"Unsolvable problem"


What is the correct construction of "unsolvable problem"? Is it
"解{と}けられない問題{もんだい}"？ **(Question 1)**

> Plain(intransitive): 解{と}ける "To be solved".
>
> Potential form(of intransitive): 解けられる "can be solved".
>
> Potential + Negated(of intransitive): 解けられない "cannot be solved"

* * *

> Plain(transitive) 解{と}く "(Some agent) solves".
>
> Potential form(of transitive): 解ける "(Some agent) can solve".
>
> Potential + Negated(of transitive): 解けない "(Some agent) cannot solve"

* * *

Does 解けられない問題 mean it intrinsically cannot be solved? **(Question 2)**

Does the transitive version "解けない" implies that some agent is unable to solve,
but does not mean it is absolutely unsolvable? **(Question 3)**

*(Agent: one who initiates and/or completes an action or event)




Neither of the current answers sit well with me at the moment, so I'm going to
risk adding to the confusion by posting another.

### Question 1 (grammar)

First, let's clarify the two verbs in question:

> 解く solve (a problem)
>
> 解ける resolve (itself)

(These are not the only definitions, but for the sake of brevity and on-topic-
ness we'll go with these.)

The difference lies in the subjects. With 解く, the subject is the person (or
agent) who actually and intentionally does the solving, but with 解ける, the
subject is the problem itself, and no agent is given:

> **私が** 問題を解いた。 I solved the problem.
>
> **問題が** 解けた。 The problem was solved (solved itself).

Note that 問題が解けた could also mean "[I] was able to solve the problem", since
解けた is both the past potential form of 解く (解く→解ける→解けた) and the past form of
解ける (解ける→解けた).

This shows why the potential form of 解く (解ける) is valid, but the potential form
of 解ける (解けられる) is not: 解けられる personifies an inanimate subject. (sawa says this
is ungrammatical; I don't know if that's the right term, since it's a
syntactically valid form of 解ける, but it's certainly not used.)

> ○ （私は）この問題が解けない。 [I] can't solve this problem.
>
> × この問題は解けられない。 This problem can't solve itself.

The second sentence might sound fine at first (from a grammatical standpoint),
but when you think about it, assigning an ability to an inanimate subject
doesn't work here. (Even in English, the better way to express the idea behind
sentence #2 is, "This problem **will not** solve itself.")

Because of this, only 解けない問題 is correct.

### Questions 2 and 3

This is an issue of context, as sawa and istrasci mentioned. For example:

> 私が解けない問題 a problem I can't solve
>
> だれでも解けない問題 a problem no one can solve

istrasci also mentioned the ～にくい suffix. I was about to agree with this, but
then I saw sawa's edit which reminded me that 解く can be read as both **と** く
and **ほど** く, and 解ける can be read as both **と** ける and **ほど** ける. (The
meanings differ between readings.) ～にくい can be attached to both ほどく and ほどける,
such as in this way:

> ほどきにくいくつひも a shoelace that is difficult to untie
>
> ほどけにくいくつひも a shoelace that won't come undone easily

…and to とく, but not to とける:

> ○ ときにくい問題 a problem that's difficult to solve
>
> × とけにくい問題

However, the ～にくい suffix does not convey the same level of "impossibility" as
だれでも解けない does.

### Bonus (more "unsolvable")

Generally, once you start throwing out kanji compounds like 解決 and 不可能, you're
in the realm of written Japanese, but these might be good to know:

> 解決できない問題 a problem that (subject) can't solve
>
> 解決不可能な問題 an unsolvable problem
>
> 解決不可能と思われる問題 a problem thought to be unsolvable




Neither of the current answers sit well with me at the moment, so I'm going to
risk adding to the confusion by posting another.

### Question 1 (grammar)

First, let's clarify the two verbs in question:

> 解く solve (a problem)
>
> 解ける resolve (itself)

(These are not the only definitions, but for the sake of brevity and on-topic-
ness we'll go with these.)

The difference lies in the subjects. With 解く, the subject is the person (or
agent) who actually and intentionally does the solving, but with 解ける, the
subject is the problem itself, and no agent is given:

> **私が** 問題を解いた。 I solved the problem.
>
> **問題が** 解けた。 The problem was solved (solved itself).

Note that 問題が解けた could also mean "[I] was able to solve the problem", since
解けた is both the past potential form of 解く (解く→解ける→解けた) and the past form of
解ける (解ける→解けた).

This shows why the potential form of 解く (解ける) is valid, but the potential form
of 解ける (解けられる) is not: 解けられる personifies an inanimate subject. (sawa says this
is ungrammatical; I don't know if that's the right term, since it's a
syntactically valid form of 解ける, but it's certainly not used.)

> ○ （私は）この問題が解けない。 [I] can't solve this problem.
>
> × この問題は解けられない。 This problem can't solve itself.

The second sentence might sound fine at first (from a grammatical standpoint),
but when you think about it, assigning an ability to an inanimate subject
doesn't work here. (Even in English, the better way to express the idea behind
sentence #2 is, "This problem **will not** solve itself.")

Because of this, only 解けない問題 is correct.

### Questions 2 and 3

This is an issue of context, as sawa and istrasci mentioned. For example:

> 私が解けない問題 a problem I can't solve
>
> だれでも解けない問題 a problem no one can solve

istrasci also mentioned the ～にくい suffix. I was about to agree with this, but
then I saw sawa's edit which reminded me that 解く can be read as both **と** く
and **ほど** く, and 解ける can be read as both **と** ける and **ほど** ける. (The
meanings differ between readings.) ～にくい can be attached to both ほどく and ほどける,
such as in this way:

> ほどきにくいくつひも a shoelace that is difficult to untie
>
> ほどけにくいくつひも a shoelace that won't come undone easily

…and to とく, but not to とける:

> ○ ときにくい問題 a problem that's difficult to solve
>
> × とけにくい問題

However, the ～にくい suffix does not convey the same level of "impossibility" as
だれでも解けない does.

### Bonus (more "unsolvable")

Generally, once you start throwing out kanji compounds like 解決 and 不可能, you're
in the realm of written Japanese, but these might be good to know:

> 解決できない問題 a problem that (subject) can't solve
>
> 解決不可能な問題 an unsolvable problem
>
> 解決不可能と思われる問題 a problem thought to be unsolvable

