Addig nézzük a falat, amíg fel nem fedezzük az elrejtett betűkódot.

They work as a pair.

• addig = until then / up to that point / for that long
• amíg = until / while / as long as

Together, addig ... amíg often means do X until Y happens.

So the structure is basically:

• Addig nézzük a falat, amíg...
  = We keep looking at the wall until...

English often uses just until, but Hungarian frequently uses both parts of the pair.

Because Hungarian has definite and indefinite verb conjugation.

Here the object is a falat = the wall, which is a definite object, so the verb takes the definite form:

• nézzük = we look at it / we are looking at the specific thing
• nézünk would be used with an indefinite or no object

Compare:

• Nézzük a falat. = We look at the wall.
• Nézünk egy falat. = We look at a wall.

It can mean either, depending on context.

For many verbs, the 1st person plural definite indicative and the 1st person plural definite imperative/subjunctive look the same.

So nézzük can be:

• we look / we are looking
• let's look

In this sentence, context tells you which one is intended. If it is a statement, it means we keep looking. If it is an instruction, it can mean let's keep looking.

The same kind of ambiguity exists with fedezzük.

Because fal is the direct object of nézzük, so it takes the accusative ending -t.

• fal = wall
• falat = wall + accusative = the wall as the thing being looked at

Hungarian often inserts a linking vowel before -t, so you get falat, not just falt.

Also, learners sometimes notice that falat can also be a completely different word meaning a bite / morsel. Here, though, it is clearly fal + -t because of the context.

For the same reason: it is the direct object of fedezzük, so it takes the accusative ending.

• betűkód = letter code
• betűkódot = letter code + accusative

Again, Hungarian uses a linking vowel here:

• -ot

So:

• az elrejtett betűkódot = the hidden letter code as the thing being discovered

Because felfedez is a verb with a verbal prefix:

• fel-
  • fedez = felfedez = discover

Hungarian verbal prefixes move around depending on the structure of the sentence.

In a normal negative sentence, you would usually say:

• nem fedezzük fel = we do not discover it

But after amíg in the sense of until, Hungarian very often uses the pattern:

• amíg fel nem fedezzük = until we discover it

So this is a special and very common pattern with until-clauses.

Because in Hungarian, amíg ... nem is a standard way to express until.

So here nem does not mean that the discovery never happens. It marks the endpoint of the action in the subordinate clause.

The meaning is:

• We keep looking at the wall up to the point when we discover the hidden letter code.

A good way to think about it is:

• amíg ... nem = until

not

• until ... not

This is one of those places where you should translate the whole pattern, not each word separately.

elrejtett means hidden.

It comes from the verb:

• elrejt = to hide

Here it functions like a participial adjective:

• elrejtett betűkód = hidden letter code

It does not change because Hungarian adjectives in front of a noun usually do not agree with the noun in case or number.

So you say:

• az elrejtett betűkódot

not something like:

• az elrejtettet betűkódot

The case ending goes on the noun, not on the adjective.

Both a and az mean the.

Hungarian uses:

• a before a consonant sound
• az before a vowel sound

So:

• a falat because falat begins with f
• az elrejtett betűkódot because elrejtett begins with a vowel sound

It is just a pronunciation-based alternation, similar to a/an in English, except both forms mean the.

Yes.

amíg can mean:

• while
• as long as
• until

Context tells you which meaning is intended.

In this sentence, the combination of:

• addig
• and especially amíg ... nem

makes the meaning clearly until.

Compare:

• Amíg esik, bent maradunk. = While / as long as it is raining, we stay inside.
• Addig várunk, amíg meg nem érkezik. = We wait until he arrives.

Yes. Hungarian very often forms compound nouns as a single word.

• betű = letter
• kód = code
• betűkód = letter code

This is very normal in Hungarian. English is less consistent: sometimes English writes similar combinations as one word, sometimes two words, sometimes with a hyphen.

To some extent, yes. Hungarian word order is quite flexible, but different orders change emphasis.

The given sentence is natural because Addig sets up the time frame right away:

• Addig nézzük a falat, amíg...

That feels like:

• We keep looking at the wall until...

A different order is possible, but it may sound more marked or put focus somewhere else.

So even though the grammar allows variation, the original version is a very normal way to say it.