Learn Hardcore Icelandic: Bóndinn á þrjá hesta og vinnur allan daginn á bóndabænum. - The farmer has three horses and works all day on the farm.

Bóndi is the basic (dictionary) form meaning farmer.
Icelandic usually shows “the” by adding an ending to the noun, not by a separate word:

bóndi = a farmer
bóndinn = the farmer

So -inn is the definite article for a masculine noun in the nominative singular.
In the sentence, Bóndinn is the subject, so it’s in the nominative case and made definite: the farmer.

What exactly does á mean in Bóndinn á þrjá hesta, and why not hefur?

Here á is the 3rd person singular of the verb að eiga (to own / to have):

ég á – I have
þú átt – you have
hann / hún / það á – he / she / it has

In many everyday contexts, á is preferred for having/owning concrete things, like animals, a house, a car, etc.

Hefur comes from the verb að hafa (to have) and is often used:

with abstract things (að hafa tíma – to have time)
in compound verb forms (hefur gert – has done)

You could say Bóndinn hefur þrjá hesta, and it’s understandable, but á þrjá hesta sounds more natural for owning animals.

Why is it þrjá hesta and not þrír hestar?

The base forms are:

þrír – three (masculine form in the nominative)
hestur – horse

In the sentence they are the direct object of the verb á, so they must be in the accusative plural (masculine):

Nominative: þrír hestar – three horses (as subject)
Accusative: þrjá hesta – three horses (as object)

So:

Bóndinn á þrjá hesta. – The farmer has three horses.
Þrír hestar hlaupa. – Three horses are running.

The number þrír and the noun hestur both change form and agree in case, gender, and number. That’s why you see þrjá hesta here.

What are the dictionary forms of the main content words, and how do they relate to the forms in the sentence?

Sentence: Bóndinn á þrjá hesta og vinnur allan daginn á bóndabænum.

Dictionary forms → forms in the sentence:

bóndi (farmer) → Bóndinn (the farmer, nominative definite)
að eiga (to own / have) → á (he has, 3rd person singular present)
þrír (three, masc. nom.) → þrjá (three, masc. acc.)
hestur (horse) → hesta (horses, masc. acc. plural)
að vinna (to work) → vinnur (he works / is working, 3rd person singular present)
allur (all, masc.) → allan (all, masc. acc. singular)
dagur (day) → daginn (the day, masc. acc. definite)
bóndi
- bær (farmer + farm/stead) → bóndabær (farmstead) → bóndabænum (on the farmstead, masc. dat. definite)

All these changes are normal Icelandic inflection: endings change for case, number, gender, definiteness, and verb person/tense.

Why is the verb vinnur and not something like vinnir, vinnar, or vinna?

Vinnur is the 3rd person singular present of að vinna (to work).

Present tense paradigm (simplified):

ég vinn – I work
þú vinnur – you work (singular)
hann / hún / það vinnur – he / she / it works
við vinnum – we work
þið vinnið – you work (plural)
þeir / þær / þau vinna – they work

So with the subject Bóndinn (he), the correct form is vinnur.
Icelandic present tense normally covers both works and is working in English, so:

Bóndinn vinnur allan daginn
= The farmer works all day / The farmer is working all day.

What does allan daginn literally mean, and why those endings?

Literally, allan daginn means all the day.

Breakdown:

allur = all (masculine adjective)
dagur = day (masculine noun)

In the sentence, this phrase expresses a duration of time, functioning as a kind of object of time, so Icelandic uses the accusative case:

allan – masculine accusative singular form of allur
daginn – masculine accusative singular definite of dagur (the day)

Compare:

Dagurinn er langur. – The day (nom.) is long.
Hann vinnur allan daginn. – He works all day (acc.).

So allan daginn = all (of) the day, i.e. all day.

Why is á used twice, and is it the same word in both places?

Spelling is the same, but the function is different:

Bóndinn á þrjá hesta
- á here is a verb form: 3rd person singular present of að eiga (to own / to have).
- Meaning: has / owns.
á bóndabænum
- á here is a preposition.
- With the dative case, it typically means on / at (location).
- Here: á bóndabænum = on / at the farmstead.

So they look the same, but you recognize which is which from the grammar:

After a subject and before an object → likely verb.
Before a noun in a case form (bóndabænum) → preposition.

How is bóndabænum formed, and what case is it in?

Bóndabænum is a compound noun in the dative definite:

First part: bóndi – farmer
- In compounds it appears as the genitive singular: bónda-.
Second part: bær – farm, town, homestead
- Bare compound: bóndabær – a farmer’s farm / farmstead.
Case and definiteness:
- Dative singular of bóndabær: bóndabæ
- Dative singular definite: bóndabænum – on/at the farmstead

The preposition á (in the sense of location on/at) takes the dative case, so we get:

á bóndabænum = at the farmstead / at the farm.

Why is there no hann (he) before vinnur? Is that allowed?

Yes, it’s normal and correct.

The subject Bóndinn (the farmer) is already established before the conjunction og (and). In Icelandic, when two clauses share the same subject, you can omit the repeated subject in the second clause:

Bóndinn á þrjá hesta og (hann) vinnur allan daginn á bóndabænum.

Adding hann is grammatically fine but sounds more heavy / redundant in such a short, simple sentence. Native speakers usually leave it out here.

Can the word order in vinnur allan daginn á bóndabænum be changed?

Yes, there is some flexibility. The basic pieces are:

verb: vinnur
time phrase: allan daginn (all day)
place phrase: á bóndabænum (at the farm)

Common orders include:

vinnur allan daginn á bóndabænum
vinnur á bóndabænum allan daginn

Both mean the same. Icelandic often prefers verb in 2nd position in main clauses, but time and place expressions can switch places after the verb. The choice can slightly affect emphasis (putting the most important/new information last), but in a neutral context they are interchangeable.