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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)

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4 Answers 4

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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

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  • In 問題が解けた, 解けた is the potential form of the verb 解く created by the affix -e-, and still has a subject. Since it is potential, the subject apprears with に, not が: (彼に)問題が解けた. Implicit or not, there is always the agent who solves a question. Animacy is irrelevant. -rare- is another affix meaning potential, and adding it to 解ける is redundant. It is completely ungrammatical. It is ungrammatical in the same way the English expression more better is ungrammatical. The meaning 'undone' belongs to a different verb 解ける (hodokeru), written the same but pronounced differently.
    – user458
    Jul 29, 2011 at 16:24
  • @sawa: 解けた is both the past potential form of 解く as well as the past form of 解ける. 問題が解けた is therefore ambiguous. I will edit to clarify this point.
    – Derek Schaab
    Jul 29, 2011 at 16:31
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    Why is ほどけにくい　acceptable while とけにくい unacceptable? Following the logic of "I cannot assign ability to an inanimate subject", it should cause both to be unacceptable since both are intransitive and by extension force the existence of the "inanimate subject"
    – Flaw ♦
    Jul 29, 2011 at 16:53
  • I do not think that ほどけられない靴ひも (that is, intransitive ほどける + られる) is correct, and I wonder why ほどけにくい靴ひも does not have the same problem. Because of this, I doubt that とけにくい問題 has the same problem as とけられない問題. とけにくい問題 still sounds strange to me, but I think that the reason is something different.
    – Tsuyoshi Ito
    Jul 29, 2011 at 17:03
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    But you can say 割れにくい, and that is instantaneous. Perhaps 〜eにくい has some sort of restriction relating to concrete vs abstract subjects?
    – Matt
    Jul 29, 2011 at 22:53
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Question 1: Either 解けない問題 or 解けられない問題 is correct. The subtlety comes in what the exact semantics are. 解けられない問題 means the problem itself it unsolvable. 解けない問題 means an Agent (as you denoted) cannot solve the problem, although the Agent is omitted. Would be clearer to use something like 私が解けない問題 ("A problem I can't solve") or whoever the Agent happens to be.

Thus I would say that 解けられない問題 is stronger because it implies that nobody can solve it.

Question 2: Yes (based on what I just said), but something like 解けにくい or 解けがたい might work as well or better without the possible confusion.

Question 3: Again, it would probably be better to specifically to include the agent, otherwise まぎらわしい. If you wanted to use this form to say "absolutely unsolvable", you'd have to say だれも解けない問題.

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    解けられない問題 sounds undoubtedly ungrammatical to my native ear. I do not know if there is any situation where form 解けられる can be used.
    – Tsuyoshi Ito
    Jul 29, 2011 at 15:36
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There is no such thing as intransitive form of 解ける used in the sense as in 解ける問題. 解けられる and 解けられない are ungrammatical. The three forms that you have under the transitive version are the correct one.

To answer Question 3, They are just a single usage. If the subject is implicit and is to be interpreted as 'anyone', then that will mean that it is absolutely unsolvable.

Edit Maybe, you are confusing 解ける (tokeru) with 解ける (hodokeru), which is an intransitve verb, does not mean 'solve', and is a completely different verb.

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  • "There is no such thing as intransitive 解く" - But there is intransitive 解ける, just like Flaw wrote in the question. And its potential form is 解けられる, is it not?
    – Lukman
    Jul 29, 2011 at 13:46
  • @Lukman It is still transitive in the sense that it takes an object. The object of 解ける takes が due to the potential morpheme -e-.
    – user458
    Jul 29, 2011 at 14:14
  • the potential form of 解く yields 解ける. Is this "解ける(potential transitive)" equivalent to the "解ける(intransitive)" ?
    – Flaw ♦
    Jul 29, 2011 at 14:32
  • @Flaw As I wrote, I have no idea what you mean by intransitive version. Do you have an example?
    – user458
    Jul 29, 2011 at 14:37
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    @sawa: “There is no such thing as intransitive 解く.” You are the only person who has mentioned “intransitive 解く.”
    – Tsuyoshi Ito
    Jul 29, 2011 at 15:44
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Only answering question #1.

In mathematical jargon, we have "決定不能問題" for an "undecidable problem", so I expect "unsolvable problem" to be worded similarly, or as "解決不能問題".

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