Musíme vyřešit hádanku.

Musíme is the 1st person plural present form of muset (to have to / must).

So:

• musím = I must
• musíš = you must
• musí = he/she/it must
• musíme = we must
• musíte = you must (plural/formal)
• musí = they must

In this sentence, musíme already tells you the subject is we, so Czech does not need a separate my.

Because Czech usually drops subject pronouns when they are clear from the verb ending.

• Musíme vyřešit hádanku. = We must solve the riddle.

The ending -íme already shows we.

You can add my if you want emphasis or contrast:

• My musíme vyřešit hádanku. = We have to solve the riddle.
  (for example, not someone else)

So the version without my is the normal, neutral one.

This is about aspect, which is very important in Czech.

• řešit = imperfective → to solve / be solving / deal with
• vyřešit = perfective → to solve completely, to finish solving

In Musíme vyřešit hádanku, the idea is that the riddle needs to be successfully solved, with a result. That is why vyřešit is natural here.

Compare:

• Musíme řešit hádanku.
  = We need to work on the riddle / deal with it.

• Musíme vyřešit hádanku.
  = We must solve the riddle (bring it to completion).

After modal verbs like muset, Czech still chooses aspect based on meaning.

Because it is the direct object of the verb, so it goes into the accusative case.

The dictionary form is:

• hádanka = riddle

But in the sentence:

• vyřešit hádanku = to solve a/the riddle

That is because hádanka is a feminine noun ending in -a, and in the singular accusative it changes to -u.

A similar pattern:

• kniha → knihu
• otázka → otázku
• hádanka → hádanku

Yes, it is a very natural phrase.

• vyřešit hádanku = solve a riddle
• rozluštit hádanku can also be used, especially with the sense of figure out / decipher
• uhodnout hádanku is less standard if you mean solve; uhodnout is more like guess correctly

So for a learner, vyřešit hádanku is an excellent and standard choice.

After muset, Czech uses the infinitive directly, with no equivalent of English to.

• Musíme vyřešit hádanku.
• literally: We must solve a/the riddle.

Not:

• Musíme to vyřešit. = We must solve it.
• Musím jít. = I must go.
• Musíte čekat. = You must wait.

So the pattern is:

muset + infinitive

Yes. Czech word order is flexible, although different orders sound more or less natural depending on emphasis.

The neutral order here is:

• Musíme vyřešit hádanku.

You could also say:

• Hádanku musíme vyřešit.
  This puts more emphasis on the riddle.

• Musíme hádanku vyřešit.
  Also possible; slightly different information structure.

Because Czech uses endings to mark grammar, word order often expresses focus, topic, or emphasis, not just basic sentence structure.

A rough pronunciation guide:

• Musíme ≈ moo-ZHEE-meh
• vyřešit ≈ VIH-rzheh-shit
• hádanku ≈ HAA-dahn-koo

A few important points:

• Czech stress is usually on the first syllable:

  • MUsíme
  • VYřešit
  • HÁdanku

• í and á are long vowels.

• ř is the famous Czech sound. It is difficult for English speakers. In vyřešit, it sounds something like a mix of r and zh.

• š sounds like English sh.

If you cannot say ř yet, an approximate rzh sound is a reasonable learner substitute.

It can mean either, depending on context.

• Musíme vyřešit hádanku.
  can be understood as:
  • We must solve the riddle.
  • We have to solve the riddle.

In many everyday contexts, Czech muset covers both ideas. The difference between English must and have to is often not marked strongly in Czech.

Not really. This sentence strongly suggests a goal: the riddle needs to get solved.

That is mainly because of vyřešit, the perfective verb. Perfective aspect points toward completion.

If you wanted more of a process meaning, Czech would more likely use řešit:

• Musíme řešit hádanku.
  = We need to work on the riddle / deal with the riddle.

So Musíme vyřešit hádanku is about the result, not just the activity.