Thursday, 18 January 2018

The development of Fire Emblem Heroes

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.


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.


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.


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.

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.

Wednesday, 6 December 2017



































另外好老婆不炒嗎?關鍵字: ACGN-stock