Learn Hardcore German: In der Hütte hängt eine alte Grafik mit einem Pfeil, der den sicheren Weg ins Tal zeigt. - In the hut there is an old graphic with an arrow that shows the safe way down into the valley.

In der Hütte hängt eine alte Grafik … (focus on the location)
Eine alte Grafik mit einem Pfeil, der den sicheren Weg ins Tal zeigt, hängt in der Hütte. (focus on the graphic)

Both are correct. The choice mainly changes what you present as the starting point (topic) of the sentence, not the basic meaning.

Why is it In der Hütte with dative, but ins Tal with accusative?

Because in is a two‑way preposition in German. It takes:

Dative when expressing a location (where?)
Accusative when expressing a direction (where to?)

In this sentence:

In der Hütte = in the hut → a place where something is → location → dative
- Feminine noun: die Hütte
- Dative singular: der Hütte → in der Hütte
ins Tal = into the valley → movement towards / into somewhere → direction → accusative
- Neuter noun: das Tal
- Accusative singular: das Tal
- in das Tal contracts to ins Tal (very common in speech and writing).

So the case difference directly reflects the difference between being somewhere and going/leading somewhere.

What’s the difference between hängt and ist here? Could I say In der Hütte ist eine alte Grafik?

You could say In der Hütte ist eine alte Grafik, and it’s grammatically correct, but the nuance is different:

hängen describes the way something is positioned: it is hanging (on a wall, from the ceiling, etc.).
sein is much more general: it just says there is / exists.

So:

In der Hütte hängt eine alte Grafik …
→ specifically tells you the graphic is hanging, probably on a wall.
In der Hütte ist eine alte Grafik …
→ simply says there is an old graphic somewhere in the hut. It could be hanging, leaning against something, lying on a table – it doesn’t specify.

In a context where it’s clearly about something on the wall, hängt is the more natural and informative choice.

Why is it eine alte Grafik and not something like ein altes Grafik?

Because Grafik is feminine in German.

Noun: die Grafik (feminine)
Here it’s the subject in the nominative:
- Indefinite article, nominative feminine: eine
With an indefinite article, attributive adjectives take -e in the nominative singular (all genders except masc./neut. without ending):
- alt → alte

So you get:

eine alte Grafik

ein altes Grafik would be wrong for two reasons:

Grafik is not neuter, so ein …es doesn’t match the noun’s gender.
Even with a neuter noun, the adjective and noun would both have to agree in gender: e.g. ein altes Bild (a(n) old picture), but not Grafik.

Why is it mit einem Pfeil and not mit einen Pfeil or mit ein Pfeil?

Because the preposition mit always takes the dative case in German.

Noun: der Pfeil (masculine)
Indefinite article, dative masculine: einem
So: mit einem Pfeil

Forms you mentioned:

mit einen Pfeil → einen is accusative masculine, wrong after mit.
mit ein Pfeil → article in the wrong case (and also wrong form for dative).

Pattern to remember:
mit + Dativ → mit einem Pfeil, mit einer Frau, mit einem Kind, etc.

In der den sicheren Weg ins Tal zeigt, what does der refer to, and why is it der and not die or dem?

der here is a relative pronoun introducing a relative clause:

mit einem Pfeil, der den sicheren Weg ins Tal zeigt
Literally: with an arrow that shows the safe way into the valley.

So der refers back to Pfeil.

Gender & number:
- Pfeil is masculine singular → the matching relative pronoun is der in the nominative.
Case inside the relative clause:
- In der den sicheren Weg ins Tal zeigt, the thing that shows is the arrow.
- So der is the subject of zeigt → nominative.
- den sicheren Weg is the direct object.

That’s why it is:

der (nominative masculine)
and not:
die (feminine / plural), because Pfeil is masculine.
dem (dative), because in the clause the arrow is not an indirect object; it’s the subject.
den (accusative), for the same reason.

Important point:
The case of the relative pronoun comes from its function inside the relative clause, not from the case of its antecedent Pfeil in the main clause (where einem Pfeil is dative because of mit).

Why is it den sicheren Weg and not der sichere Weg or dem sicheren Weg?

Here Weg is the direct object of the verb zeigt (shows), so it must be in the accusative case.

Noun: der Weg (masculine)
Definite article, accusative masculine: den
With a definite article in masculine accusative, the adjective ending is -en:
- sicher → sicheren

So:

den sicheren Weg = the safe way (as an object).

Comparisons:

Der sichere Weg ist lang.
→ Here der sichere Weg is the subject (nominative masculine).
Sie folgen dem sicheren Weg.
→ After some verbs or prepositions you might get dative: dem sicheren Weg (dative masculine).

In this sentence, because Weg is what is being shown, accusative den sicheren Weg is required.

Why is there a comma before der den sicheren Weg ins Tal zeigt?

Because German always separates relative clauses with commas.

mit einem Pfeil, der den sicheren Weg ins Tal zeigt
The part der den sicheren Weg ins Tal zeigt is a Nebensatz (subordinate clause), specifically a Relativsatz (relative clause) that describes Pfeil.

Rule:
Any relative clause introduced by a relative pronoun (der, die, das, welcher, etc.) is set off by commas:

Das ist der Mann, der neben mir wohnt.
Ich lese ein Buch, das sehr spannend ist.

Unlike in English, the comma is not optional here; it is mandatory.

Why is zeigt at the very end of the relative clause?

Because relative clauses in German are a type of subordinate clause, and in subordinate clauses the finite verb goes to the end.

Structure here:

der (subject)
den sicheren Weg ins Tal (object phrase)
zeigt (finite verb, at the end)

Pattern:

[relative pronoun] + [other elements] + [verb]

More examples:

…, der mir gestern geholfen hat.
…, die du mir empfohlen hast.
…, das ich noch nie gesehen habe.

So zeigt being at the end is simply standard subordinate‑clause word order.

What exactly is the nuance of Grafik here? Could I use Bild or Poster instead?

Grafik in German usually suggests something like:

a print, etching, drawing, or technical illustration,
often more like a line drawing or a map‑like picture, rather than, say, a photograph.

You could use:

Bild = very general “picture / image”
- In der Hütte hängt ein altes Bild mit einem Pfeil, der …
- This is the most neutral/colloquial.
Poster = specifically a poster (usually large, modern, printed on paper)
- In der Hütte hängt ein altes Poster mit einem Pfeil, der …
- This changes the mental image (not an artwork/print but a poster).

Grammatically, all those nouns work the same way (you’d still say mit einem Pfeil, der … because the arrow is still masculine Pfeil). The choice mainly affects what kind of object the reader imagines.

Could I rephrase the sentence as Eine alte Grafik mit einem Pfeil, der den sicheren Weg ins Tal zeigt, hängt in der Hütte? Does that change the meaning?

Yes, that alternative word order is fully correct:

Eine alte Grafik mit einem Pfeil, der den sicheren Weg ins Tal zeigt, hängt in der Hütte.

The meaning is essentially the same. The difference is:

Original: In der Hütte hängt eine alte Grafik …
→ Puts the hut / location first; you’re talking about the hut and then saying what’s in it.
Alternative: Eine alte Grafik … hängt in der Hütte.
→ Puts the graphic first; you’re talking about the graphic and then saying where it is.

In both versions, the verb hängt is still in second position, so both follow the V2 rule correctly.

What’s the difference between den sicheren Weg ins Tal and something like den sicheren Weg zum Tal?

Both are possible, but the nuance is different:

den sicheren Weg ins Tal
- ins = in das = into the
- Suggests a path that leads down into / inside the valley, crossing its boundary.
- Emphasis on entering the valley as a space.
den sicheren Weg zum Tal
- zum = zu dem = to(wards) the
- Suggests a path leading to the valley, maybe to its edge or entrance.
- Emphasis on approaching the valley as a destination/point, not necessarily entering deeply into it.

In many contexts, Weg ins Tal is the more typical collocation when you imagine actually going down into the valley itself.