Arvoitus oli vähällä ratketa ilman viimeistä johtolankaa.

It uses the structure olla vähällä + infinitive, which means to be close to doing something, to almost do something, or to be on the verge of happening.

So:

• oli = was
• vähällä = part of the fixed expression
• ratketa = to get solved / to be solved

Together, oli vähällä ratketa means was close to being solved or almost got solved.

This is a very common Finnish pattern:

• Olin vähällä kaatua. = I almost fell.
• Hän oli vähällä unohtaa avaimet. = He/She almost forgot the keys.

Not really in a word-for-word way. In this sentence, vähällä is best learned as part of the fixed idiom olla vähällä.

Historically, vähällä is a case form related to vähä/vähän (little), but in modern Finnish learners usually just treat olla vähällä as one chunk meaning:

• to almost
• to be close to
• to be on the verge of

So it is better to understand:

• Arvoitus oli vähällä ratketa
  = The mystery almost got solved

rather than trying to interpret vähällä separately every time.

Because ratketa and ratkaista are different verbs.

• ratkaista = to solve something
  This is transitive, so it takes an object.
• ratketa = to get solved / to be solved
  This is intransitive, so the thing itself is the subject.

Compare:

• Poliisi ratkaisi arvoituksen.
  = The police solved the mystery.

• Arvoitus ratkesi.
  = The mystery got solved / was solved.

In your sentence, arvoitus is not the object of someone else's action. It is the thing that was close to becoming solved, so ratketa is the natural choice.

It is the basic infinitive form of the verb, the form you see in dictionaries.

After olla vähällä, Finnish uses this infinitive:

• olla vähällä tehdä = to almost do
• olla vähällä kaatua = to almost fall
• olla vähällä ratketa = to almost get solved

So ratketa is not conjugated here because the finite verb is already oli.

Because the preposition ilman (without) requires the partitive case.

So:

• johtolanka = clue
• partitive singular: johtolankaa

And because the adjective must match the noun:

• viimeinen johtolanka = the last/final clue
• partitive singular: viimeistä johtolankaa

This is just normal case government after ilman:

• ilman rahaa = without money
• ilman autoa = without a car
• ilman viimeistä johtolankaa = without the last clue

Because viimeinen is one of the common -nen words, and those change their stem when they are declined.

So:

• dictionary form: viimeinen
• partitive singular: viimeistä

This is a very common pattern:

• nainen → naista
• ihminen → ihmistä
• punainen → punaista
• viimeinen → viimeistä

So the change is not special to this sentence; it is just normal declension for -nen words.

Often yes, in translation it can correspond to English be solved, but grammatically Finnish is doing something a little different.

English often says:

• The mystery was solved.

Finnish can express that idea with an intransitive verb:

• Arvoitus ratkesi.

That is not a Finnish passive form. It is an ordinary verb whose meaning is naturally become solved / get solved.

So in this sentence:

• Arvoitus oli vähällä ratketa
  literally feels like The mystery was close to getting solved

This is very natural Finnish.

They are similar, but not always identical.

• melkein = almost / nearly
• olla vähällä + infinitive = to almost do, often with a sense of coming very close to an event

Very often both are possible:

• Arvoitus melkein ratkesi.
• Arvoitus oli vähällä ratketa.

But olla vähällä can sound a bit more dramatic or event-focused, as if something nearly happened. It is especially common with things that almost happened unexpectedly:

• Olin vähällä pudota. = I almost fell.

In your sentence, oli vähällä ratketa gives a strong sense of it was just about to be solved.

Yes, Finnish word order is flexible, but the given order is neutral and natural.

Arvoitus oli vähällä ratketa ilman viimeistä johtolankaa.

This order presents the sentence clearly:

1. Arvoitus = topic/subject
2. oli vähällä ratketa = what nearly happened
3. ilman viimeistä johtolankaa = under what condition

You could move elements around for emphasis, for example:

• Ilman viimeistä johtolankaa arvoitus oli vähällä ratketa.

This puts extra focus on without the last clue.

So the original sentence is not the only possible order, but it is a very standard one.

It is a compound noun:

• johto can relate to leading, guidance, or direction
• lanka = thread

So the literal image is something like a guiding thread or leading thread, which matches the idea of a clue.

Finnish uses a lot of compounds like this, and learning to notice them can be very helpful.

Yes. It suggests that the last/final clue would normally seem important, but even without it, the mystery was very close to being solved.

So the sentence has a nuance like:

• the final clue was missing
• even so, the solution was almost reached

That gives the sentence a slightly dramatic feel: the mystery was nearly solved despite lacking one crucial piece of information.