹到別的勢力的話可是不太妙呢。」

「票也不用太多每季10張就好了啦，反正中價區價格彈性比較大，要調節也比較不會被投訴。」

「我就真的沒有餘裕能把票分出來了……按照我的模型來看如果我家公司的票再減三張左右的話高價循環就會崩潰，能套錢的地方少了一個的話會十分不方便。對面也盯票流盯得很緊，我換過來應該很快就會被發現了。」

「嗯……反正我這邊有的是人，來交易一下如何？我把你加進換票網絡裡，這樣對手會比較難鎖定你的動向；相應地我希望你能放5-8張票在我們這邊公司，這方面同樣可以透過網絡進行。」要比喻的話換票網絡大概就像洋蔥網絡(TOR)一樣，將點對點的連結遮蔽起來。

「這個可以。不過我在明處不需要太多小動作，放20票進網絡就夠了。你公司需要的票數我會從我手下分出來。」

「真夠精打細算呢。你的目標定好了嗎？」

「其實我沒有，不過手下們可是有一大串目想炸的人呢。有一位想搶新的碧藍艦娘經理不但被守下來，對方還真接把公司炸爛，每次提到那公司他都氣得牙癢癢的呢。反正現在是第一個賽季，多收攏人心才是最重要的。」

「多收下線準備下季再大幹一場嗎？」

「老了肝不起來了啦。有互助圈子才能確保大家不會餓死也不會輕易被攻擊，玩下去就當多認識朋友好了……總之你幫我將票吃掉，我幫你經營集火，這樣沒問題吧。」

「沒問題，那就拜托你囉。」

「合作愉快。」

「Cheers」

---------------------------

忙完手上的事情，聖誕大概就有空能寫作了。到時看上面能不能拓展成外傳囉。

另外好老婆不炒嗎？關鍵字： ACGN-stock

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