Various Interpretations of Mathematical Standard Methods and Solutions 🧩

I have been writing articles about standard methods again recently.

Having looked at various things lately, I've realized something.

Perhaps because the concept of 'standard methods' has become quite widespread thanks to Isshiki-san, theory, and Tetsuryoku, or perhaps precisely because it is starting to permeate,

much like self-study, essence, rote memorization, and volume of practice, the meaning of 'standard method' itself has come to have a very wide range of interpretations.

When this happens, a familiar phenomenon occurs:

The interpretation shifts between the sender and the receiver, leading to a misalignment in what is being referred to, so even if one practices it, they don't improve.

Because both sides are discussing things while the definitions they are pointing to are subtly different and misaligned, they remain at a stalemate forever.

This is what is happening.

I explain this in my articles, but basically, it's about 'doing X when Y' or 'doing X when you see Y'.

It is a style that clarifies the usage context for each characteristic structure based on phenomena, such as 'characteristic structure and its analysis method'.

My interpretation is that if the usage context is not clear, it cannot be called a standard method unless you at least write down guidelines or tendencies, such as 'it is often used in these situations' or 'it is weak/strong against this'.

This is written in a math book that is called a 'god-tier book' but is actually overrated.

Solutions for integer problems: (i) Narrow down the range, (ii) Divide by remainders, (iii) Factorize.

To me, it is highly questionable whether this can be called a standard method.

The way standard methods are summarized varies from person to person. There are listing types like this one, and others that number them in the order to try them.

In my view, the current 'solution enumeration' type is not what I would call a standard method, or it falls under the category of low-quality standard methods.

As for terrible examples other than this overrated book,

they claim to handle standard methods, but they just deal with problems without even verbalizing the solution method itself.

They just write 'divide by remainders,' leaving the method exposed without any intention of organizing it; it's just the final result of a selection among multiple candidates, with no explanation of the process.

Sometimes it's divided by remainders, sometimes you just keep doing it, and sometimes you do it this way—it ends up being a divergent explanation, and sometimes that happens.

Conversely, let's look at a good example. Please take a look at Issiki-san's guide to difficult problems regarding narrowing down integers using inequalities.

Can you see the difference?

In a certain 'god-tier' book, it was nothing more than guesswork, wondering if it's this or that, based on atmosphere and feeling.

Issiki-san's work provides follow-up in an easy-to-use format for guidelines and trends, such as whether it involves two or three or more characters, or tools for proof problems, when the usage scenarios are not clearly defined.

Of course, Koyama-sensei also explains this.

It's from around the 20-minute mark.
※Since the video for free member registration has been changed and the place where it was posted can now be viewed, I will delete it from here.

This might look like a small difference, but if you think about solving problems, it's a pretty big difference.

Or rather, regarding this integer part, the arrangement itself is strange to begin with. As expected of an overrated book. If you think about the ease of solving problems, it should at least be arranged like Issiki-san's or Koyama-sensei's.

And when there is a summary, people immediately come out saying 'It's summarized! It's summarized!', but the quality is completely different.

There are too many people who don't understand this point.

That is why I think strange opinions like 'Master Key and Blue Chart are the same' come out. In addition, isn't that why overrated books like S or E and S·K are being produced?

Seeing overrated books being called 'god-tier' makes me think that
while Issiki-san's summary is sufficiently appreciated, it is actually underrated relative to its quality.

There are other things I think are different, too.

When people say 'standard method,' they often think it only consists of geometry problems, integer problems, complex numbers, identities, recurrence relations, the Law of Sines, the Law of Cosines, and integral calculations.

If they are a bit more sensible, they might also include limΣ, maximum/minimum, loci/regions, and probability recurrence relations.

This is probably the same as thinking 'formula = point'.

Believing in the myth that the Blue Chart has comprehensiveness and contains everything,
they think that only what is included is a standard method.

Or, because these are the things that come up first when you search for 'standard method,' they think only these are standard methods.

I think that is the current state of things.

This area seems likely to be corrected due to the influence of theories and master keys.

You might ask, 'Is there really anyone like that?' but to those reading this a few years from now: they really did exist.

It is likely due to the influence of solution theories, master keys, Sekakyo, Shoaku, Must, and external users of Tetsuryokukai, but I have a sense that the number of people who are quite good at mathematics has increased rapidly over the past few years.

I feel it is a very good thing that the number of victims of the 'it's all about the essence' idealism is decreasing.

Answers to frequently asked questions

I will answer them all at once here.

1. Does this also apply to probability? Yes, it does.

2. But it's only for probability recurrence relations, right? No, it is not.

3. Are you talking about the Blue Chart? Partly, but basically no.

4. Standard methods only work for simple problems, right? I don't know what level you consider 'simple,' but if you mean problems you can solve at a level that won't leave you behind other examinees, then yes, they work. Except for the University of Tokyo, in most cases, you will be in a position to gain a significant advantage.

5. There's no point in just memorizing standard methods, right? I never said to just memorize standard methods.

6. I can't create them. You don't need to create them yourself at first.

7. I can't master them. Aren't you learning from incomplete and difficult-to-use books, like those overrated ones? If you aren't succeeding with those, let's change the books.