Ratkomme vaikeaa pulmaa.

The ending -mme marks 1st person plural in the present tense, so ratkomme means we solve / we are solving.

Because Finnish verb endings already show the subject, you do not need a separate word for we.
So:

• ratkomme = we solve / we are solving
• me ratkomme = also we solve, but with an explicit pronoun, often for emphasis or contrast

The dictionary form is ratkoa.

This verb is related to ratkaista, which is another common verb meaning to solve.
However, ratkoa often has a more process-oriented feel: it can suggest working through, trying to solve, or solving piece by piece.

So ratkomme vaikeaa pulmaa sounds a bit like:

• we are working on a difficult puzzle/problem
• we are trying to solve a difficult puzzle/problem

rather than strongly emphasizing the completed result.

They are in the partitive singular because the object is being presented as incomplete, ongoing, or not fully bounded.

In Finnish, objects often appear in the partitive when:

• the action is ongoing
• the result is not yet complete
• the verb itself tends to describe an unbounded process

That fits well with ratkoa, which often suggests working on solving rather than clearly finishing.

So:

• vaikeaa pulmaa = partitive singular
• this gives the sense of working on a difficult puzzle/problem

Because in Finnish, adjectives usually agree with the noun they describe.

Here, pulmaa is in the partitive singular, so the adjective vaikea must also be in the partitive singular:

• vaikea pulma = a difficult puzzle/problem
• vaikeaa pulmaa = partitive singular

So both words change together.

The dictionary forms are:

• vaikea = difficult
• pulma = puzzle, problem

In this sentence they appear as partitive singular forms:

• vaikea → vaikeaa
• pulma → pulmaa

The double a is normal here. Finnish often adds the partitive ending so that two vowels come together.

Yes. That would also be grammatical, but the nuance changes.

• Ratkomme vaikeaa pulmaa = we are working on solving a difficult puzzle/problem; the process is emphasized
• Ratkaisemme vaikean pulman = we solve / will solve the difficult puzzle/problem; the result or completion is emphasized more strongly

Also notice the object changes:

• vaikeaa pulmaa = partitive object
• vaikean pulman = total object form, suggesting a completed whole

So the choice is not random; it affects the meaning.

Finnish often leaves subject pronouns out when the verb ending already makes the subject clear.

Here, ratkomme already tells you the subject is we.

So both of these can work:

• Ratkomme vaikeaa pulmaa.
• Me ratkomme vaikeaa pulmaa.

The version without me is very normal.
Adding me may sound more emphatic, for example if you want to contrast we with someone else.

Finnish has no articles like English a, an, or the.

So pulma can mean:

• a puzzle/problem
• the puzzle/problem

The exact meaning depends on context.

That is why vaikeaa pulmaa does not contain any separate word corresponding to a or the.

Yes. This is a normal Finnish sentence.

A full version with the pronoun would be:

• Me ratkomme vaikeaa pulmaa.

Since the verb ending already shows the subject, Finnish often drops me, leaving:

• Ratkomme vaikeaa pulmaa.

So even though the sentence begins with the verb, it is still completely natural.